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Phase transitions in the neighbourhood of mean field theory Timmers, Dennis 2012

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Phase transitions in the neighbourhood of mean field theory  by Dennis Timmers B.Sc., University of Technology Eindhoven, 2005 M.Sc., University of Technology Eindhoven, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Mathematics)  The University Of British Columbia (Vancouver) December 2012 c Dennis Timmers, 2012  Abstract We study systems consisting of interacting spin particles which can have a positive or negative spin. We consider an Ising model and a type of Widom-Rowlinson (WR) model. The interactions between spin particles are regulated by Kac potentials which carry a parameter γ. It is known that in the Kac limit, as γ tends to zero, models with Kac potentials become mean field theory. Mean field theory is known to exhibit phase transitions. The focus of this work is to prove phase transitions not only in the Kac limit but also near the Kac limit i.e., for γ small but strictly positive. Placing the Ising and WR model in a rectangular box with side-length L and periodic boundary conditions defines finite volume Gibbs measures. The infinite volume Gibbs state ν is the limit of the finite volume Gibbs measures as L tends to infinity. A particle system exhibits a phase transition if ν is a mixture of ergodic states. The main achievement of this thesis is the development of a new method to prove phase transitions. We first apply the Kac-Siegert transformation which reformulates the particle system by introducing an auxiliary field. The spin-spin interactions are replaced by interactions of the spin particles with the auxiliary field. The main idea of this dissertation is to study the mean auxiliary field. In principle it should be easier to work with the mean field because, as we will show, it is approximately Gaussian. By a new expansion around mean field theory we prove that for γ strictly positive but small the infinite volume Gibbs state for the auxiliary field, for both the Ising and the WR model, is a mixture of two ergodic states. It is shown that this implies that the infinite volume Gibbs state for both the Ising and WR model is a mixture of two ergodic states. One Gibbs state predominantly has positive spin particles, the other Gibbs state predominantly has negative spin particles. The new expansion is related to the Glimm Jaffe Spencer expansion around mean field theory.  ii  Preface Except for Chapter 3 all the work in this thesis is original work conducted solely by myself under the supervision of my advisor Professor David Brydges. Chapter 3 is based on the thesis of G. Guadagni [31, Chapter 5]. The research program was originally suggested by Professor Brydges.  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  History of the theory of phase transitions . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Kac model near mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . .  5  1.3  Method and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6  Models and phase coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  2.1  Models and finite volume measures . . . . . . . . . . . . . . . . . . . . . . . . .  8  2.1.1  LMP continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  2.1.2  Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10  2.1.3  Widom-Rowlinson model . . . . . . . . . . . . . . . . . . . . . . . . . .  12  2.2  Local convergence and DLR measure . . . . . . . . . . . . . . . . . . . . . . . .  15  2.3  Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  18  2.3.1  LMP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  18  2.3.2  Ising and WR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19  Kac-Siegert transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21  2.4.1  Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22  2.4.2  Widom-Rowlinson model . . . . . . . . . . . . . . . . . . . . . . . . . .  24  A cluster expansion for Gaussian fields . . . . . . . . . . . . . . . . . . . . . . . . .  28  3.1  28  2  2.4  3  Finite range decompositions and polymer models . . . . . . . . . . . . . . . . . . iv  3.2  4  log Z (n)  33  3.3  The formal power series for  . . . . . . . . . . . . . . . . . . . . . . . . .  34  3.4  Convergence of the cluster expansion . . . . . . . . . . . . . . . . . . . . . . . .  37  3.4.1  Proof of Theorem 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . .  40  Perturbations of a Gaussian measure . . . . . . . . . . . . . . . . . . . . . . . . . .  50  4.1  Setup and conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  50  4.2  Phase coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  4.2.1  Conditions on the Radon-Nikodym derivative . . . . . . . . . . . . . . . .  55  4.2.2  Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  Contours and proof strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  4.3.1  Definition and Peierls estimate . . . . . . . . . . . . . . . . . . . . . . . .  58  4.3.2  Contour classification  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  60  Gaussian polymer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  61  4.4.1  Gaussian measure and FRD. . . . . . . . . . . . . . . . . . . . . . . . . .  62  4.4.2  Restricted contour ensembles . . . . . . . . . . . . . . . . . . . . . . . .  64  4.4.3  Unrestricted contour ensembles . . . . . . . . . . . . . . . . . . . . . . .  72  Cluster expansion bounds for a generalized polymer model . . . . . . . . . . . . .  74  4.5.1  Proof of Theorem 4.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . .  77  4.5.2  Proof of bound (4.36) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  84  4.5.3  Proof of the pre-Peierls estimate . . . . . . . . . . . . . . . . . . . . . . .  85  4.3  4.4  4.5  5  Convergence of the pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.6  Proof of the Peierls estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102  4.7  Phase coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.7.1  Reduction and restatement of theorem . . . . . . . . . . . . . . . . . . . . 106  4.7.2  Finite volume measures νn± and local convergence . . . . . . . . . . . . . 107  4.7.3  Decay of correlations and concentration of Gaussian averages . . . . . . . 113  4.7.4  Equivalent definition of ν ± . . . . . . . . . . . . . . . . . . . . . . . . . . 116  Proof of phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1  Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1.1  Well-adapted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121  5.2  WR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123  5.3  WR model well-adapted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.3.1  Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127  5.3.2  Derivatives as moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 128  5.3.3  Computing moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129  5.3.4  Proof of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137  5.3.5  Proof of Theorem 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 v  6  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 A Results for Gaussian measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A.1 Finite range decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A.2 Positive definite minimizers with boundary conditions . . . . . . . . . . . . . . . . 151 A.3 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 A.3.1 Application to Gaussian averages . . . . . . . . . . . . . . . . . . . . . . 154 B Combinatorial result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 C Variational problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 C.1 Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 C.2 WR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160  vi  List of Figures Figure 1.1  The van der Waals equation of state and Maxwell’s correction . . . . . . . . .  2  Figure 3.1  Blue (dashed) edges form a tree on Ui ∈ A j , red edges form a tree on {A j } . . .  43  Figure 3.2  Blue (dashed) edges form a tree on xˆi , red edges form a tree on {U j }. . . . . .  45  Figure 4.1  A double-well function with parabolic lower-bounds . . . . . . . . . . . . . .  54  Figure 4.2  Example in d = 1 where I0 (ω) consists of two connected sets . . . . . . . . .  66  Figure 4.3  Unrestricted contours versus restricted contours . . . . . . . . . . . . . . . . .  66  Figure 4.4  70  Figure 4.5  The cluster rule with three clusters: X1 , X2 and X3 . . . . . . . . . . . . . . . . ˚ X ∗ and ∂˚ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The sets X,  Figure 4.6  A contour Γ with its σ -contours. . . . . . . . . . . . . . . . . . . . . . . . . .  88  Figure 4.7  Contours for a local function G . . . . . . . . . . . . . . . . . . . . . . . . . . 109  Figure 4.8  Example of G-contour which contains sp(G) . . . . . . . . . . . . . . . . . . 113  vii  79  List of Symbols −  smallest length-scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 51  1  length-scale of interaction  +  largest length-scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 28, 51  1  = γ −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 51  cube of side-length  −  centered at x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51  Cx  cube of side-length  1  centered at x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10  +  Cx  cube of side-length  +  centered at x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51  m˚ x mxˇ  site average of spins σ in Cx(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11  •  block average of spins σ in Cx(1) ˇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  ρxˇ  average number of particles Cxˇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  ξ ψ˚ x ψx  auxiliary field of Gaussian random variables . . . . . . . . . . . . . . . . . . . . . . . . 22  A  generic symbol for a set of scaled polymers . . . . . . . . . . . . . . . . . . . . . . . . 35  K (n) (X, ξ )  polymer activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  δ K (n) (X, s)  scaled polymer activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  ❑  weight for a set of scaled polymers A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  Am (X)  family of sets of scaled polymers which intersect X . . . . . . . . . . . . . . . . . 37  Am (X,Y )  family of sets of scaled polymers which intersect both X and Y . . . . . . . 37  P(n) (X)  power series in powers of polymer activities K . . . . . . . . . . . . . . . . . . . . . . 37  P(n) (X,Y )  power series in powers of polymer activities K . . . . . . . . . . . . . . . . . . . . . . 37  ∂ (X)  external boundary of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51  X  union of X with ∂ (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51  Fm (X, Ω)  family of admissible polymer fillings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71  Cx (1)  •  (n) (A)  site average of auxiliary field ξ in Cx(1) . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 52 block average of auxiliary field ξ in Cx(1) . . . . . . . . . . . . . . . . . . . . . . . . 24, 52  viii  Acknowledgements I want to thank my supervisor Professor David Brydges for his patient and thoughtful guidance. His resourceful ideas helped me through the many difficult times while I was stuck working on a problem. His careful reading of this manuscript improved it greatly. I would like to thank Bill Faris, Joel Feldman and Gordon Slade for their comments and suggestions on this work. My family and friends have my gratitude for their encouragement and support during these years of study. I especially want to thank my parents for their enthusiasm and in supporting me to pursue my dream. Finally, a special thanks goes to Yuri. Sharing our experiences and her support made this work possible.  ix  Chapter 1  Introduction The chapter gives an introduction to the context and structure of this thesis. Section 1.1 provides a general overview of the developments in the study of phase transitions. In Section 1.2 the focus lies on the specific research problem of this thesis: the study of phase transitions near mean field theory. The section presents the existing work on this specific topic. In Section 1.3 we propose a new method to prove the existence of a phase transition and provide an outline of the thesis.  1.1  History of the theory of phase transitions  Phase transitions such as the liquid-vapour phase transition abound in nature. However, deriving the existence of phase transitions from the first principles of statistical mechanics proves to be a difficult problem. The first step towards a theory of liquid-vapour phase transitions was taken by Maxwell [44]. He studied a model with N particles in a finite volume Λ which experience a short-range repulsion and a pairwise attractive force. The attractive force was assumed to be the same for all pairs of particles, regardless of the distance between particles. This is called a mean field attraction. The model was originally introduced by van der Waals [59] who derived the equation of state Pw (ρ, T ) =  ρkT − aρ 2 , 1 − ρb  where Pw is the pressure, k the Boltzmann’s constant, T the temperature, ρ =  (1.1) N |Λ|  the particle density  and a and b are positive constants who regulate the attractive and repulsive part of the interaction, respectively. When T exceeds the critical temperature Tc =  8a 27kb ,  then the van der Waals equation of state  (1.1) gives a good qualitative representation of the isotherms of a real fluid. An isotherm is the graph of the pressure against the inverse particle density at a fixed temperature. Maxwell noticed the unphysical behaviour for T < Tc : there exists an interval where the pressure Pw increases whereas 1  the particle density decreases, see Figure 1.1(a). Maxwell interpreted the unphysical behaviour as an indication of the coexistence of a liquid and a vapour (or gas) phase. Under the same pressure and temperature these phases have different densities ρ and ρg . When the gas is compressed at constant temperature the coexistence of phases avoids the unphysical behaviour: the condensation of gas into liquid allows the pressure to be kept constant while the density changes from ρg to ρ . The Maxwell equal area rule depicted in Figure 1.1(b), determines the liquid-gas coexistence pressure Pco . For T < Tc this leads to Maxwell’s correction of the van der Waals equation of state (1.1)  P (ρ, T ) if v < v or v > v , w g P(ρ, T ) = P if v < v < vg , co  (1.2)  where v = ρ −1 . Maxwell’s correction to the van der Waals pressure shows that at fixed temperature T < Tc the derivative of pressure has singularities. Hence, phase transitions can be described by singularities of the (higher-order) derivatives of thermodynamic functions such as the pressure.  (a) van der Waals equation of state  (b) the two shaded regions have equal area  Figure 1.1: The van der Waals equation of state and Maxwell’s correction  Gibbs distributions. About 1870 work of Boltzmann initiated a new branch of physics: statistical mechanics. Statistical mechanics applies probability theory to study the thermodynamic behaviour of systems which consist of a large number of interacting components. The main goal of statistical mechanics is to explain the macroscopic (thermodynamic) properties of matter from the microscopic description of the particle system. In what follows we do not aim to give a rigorous treatment of statistical mechanics. Formal definitions are avoided. Instead of describing the concepts in full generality we choose to present concepts by using a basic model: the Ising model. The finite volume Ising model has spin configurations σ which assign a spin ±1 to each lattice site in a finite set Λ ⊂ ❩d . Ising models are isomorphic to lattice particle models which place at 2  most one particle at each lattice site. Hence, results derived for the Ising model also hold for the corresponding lattice particle model. Statistical mechanics models the physical systems by probability distributions. A given Hamiltonian H assigns a potential energy H(σ ) to each configuration σ . For example the Ising model on a finite set Λ has the Hamiltonian HΛ (σ ) = −  ∑  J(i, j)σi σ j − h ∑ σi ,  i, j∈Λ  (1.3)  i∈Λ  where h is a real parameter and J(i, j) > 0 is called the two-body potential. The potential J(i, j) is assumed to be of finite range, J(i, j) = 0 if the distance between i and j is larger than some finite length L. The classical Ising-Lenz model was introduced by Ising and has J(i, j) = 1 if sites i and j are adjacent and J(i, j) = 0 otherwise. The theory of statistical mechanics states that a physical system with Hamiltonian HΛ is described by the probability measure µΛ (σ ) = ZΛ−1 e−β HΛ (σ ) .  (1.4)  The parameter β is proportional to inverse temperature and ZΛ is the normalizing constant called the grand partition function. The measure µΛ is called a finite volume Gibbs measure. In the early 1900s there was a widespread belief that Gibbs distributions could not be used to describe a phase transition. The argument is that in finite volumes the thermodynamic functions of a system described by a Gibbs distribution are expressed as finite sums of factors e−β H . The sum of analytic functions is still analytic and has no singularities. If the thermodynamic functions cannot have singularities, then the physical system will not undergo a phase transition. However, in the thermodynamic limit as Λ  ❩d the infinite sums can lead to singularities. Hence, the correct way  to find a phase transition is to study the finite volume Gibbs measures in the thermodynamic limit as Λ  ❩d .  Ising [35] studied the classical Ising-Lenz model. He solved the model exactly and showed that there is no phase transition for the Ising model in dimension d = 1. Peierls (1936) developed a method to show that for d ≥ 2 the Ising model exhibits a phase transition. His proof was not quite rigorous and careful proofs were later given by Griffiths [29] and independently by Dobrushin [17]. Onsager [48] gave the first rigorous proof of a phase transition by solving the the Ising model in dimension d = 2 with h = 0 exactly. Kac models. A next step in the theory of phase transitions was the development of the Kac model. Kac [36] introduced a one-dimensional continuum particle model with a parameter γ. Kac, Uhlenbeck and Hemmer [37] placed the latter model in a bounded box Λ ⊂ ❘d . They established the validity of the Maxwell equal-area rule (1.2) when first taking the thermodynamic limit Λ  3  ❘d  followed by the Kac limit γ → 0. The van der Waals pressure (1.1) had to be replaced by the mean field theory of the Kac model. Lebowitz and Penrose [42] extended the result of Kac, Uhlenbeck and Hemmer to any dimension and for a general class of potentials called the Kac potentials. The Kac Ising model has pairwise interactions which only depend on the distance between pairs of particles Jγ (x, y) = γ d v(γ|x − y|).  (1.5)  In the context of this thesis v is a non-negative function which is integrable, continuous at 0 and compactly supported. However, the result holds for a larger class of functions w. Moreover, the result of Lebowitz and Penrose extends to continuum models, where particles are located in ❘d , by including a hardcore condition: there exists some r0 > 0 such that any pair of particles cannot be within distance r0 of one another. For γ small the Kac potentials model systems where particles experience long-range forces. The attractive potential Jγ (x, y) has a small but rather long-range tail. The key observation is that in the Kac limit γ → 0 the model becomes mean field theory. DLR measures. Phase transitions do not occur in finite volumes. Hence, to study phase transitions one defines the Gibbs measures in finite volumes Λ and takes the thermodynamic limit as Λ  ❩d .  This leaves the question whether it is possible to define infinite volume Gibbs measures, also called Gibbs states, directly in the infinite volume  ❩d . The finite volume Gibbs measures (1.4) do not  suffice because these measures are only well-defined on finite sets. Around the same time Dobrushin [18, 19] and Lanford-Ruelle [40] proposed to define a Gibbs state µ by prescribing its conditional distributions for any finite set Λ. Consider the specific case of the Ising model. For any finite set Λ ⊂ ❩d and a fixed spin configuration σ defined on Λc define the conditional Hamiltonian HΛ (σ |σ ) = −  1 ∑ J(i, j)σi σ j − ∑ J(i, j)σi σ j . 2 i∈Λ, j∈Λ i∈Λ, j∈Λ /  (1.6)  The Hamiltonian HΛ (σ |σ ) represents the energy of the subsystem Λ in a fixed auxiliary field σ . The Hamiltonian is well-defined because J has finite range. For any finite Λ ⊂ ❩d and any σ defined on Λc define the conditional discrete probability density µΛ (σ |σ ) = ZΛ (σ )−1 e−β HΛ (σ |σ ) ,  (1.7)  where ZΛ (σ )−1 is the normalization constant. Given a spin configuration σ on ❩d and Λ ⊂ ❩d let σ Λ denote the restriction of σ to Λ. Dobrushin-Lanford-Ruelle call a measure µ on  4  ❩d a Gibbs  state for the potential J if µ(σ |σ ) = µΛ (σ Λ |σ ), for all finite sets Λ and every boundary condition σ in Λc . Given a potential J let G (J) denote the set of all possible Gibbs states for the potential J. The Gibbs states should provide a description of all possible phases of a physical system. Nonuniqueness of Gibbs states would imply that the system can choose between multiple phases. Hence, non-uniqueness could be considered a characteristic of a phase transition. We adopt the commonly used terminology: Definition 1.1. A potential J is said to exhibit a phase transition if |G (J)| > 1.  1.2  Kac model near mean field theory  The main motivation for this work is the question whether the theory of Kac potentials can be extended to prove phase transitions not only in the Kac limit but also near the Kac limit, i.e., for γ small but strictly positive. In dimension d = 1 the answer is negative. Kac models with γ positive have finite range interactions. It has been long known that in dimension d = 1 any Ising model with finite range interactions has no phase transition [56]. In this section we describe the work which has been done for dimensions d ≥ 2. Ising model.  Bricmont and Fontaine [7] studied an Ising model in dimension d = 2 with a potential  J(i, j) which is a Kac potential. The chosen potential is of infinite range, J(i, j) = 0 for any i, j ∈ ❩d , but decays exponentially with the distance between i and j. They were able to prove a phase transition for γ small but positive by considering the spin-spin correlation σo σx = lim Λ  σo σx dµΛ (σ ),  ❩d  where o is the origin and x ∈ ❩d . They showed that there exists βc such that for β < βc the correlations σo σx decay exponentially with the distance of x to the origin o. For β > βc the correlations have long-range order σ0 σx i.e., σo σx → 0 as |x| → ∞. Cassandro and Presutti [16] considered the class of Ising models in dimension d ≥ 2 which have a finite range Kac potential. They showed that if β > 1, then the Ising model admits at least two distinct Gibbs states for γ small but positive. This has been generalized to Ising models with a Kac potential and a fixed finite range potential by Bodineau and Presutti [4]. We use the equivalent lattice particle model description of the Ising model to state the results. For a given x ∈ ❩d let Cx be a large cube centered at x. Define the particle density ρx to be the number of particles in Cx divided by the volume |Cx |. In [4, 16] they prove the coexistence of at least two Gibbs states µ ± 5  by studying particle densities. The particle densities in the Gibbs states µ ± are close to ρ ± which satisfy 0 < ρ − < ρ + . Intuitively, the Gibbs state µ − corresponds to the gas phase while the Gibbs state µ + corresponds to the liquid phase. In the analysis of the Ising model symmetry plays an important role: the interactions are left unchanged when simultaneously flipping all the spins. Continuum models.  The next step up are continuum particles models. In continuum particle mod-  els the particle configurations q represent the positions of the particles in ❘d . The sets q are locally finite, i.e., only finitely many points are contained in any bounded region X ⊂ ❘d . Proving phase  transitions in continuum models has been an enduring challenge. The first attempt would be to extend the results for discrete models to the continuum by taking a lattice particle model and letting the lattice spacing tend to zero. However, the phase coexistence results for the lattice model give a nonsensical limit. In the lattice model the particle densities ρ ± for the Gibbs states µ ± are strictly positive. The resulting continuum model would have a positive density of particles in each bounded set of  ❘d . Every bounded set would thus have uncountable  many particles. Widom and Rowlinson [60] proposed a simple way to recreate the symmetry of the discrete Ising model in a continuum setting. The WR model has two types of continuum spin particles. The spin particles can have either a positive or negative spin. The only interaction is a hard-core repulsion between particles which have a different spin. There are no interactions between particles which have the same spin. Ruelle [51] was the first to show a coexistence of Gibbs states. One Gibbs state predominantly has positive spin particles, the other Gibbs state predominantly has negative spin particles. An extension to systems where the symmetry is absent has been considered in [8, 9]. They consider systems with r different types where the particles of different types experience a hardcore repulsion. Finite range interactions are allowed but only between particles of the same spin. A major achievement was the work of Lebowitz, Mazel and Presutti [41]. They introduced the LMP continuum model with Kac potentials and proved a coexistence of phases for non-zero γ. They proved a long-standing conjecture of the existence of a liquid-vapour phase transition in  ❘d for d ≥ 2. In contrast to the WR model and Ising model, the LMP model does not have any symmetries. The LMP model will be discussed in more detail in Chapter 2.  1.3  Method and outline  The proofs of phase transitions in [4, 6, 16, 41] are all based on an analysis of the particle density. The main accomplishment of this thesis is the development of a new method to prove phase transitions. The new method uses a tool called the Kac-Siegert transformation. The Kac-Siegert transformation reformulates a system of spin particles interacting through a two-body potential into a system where independent particles interact with an auxiliary field. The main idea of the thesis is 6  to analyze the auxiliary field which in the Kac limit becomes the mean field. In principle it should be easier to work with the auxiliary field because, as we shall see in Chapter 4, it is approximately Gaussian. The technique is inspired by the work of Glimm, Jaffe and Spencer [27, 28]. Bricmont and Fontaine [7] did use the Kac-Siegert transformation. However, after applying the transformation the model only has interactions between adjacent sites in the lattice  ❩d . They use  infrared bounds to prove a phase transition. Infrared bounds can only be applied to systems with nearest-neighbour interactions. This proof does not apply to Kac potentials which cause long-range interactions. We develop our method by first tackling an Ising model. The Ising model has a Kac potential which is the discrete equivalent of the potential used in the continuum LMP model. Strictly speaking the model does not fall into the class of Ising models for which phase transitions were proved by Bodineau and Presutti [4]. However, the methods of Bodineau and Presutti would be able to handle this particular Ising model. The second model is a WR model which is a step towards a continuum model. The WR model is defined on continuum spin particle configurations. However, a discretized Hamiltonian is chosen such that in essence the WR model becomes a lattice model. To our knowledge, Widom-Rowlinson models with Kac potentials have not yet been studied in the existing literature. Outline. Chapter 2 introduces the Ising model and WR model and in Section 2.3.2 we state our main result that both models exhibit a phase transition. It is shown how to include an auxiliary field into the particle models via the Kac-Siegert transformation. We also present the LMP model and their result of a phase transition. Chapter 3 describes a technique to study the measures which are approximately Gaussian: a cluster expansion for Gaussian fields. This is a standalone result and is described in full generality without making mention of the particle models. Chapter 4 proves that the auxiliary field exhibits a phase transition. The proofs for phase transitions in the Ising model and WR model are completed in Chapter 5. There it will be shown that a phase transition in the auxiliary field implies a phase transition in the particle models.  7  Chapter 2  Models and phase coexistence In this chapter we describe the LMP continuum model in more detail. In the next sections we introduce the two models which were mentioned in Chapter 1. The first model is an Ising model. The second model falls into the class of Widom-Rowlinson (WR) models. For the latter two models we show how the Kac-Siegert transformation can be used to rewrite the Gibbs measures in terms of Gaussian integrals.  2.1  Models and finite volume measures  In the next sections we introduce three models: the LMP model, the Ising model and a WidomRowlinson model. The interactions between the particles are regulated by Kac potentials.  2.1.1  LMP continuum model  The continuum LMP model considers particles living in ❘d for d ≥ 2. A subset q of ❘d is called  locally finite if any bounded region of ❘d only contains finitely many points. In continuum particle models the particle configurations q = {qi } are locally finite subsets of ❘d . The set q = {qi } repre-  sents the locations of the particles. Let Σ be the set of all locally finite sets in ❘d while for a subset Λ ⊂ ❘d let Σ(Λ) = {q ∈ Σ : q ⊂ Λ}.  Hamiltonian. The Hamiltonian of a given particle configuration q represents the total energy of q generated by the multi-particle interactions between particles. Let Bγ (ri ) be the ball centered at ri with volume γ −d where γ is the Kac parameter. The specific choice for the n-body Kac potential in the LMP model is  n  (n)  Jγ (r1 , . . . , rn ) = γ nd  Bγ (ri ) , i=1  8  (2.1)  (n)  where | · | denotes Lebesgue measure. Hence, Jγ (r1 , . . . , rn ) is the volume of the intersection of balls Bγ (r1 ), . . . , Bγ (rn ) multiplied by a factor γ nd . As γ → 0, the n-body potentials become weaker and long-range. The Hamiltonian Hγ,υ (q) assigns an energy based on the interaction energy between the particles in q. LMP consider the continuum model with the Hamiltonian Hγ,υ (q) = −υ|q| −  1 1 (2) (4) Jγ (qi1 , qi2 ) + Jγ (qi1 , qi2 , qi3 , qi4 ), ∑ 2! i1 =i2 4! i1 =i2∑ =i3 =i4  (2.2)  where υ ∈ ❘ is called the chemical potential and |q| denotes the total number of particles in configuration q. The signs in front of the multi-body potentials are chosen such that the Hamiltonian has an attractive two-body potential and a repulsive four-body potential. R EMARK 2.1. We next show that J (2) (r1 , r2 ) is indeed a Kac potential. Let Rd be the constant such that the ball of radius Rd centered at the origin has unit volume J (2) (x, y) = γ 2d  ❘  d  ✶{γ|r−y|≤Rd } ✶{γ|r−x|≤Rd } dr = γ 2d  ❘d  ✶{γ|r|≤Rd } ✶{γ|r−(x−y)|≤Rd } dr.  Make the change of variables r → γ −1 r J (2) (x, y) = γ d  ❘d  ✶{|r|≤Rd } ✶{|r−γ(x−y)|≤Rd } dr.  The two-body potential satisfies J (2) (x, y) = γ d v(γ(x − y)) where v is the convolution of the function f (r) = ✶{|r|≤Rd } with itself.  ♦  Fix a bounded set Λ ⊂ ❘d and let q ∈ Σ(Λ) while q ∈ Σ(Λc ). The conditional Hamiltonian HΛ,γ,υ (q|q) represents the energy of q in the field generated by q HΛ,γ,υ (q|q) = −υ|q| −  1 1 (2) (4) Jγ (qi1 , qi2 ) + Jγ (qi1 , qi2 , qi3 , qi4 ). ∑ ∑ 2! i1 =i2 4! i1 =i2 =i3 =i4  (2.3)  By convention the sum runs over sets of labels of particles in q ∪ q such that at least one particle is located in the region Λ. The sum HΛ,γ,υ (q|q) is finite because the potentials have finite range. Gibbs measures. Fix a bounded set Λ ⊂ ❘d and a boundary configuration q ∈ Σ(Λc ). The energy of a spin particle configuration q ∈ Σ(Λ) is HΛ,γ,υ (q|q). The free particle system is defined to be the Poisson process on Λ with constant intensity. The finite volume Gibbs measure is the probability measure on Σ(Λ) which is absolutely continuous w.r.t. the Poisson process and has Radon-Nikodym derivative proportional to e−β HΛ,γ,υ (q|q) . The details are in the next paragraphs.  9  For a function f on Σ(Λ) define the free measure dν Λ (q) by ∞  dν Λ (q) f =  1  ∑ n!  n=0  Λn  dq1 · · · dqn f ({q1 , . . . , qn }).  If properly normalized, the measure dν Λ (q) is the Poisson process measure on Λ with intensity 1. Let β be a parameter proportional to the inverse temperature. The finite volume Gibbs measure for the LMP continuum model is the probability density defined by µΛ,γ,υ,β (dq|q) =  e−β HΛ,γ,υ (q|q) dν Λ (q) , Zγ,υ,β (Λ|q)  (2.4)  where the grand partition function Zγ,υ,β (Λ|q) is the normalization constant Zγ,υ,β (Λ|q) =  2.1.2  dν Λ (q)e−β HΛ,γ,υ (q|q) .  (2.5)  Ising model  Consider the Ising model with periodic boundary condition by placing the model on a torus ❚(n) .  ❚(n) for n ∈ ◆ is the graph with vertices [−L(n) , L(n) ]d ∩ ❩d where opposite faces are identified. The length L(n) is equal to n + where + is an integer and + γ −1 . Vertices x, y in ❚(n) are connected by an edge if the supremum norm x − y ∞ = 1. The notation ❚(n) may refer to either the graph itself or the vertex set of ❚(n) . Based on the context it will be clear which interpretation  The torus  to use. Ising spin configurations are elements of {−1, +1}❚ . A spin configuration places a spin on (n)  each lattice point in  ❚(n) which can either be positive or negative. The interactions between the  spins are regulated by the Hamiltonian. Hamiltonian. Distances dist(x, y) for x, y ∈ ❚(n) are measured in the graph distance norm on ❚(n) , i.e., dist(x, y) is the number of edges in a shortest path connecting x and y. Let the set Cx(1) to be the cube of side-length  1  1  = γ −1 and define  on the torus ❚(n) centered at x.  Cx(1) = {y ∈ ❚(n) : dist(x, y) ≤  1  2  }.  The two-body potential in the Ising model is Jγ (x, y) =  |Cx(1) ∩Cy(1) | . |C(1) |2  10  (2.6)  Hence, it is the lattice equivalent of the two-body LMP potential (2.1). The energy of a spin configuration σ ∈ {−1, +1}❚  (n)  is (n)  Hγ (σ ) = −  1 σx σy Jγ (x, y), 2∑ x,y  where the sum runs over all pairs (x, y) in ❚(n) × ❚(n) . Remark 2.1 shows that the potential Jγ (x, y) is indeed a Kac potential. The Hamiltonian has an alternative expression in terms of magnetizations. Given a spin configuration σ ∈ {−1, +1}❚ , define the magnetization at location x by (n)  m˚ x = m˚ x (σ ) =  1 |C(1) |  ∑  (2.7)  σy  (1) y∈Cx  The two-body potential can be expressed in terms of the magnetization.  ∑  x,y∈❚(n)  σx σy Jγ (x, y) = =  σx σy ✶{z∈Cx(1) } ✶{z∈Cy(1) } |C(1) |2 z∈∑ ❚(n) x,y∈❚(n)  ∑  1 (1) 2 |C | z∈❚(n)  ∑  ∑  x∈❚(n)  σx ✶{x∈C(1) } z  The Hamiltonian satisfies (n)  Hγ (σ ) = −  ∑  y∈❚(n)  σy ✶{y∈C(1) } = z  ∑  z∈❚(n)  m˚ z (σ )2  1 m˚ x (σ )2 . 2 x∈∑ ❚(n)  (2.8)  Gibbs measure. The finite volume Gibbs measures Pn,γ,β (σ ) on the torus ❚(n) is easily defined. (n)  Pn,γ,β (σ ) =  e−β Hγ  (σ )  (2.9)  (n)  Zγ,β  The normalization constant, also called the grand partition function, is (n)  (n)  Zγ,β =  ∑  (n) σ ∈{−1,+1}❚  e−β Hγ  (σ )  .  (2.10)  The finite volume measures are invariant under simultaneously flipping the signs of the spins. The latter symmetry is called the flip-symmetry. R EMARK . In comparison with the continuum Hamiltonian (2.2), the Ising Hamiltonian does not include a term with the chemical potential and is missing the repulsive four-body potential. Firstly, the function of the four-body potential in the continuum LMP model is to guarantee stability of the system. However, in the Ising model stability is ensured by the condition that there can be at most one spin particle per lattice site. In other words, stability is ensured by a hardcore potential. 11  Secondly, in the definition of the Hamiltonian (2.8) the chemical potential υ is set to zero. A phase transition is to be expected because of the symmetry in the model. For υ = 0 the symmetry ♦  breaks down and indeed in this case there is no phase transition [43, 52, 53].  2.1.3  Widom-Rowlinson model  We will define a continuum Widom-Rowlinson model with periodic boundary condition by placing the model on the torus T (n) = [−L(n) , L(n) ]d where opposite faces are identified. The length L(n) is equal to n  +  where  +  is an integer and  +  γ −1 . The WR model has two types of spin particles  which can have either a positive or negative spin. The spin particle configurations are represented by sets qσ = {qσi } = {(qi , σi )}. The set {qi } is a locally finite subset of T (n) which represents the locations of the particles while σi ∈ {−1, +1} represents the spin or type of the of the ith particle. For a given qσ let q be the corresponding particle configuration. Let Σσn be the set of all spin particle configurations in T (n) . The WR model is defined for continuum spin particle configurations. However, our method is not yet able to handle a purely continuum system. Partition the torus T (n) into large cubes of side-length  γ −1 such that one cube is centered at the origin. The side-length  −  −  is integral. We  consider a discretized Hamiltonian which only depends on the number and the signs of the particles in each open cube in the partition. In other words, the Hamiltonian does not depend on the explicit (n)  locations of the particles inside the open cubes. Let To at scale  −  ⊂ T (n) denote the union of all open cubes  which are contained in T (n) .  Hamiltonian. Recall the definitions of the discrete torus  ❚(n) ⊂ T (n) and the distance dist(x, y)  given in Section 2.1.2. The cube Cx(1) is defined by Cx(1) = {y ∈ ❚(n) : dist(x, y) ≤  1  2  }.  Let Cn− be the partition of ❚(n) into cubes C ⊂ ❚(n) of side-length set of the cube-centers of all cubes C contained in X. Given x ∈  . For a set X let Vˇ (X) be the  −  (n) To  let v(x) ˇ denote the center of the unique open cube which contains x. The center v(x) ˇ will be an element in Vˇ (❚(n) ). (n)  Given x, y ∈ To  define the two-body potential (1) (1) |Cv(x) ˇ ∩Cv(y) ˇ |  J(x, y) = where |C(1) | =  d 1  |C(1) |2  is the volume of a cube at scale  1.  ,  LMP had to introduce a repulsive four-body  potential to ensure stability of the continuum system. In the WR model stability is imposed by a strictly repulsive two-body potential at scale  (n)  . For x, y ∈ To  −  12  the repulsive two-body potential is  defined by J˜γ (x, y) = where Cv(x) is the cube of side-length ˇ (n)  For qσ ∈ Σσn with q ⊂ To  −  |Cv(x) ˇ ∩Cv(y) ˇ | , 2 |C|  centered at v(x) ˇ with volume |C| =  d. −  define the Hamiltonian  σ Hn,γ,υ,r (qσ ) = −υ|q| −  1 r σi σ j Jγ (qi , q j ) + ∑ J˜γ (qi , q j ) ∑ 2 i, j 2 i, j  (2.11)  where υ is the chemical potential, |q| the number of particles and r a positive constant. Remark 2.1 shows that both potentials Jγ and J˜γ are Kac potentials. The potential Jγ introduces an attractive force between particles of the same type while particles of opposite types experience a repulsive force. The potential J˜γ guarantees stability of the system by incorporating a pairwise repulsive force between any pair of particles (regardless of their types). The repulsion parameter r controls the strength of the repulsive force. R EMARK . One would expect the term υA ∑ ✶{σi =+1} + υB ∑ ✶{σi =−1} , i  i  instead of υ|q|. However, a phase transition is only expected to occur on the curve υA = υB and we ♦  have thus chosen υA = υB = υ/2.  Gibbs measure. The free spin particle system is defined to be the union of two Poisson processes on T (n) with equal constant intensity. Each Poisson process represents one of the species of particles. The Gibbs measure is the probability measure on Σσ (T (n) ) which is absolutely continuous w.r.t. the union of the Poisson processes and has Radon-Nikodym derivative proportional to e−β Hn,γ,υ,r (q ) . The details are in the next paragraphs. σ  σ  For a function f on Σσn define the measure dνσn (qσ ) by 2−m m=0 σ ∈{−1,+1}m m! ∞  dνσn (qσ ) f =  ∑  ∑  T (n)  dq1 · · ·  T (n)  dqm f ({(q1 , σ1 ), . . . , (qm , σm )}).  If properly normalized the above measure corresponds to the union of two independent Poisson processes of intensity 21 . The sum of the intensities adds up to 1. σ For sake of definition let Hn,γ,υ,r (qσ ) be zero when not all the particles in qσ are located inside  the open unit cubes. The Gibbs measure for the WR continuum model is the probability density defined by e−β Hn,γ,υ,r (q ) dνσn (qσ ) σ  µn,γ,υ,r,β (dqσ ) =  σ  (n)  Zγ,υ,r,β 13  (2.12)  where the grand partition function Zγ,υ,r,β (T (n) ) is the normalization constant (n)  dνσn (qσ )e−β Hn,γ,υ,r (q ) . σ  Zγ,υ,r,β =  σ  (2.13)  Alternative lattice description (n)  The energy of a particle configuration qσ ⊂ To  only depends on the signs of the particles and  the locations v(q ˇ i ). Hence, to get a complete description of the WR model it suffices to know ˇ (n) = Vˇ (❚(n) ) and the number of A and B particles in each cube C of the partition Cn− . Let ❚  ˇ (n) ) denote the number of particles in each open cube in T (n) . Let given qσ let k = (kxˇ , xˇ ∈ ❚ ˇ (n) , 1 ≤ i ≤ kxˇ ) record the number of particles and their spins in each open cube in σ = (σ i , xˇ ∈ ❚ xˇ  (n)  To  ˇ (n) . We compute the marginal distribution of σ . centered at xˇ ∈ ❚  The set of lattice spin configurations is ˇ (n) ) = {σxˇi : xˇ ∈ ❚ ˇ (n) , σxˇi = ±1, 1 ≤ i ≤ kxˇ with kxˇ ∈ ◆0 }. Σσ (❚ ˇ (n) ) define the magnetization mxˇ = mxˇ (σ ) and particle density ρxˇ = ρxˇ (k) by For σ ∈ Σσ (❚ •  •  mxˇ =  1 |C(1) |  ∑  •  σyˇi ,  ∑  and  ρxˇ =  y∈ ˇ Vˇ (Cxˇ ) 1≤i≤kyˇ (1)  kxˇ . |C|  ˇ (n) . The next lemma shows that the marginal distribution of σ is a Gibbs measure on the lattice ❚ Lemma 2.2. Define the Hamiltonian L Hn,γ,υ,r (σ ) = −υ  kxˇ −  ∑  ˇ x∈ ˇ ❚  (n)  1 |C| 2  ∑  mxˇ (σ )2 + •  ˇ x∈ ˇ ❚  (n)  r |C| 2  ∑  ˇ x∈ ˇ ❚  ρxˇ (k)2 .  (2.14)  (n)  The marginal distribution of σ is  Pn,γ,υ,r,β (σ ) =  1 (n)  Zγ,υ,r,β  L |C|kxˇ −β Hn,γ,υ,r (σ ) , ∏ 2kxˇ · kxˇ! e ˇ (n)  (2.15)  x∈ ˇ ❚  (n)  where the normalization constant Zγ,υ,r,β is the grand partition function. Proof. The Hamiltonian (2.11) in term of the spins and location of the particles becomes L Hn,γ,υ,r (σ ) = −υ ∑ kxˇ −  xˇ  1 2  ∑  ˇ y) ˇ + σxˇi σyˇj Jγ (x,  j σxˇi ,σyˇ  r 2  ∑  J˜γ (x, ˇ y), ˇ  j σxˇi ,σyˇ  where the sum runs over all pairs of spin particles in σ . The Hamiltonian can be expressed in terms  14  of magnetization and particle densities. Given σ let σxˇ = ∑1≤i≤kxˇ σxˇi . The potential Jγ satisfies  ∑  ∑  ∑  ˇ (n) 1≤i≤kxˇ 1≤ j≤kyˇ x, ˇ y∈ ˇ ❚  σxˇi σyˇj Jγ (x, ˇ y) ˇ = =  ∑  ∑  ∑  1 |C(1) |2 z∈❚(n)  ∑  σxˇi σyˇj  ˇ (n) 1≤i≤kxˇ 1≤ j≤kyˇ x, ˇ y∈ ˇ ❚  ∑  = |C|  ∑  ˇ x∈ ˇ ❚  |C(1) |2 z∈∑ ❚(n)  σxˇ ✶{x∈C ˇ v(z) ˇ }  (n)  ✶{z∈Cv(x) ✶ ˇ } ˇ } {z∈Cv(y)  ∑  ˇ y∈ ˇ ❚  σyˇ ✶{y∈C ˇ v(z) ˇ }  (2.16)  (n)  2  •  mzˇ(σ ) .  ˇ (n) zˇ∈❚  A similar computation shows that the repulsive potential can be expressed in terms of particle densities  ∑  J˜γ (x, ˇ y) ˇ = j  σxˇi ,σyˇ  ∑  kxˇ kyˇ  ˇ (n) x, ˇ y∈ ˇ ❚  |Cxˇ ∩Cyˇ | = |C| ∑ ρxˇ (k)2 . |C|2 x∈ ˇ Vˇ (❚(n) )  (2.17)  Substitution of the (2.16) and (2.17) yields the Hamiltonian (2.14). By integrating out the positions (n)  of the particles in the open cubes of To  2.2  we obtain factors |C|kxˇ .  Local convergence and DLR measure  In this section we present, without proof, some well-known facts about Gibbs measures on lattice models. The section is completely self-contained but many details are omitted. We refer to Chapters 4, 7 and 14 of [24] for a comprehensive treatment. We present the DLR definition of Gibbs states for lattice models and define Gibbs states for the continuum LMP model. In this thesis we will not work directly with the DLR definition of Gibbs states. Instead we obtain infinite volume measures via the local limit. The theory of local limits provides sufficient conditions for a local limit measure to be a Gibbs state. This will be the topic for the second half of this section. Lattice DLR measures. Consider spin configurations {σx , x ∈ ❩d } where spins σx take values in a  measure space (E, E ). In this thesis we consider three cases E = {−1, +1}, E = {(n− , n+ ) : n± ∈ ◆} and E = ❘.  Let Λ be any finite subset of ❩d . The first step is to define a Hamiltonian HΛJ with potentials J.  A potential J = (JA ), where A ranges over the subsets of ❩d is a collection of measurable functions, such that HΛJ (σ ) =  ∑ A:A∩Λ=0/  15  JA (σ )  exists. A straightforward example would be the classical Ising model.  σ σ x y JA (σ ) = 0  if A = {x, y} and x, y adjacent,  (2.18)  else. c  Next define the Hamiltonian for a region Λ with a fixed boundary condition σ Λ in Λc . Given a c  c  spin configuration σ Λ in Λ and σ Λ let σ Λ ⊕ σ Λ denote the spin configuration whose restrictions c  to Λ and Λc are given by σ Λ and σ Λ , respectively. Define the Hamiltonian with fixed boundary c  condition σ Λ by c  c  HΛJ (σ Λ |σ Λ ) =  JA (σ Λ ⊕ σ Λ ).  ∑ A:A∩Λ=0/  c  J  The finite volume Gibbs measure µΛJ (σ Λ |σ Λ ) is a probability measures of the form e−β HΛ (σ An infinite volume Gibbs measure is a probability measure  µJ  on spin configurations on  Λ |σ Λc )  .  ❩ that d  satisfies the DLR equations: c  c  µ J (σ Λ |σ Λ ) = µΛJ (σ Λ |σ Λ ),  c  for all σ Λ .  (2.19)  When E = ❘ replace the restriction for all σ Λ to µ J almost all σ Λ . c  c  For the LMP continuum model the finite volume measures are defined  Continuum DLR measures.  by (2.4). Given a particle configuration q ∈ Σ denote its restriction to Λ by qΛ = q ∩ Λ. An infinite volume Gibbs measure is a probability density µγ,υ,β on Σ that satisfies the DLR equations: c  c  c  µγ,υ,β (dqΛ |qΛ ) = µΛ,γ,υ,β (dqΛ |qΛ ),  for µγ,υ,β almost all qΛ .  c  The measure µγ,υ,β (dqΛ |qΛ ) is the conditional probability measure when the system is immersed c  c  in the external field qΛ . The measure µΛ,γ,υ,β (dq|qΛ ) is the finite volume measure (2.4) of the LMP model. Local convergence. From here on we focus on the lattice model on the torus [−L(n) , L(n) ]d ∩ ❩d where opposite faces are identified. The length L(n) tends to ∞ as n → ∞. Let {µnJ } be a sequence of finite volume Gibbs measures on the tori {❚n }. Gibbs states in G (J) can be constructed via the notion of local convergence. In this survey the superscript J in the notation of the measures µn = µnJ is repressed. We consider the case when the potentials JA are translation invariant and have finite range.  16  Definition 2.3. A potential J has finite range if sup{diam(A) : JA = 0} < ∞, A  where diam(A) is the diameter of A. As an example consider the classical Ising model (2.18) for which the above supremum equals 1. A local function F is a bounded function which only depends on the spins {σx , x ∈ sp(F)} where sp(F) is a finite subset of ❩d . Finite volume measures {µn } converge locally to a Gibbs state µ∞ if lim µn (F) = µ∞ (F),  n→∞  for all local functions F. For lattice models it is known that µ∞ is a well-defined measure on the sigma algebra generated by the cylinder functions. If J is translation invariant and has finite range, then the measure µ∞ is a DLR measure in G (J) [24, p.70]. We list some important results for Gibbs states obtained by a local limit. The set of Gibbs states G (J) is convex. A Gibbs state µ is called pure or extreme if it cannot be expressed as a strictly convex combination: µ = sν + (1 − s)ν for all 0 < s < 1 and ν, ν ∈ G (J) with ν = ν . Pure states are important because of the following theorem. Theorem 2.4 (Ergodic decomposition). Any measure µ ∈ G (J) can be uniquely expressed as a convex combination of pure states. Translation invariance of the measures µn provides a way to check whether a Gibbs state is extreme. Theorem 2.5. Let J be translation invariant. A Gibbs state in G (J) is pure if and only if it is ergodic. What remains is to find a test which tells whether a measure is ergodic. After introducing some notation the next theorem will state that a measure in G (J) whose correlations decay exponentially is ergodic. Definition 2.6. A measure µ has an exponential decay of correlations if there exists constants a1 , a2 such that |µ(FG) − µ(F)µ(G)| ≤ a1 (|sp(F)|, |sp(G)|)e−a2 dist(sp(F),sp(G)) , for any pair of local functions F, G with dist(sp(F), sp(G)) > L for some finite L. Cylinder events are events of the type {σΛ ∈ A} where σΛ is the spin configuration in a finite set Λ and A ∈ E Λ . Given x ∈ ❩d let θx denote the transformation which translates the spin configuration by x. The next theorem implies that if µ has exponential decay of correlations, then µ is ergodic. 17  Theorem 2.7. Let (Λk )k≥1 be any sequence of cubes in  ❩d such that |Λk | → ∞ as k → ∞. The  measure µ is ergodic if and only if 1 ∑ µ(A ∩ θx B) = µ(A)µ(B), k→∞ |Λk | x∈Λ k lim  for any pair of cylinder events A and B.  2.3  Phase transition  In this section we state the result obtained by Lebowitz, Mazel and Presutti about the phase coexistence in the continuum LMP model. We also state two important results which will be achieved in this thesis, the coexistence of phases in the Ising and WR model. In this section we freely use the results and terminology of Section 2.2.  2.3.1  LMP model  LMP show that, for γ small enough, the phase diagram in the (υ, β ) positive half-space has a critical ± curve υ(γ, β ) on which there exist at least two Gibbs states µγ,υ(γ,β ),β . Intuitively one Gibbs state represents a liquid phase while the other one represents a gas phase. Let C˜r(1) ⊂ ❘d be the cube of  side-length  1  centered at r ∈ ❘. Given a configuration q define the particle density ρr (q) =  |q ∩ C˜r(1) | . |C˜r(1) |  The numerator counts the number of particles in C˜r(1) . The denominator is the volume of C˜r(1) which is  d. 1  ± The two states µγ,υ(γ,β ),β can be distinguished because the expected particle density in the  liquid phase is strictly larger than the particle density in the gas phase. Theorem 2.8. Let d ≥ 2, there exist βc , β0 , γ0 (β , d) > 0 and υ(γ, β ) such that for β in the nonempty interval (βc , β0 ) and for 0 < γ < γ0 (β ) ± (i) there exist at least two sequences of finite volume measures µΛ,γ,υ(γ,β ),β which converge to ± distinct DLR measures µγ,υ(γ,β ),β . ± (ii) The Gibbs states µγ,υ(γ,β ),β are translation invariant, ergodic and have an exponential decay  of correlations. ± ± − + (iii) The expected particle density in Gibbs state µγ,υ(γ,β ),β is ργ with ργ < ργ .  The phase diagram for the mean field theory of the LMP model has the following description: there exists βc such that for β < βc there is a unique phase. For β > βc there exists a critical curve  18  υ = υ(β ) where there is a phase coexistence. Off the critical curve υ(β ) there exists a unique phase. One expects a similar phase diagram for γ positive but small. However, the critical curve would become a function of both β and γ. Lebowitz, Mazel and Presutti uncovered a part of the phase diagram for positive γ. The inverse temperature βc mentioned in Theorem 2.8 is the mean field critical inverse temperature. The constraint β < β0 is a technical limitation of the proof and on the curve υ(β , γ) a phase coexistence is expected for β ≥ β0 as well. Off the critical curve υ(γ, β ) and for β < βc it is expected that there is a unique DLR measure. However, to our knowledge this has not yet been proved.  2.3.2  Ising and WR model  Our results for the Ising and WR model are alike. In both cases the finite volume measures are invariant under the flip-symmetry where the all the signs of the spins are simultaneously flipped. The flip-symmetry suggests the existence of a phase coexistence. We first state the result for the Ising model. For the Ising model there exist two distinct Gibbs  P±γ,β . These states can be distinguished by considering the expected value for the spins σx . − For P+ γ,β the expected spin at location x is strictly positive while for Pγ,β it is strictly negative. Let ❊+γ,β denote the expectation with respect to the measure P+γ,β . states  Theorem 2.9 (Ising model). If the dimension d ≥ 2 and β > 1, then there exists γ0 (β , d) such that for all 0 < γ < γ0 (β , d) (i) the measures Pn,γ,β converge locally to P∞,γ,β = (ii) The measures  1 2  P+γ,β + 21 P−γ,β .  P±γ,β are ergodic, translation invariant with an exponential decay of correla-  tions. (iii) There exists ξˆ = ξˆ (β ) > 0 such that for x ∈ ❩d √  ❊  ± γ,β [σx ]  √ ˆ − e− β ξ √ =± √ ˆ + O(γ d/3 ). βξ − β ξˆ e +e e  β ξˆ  R EMARKS 2.10. • The critical inverse temperature for the mean field model of the Ising model is βc = 1. For β < βc a unique DLR measure is expected but not proved in this work. • Bodineau and Presutti [4] consider a general class of Ising models where the interactions are governed by a potential Jγ (x, y). The potential Jγ is a Kac potential (1.5) with w a nonnegative, smooth, integrable and compactly supported function. They show that for β > 1 there exists γ(β ) > 0 such that for all γ in (0, γ(β )) there exist at least two distinct Gibbs 19  states. Moreover, for β < 1 there exists a unique DLR measure. Remark 2.1 shows that the function v is a convolution of indicator functions which is not smooth. Strictly speaking the Ising model considered here does not fit exactly into the framework of Bodineau and Presutti. However, with their method it would be possible to prove the existence of a phase transition. Bodineau and Presutti define finite volume measures with fixed boundary conditions as opposed to the periodic boundary conditions we consider in this work. In particular, they con˜ n,γ,β for the cube T (n) = [−L(n) , L(n) ]d ∩ ❩d without identifying sider finite volume measures P the faces. For each cube T (n) they pick two boundary conditions σ ± in {−1, +1}❩  d −T (n)  . This  (σ |σ ± )  defines the conditional Hamiltonian Hn as in (1.6). This in turn yields finite volume (n) ± ± T ˜ as defined by (1.7). The infinite volume Gibbs states measures Pn,γ,β (σ |σ ) on {−1, +1} ˜ ± (σ |σ ± ). mentioned in the result by Bodineau and Presutti are the local limits of P ♦ n,γ,β  The result for the Widom-Rowlinson model is almost the same except there are some more parameters in play. Given a WR spin configuration σ define σxˇ = ∑1≤i≤kxˇ σxˇi and the spin average σ xˇ at scale  −  by σ xˇ =   kxˇ  k−1 ∑ σ i xˇ xˇ  if kxˇ = 0,    0  else.  i=1  For the WR model there exist two distinct Gibbs states P± γ,υ,r,β . These states can be distinguished by considering the expected value of σ xˇ . For P+ γ,υ,r,β the expected spin average is strictly positive  P−γ,υ,r,β it is strictly negative. Let ❊±γ,υ,r,β denote the expectation with respect to the measure P± γ,υ,r,β .  while for  Theorem 2.11 (WR model). There exists υ0 (r, β ) such that for all d ≥ 2, β > 0, r > 1 and υ > υ0 there exists γ0 (υ, r, β , d) such that for all 0 < γ < γ0 (i) the measures Pn,γ,υ,r,β converge locally to P∞,γ,υ,r,β =  1 2  P+γ,υ,r,β + 12 P−γ,υ,r,β .  (ii) The measures P± γ,υ,r,β are ergodic, translation invariant with an exponential decay of correlations. (iii) There exists ξˆ = ξˆ (υ, k, β ) > 0 such that √ ˆ √ ˆ βξ βξ e − e √ ˆ + O(γ d/3 ). ❊±γ,υ,r,β [σ xˇ] = ± √ ˆ e β ξ + e− β ξ R EMARKS 2.12. • The existing work on phase transitions in continuum and lattice Widom-Rowlinson models [8, 9, 51] considers models with k different types of particles. Pairs of particles of a different type experience a hard-core repulsion: there exists some r0 > 0 such that any pair of particles 20  cannot be within distance r0 of one another. In lattice models the hard-core repulsion puts a restriction on spin particle configurations: there can be at most one particle per lattice site. Finite range interactions are allowed but only between pairs of particles of the same type. A coexistence of k phases is proved when the chemical potential is large enough. Our WR model is a variation on the existing work. In particular, it does not have a hard-core repulsion. Instead there are finite range interactions between any pair of particles, regardless of the type. The model has the extra complication that there can be an integral number of particles at each lattice site. Moreover, the particles at a lattice site can be of any type. • In essence the WR model is a lattice model. However, the work of Lebowitz, Mazel and Presutti [41] for the continuum LMP model shows that the WR model is an intermediate step towards a true continuum model. LMP partition ❘d into open cubes of side-length − where − γ −1 is an integral number. Let Vˇ denote the set of cube-centers in ❘d . Given a particle configuration q and xˇ ∈ Vˇ they define the particle densities ρxˆ (q) =  |q ∩Cxˇ | d  ,  −  where the numerator counts the number of particles in the open cube Cxˇ centered at x. ˇ To prove a phase transition LMP study the marginal distribution of the random variables {ρxˆ }x∈ ˇ Vˇ . Thus they integrate out the positions of the particles in the open cubes and study the measure which only depends on the number of particles in each open unit cube. Hence, our WR model can be seen as an approximation to the marginal distribution of a true continuum model.  2.4  ♦  Kac-Siegert transformation  The Kac-Siegert transformation reformulates a system of particles interacting through two-body potentials into a system where independent particles interact with an auxiliary field ξ . In the particular case of the Ising and WR model the transformation allows to rewrite the finite volume measures in terms of Gaussian integrals of field averages. The majority of the work will be to show that the Gaussian averages concentrate around one of two possible values ±ξˆ . Once that has been achieved the information about the Gaussian averages can be readily translated back into the language of Ising spin models or the WR spin particle model. The Kac-Siegert transformation was independently discovered by Kac [36] and Siegert [54] who applied it in continuum systems. The transformation is also known as the Hubbard-Stratonovich transformation. It has been credited to Stratonovich [55] and was popularized by Hubbard [32]. The full generality of the Kac-Siegert transformation is explored in [10, Chapter 2].  21  In this section we show how the Kac-Siegert transformation can be applied to the Ising and WR model. We give an heuristic argument as to why the Gaussian averages are expected to concentrate around two possible values ±ξˆ . The main objective of the thesis then becomes to validate the claim of a concentration of Gaussian averages. This will be achieved in Chapter 4.  2.4.1  Ising model  Let ξ = {ξx , x ∈ ❚(n) } be i.i.d. standard Gaussian random variables. Gaussian averages are defined in terms of coefficients w(x, ˚ y) = w˚ n (x, y). Let o be the origin, then w(x, ˚ y) = ✶{y∈C(1) } ,  w˚ =  x  w(o, ˚ y).  ∑  (2.20)  y∈❚(n)  The normalization constant w˚ is the number of lattice points in a cube C(1) . Define the averaging operator ˚ x [ξ ] = 1 ∑ ξy w(x, Av ˚ y), w˚ y∈❚(n) ˚ x [ξ ]. The first result shows that the finite volume measures can be rewritten in terms and set ψ˚ x = Av ˚ Let ❊(n) denote the expectation with respect to ξ . of Gaussian averages ψ. Lemma 2.13. The finite volume measures Pn,γ,β satisfy  Pn,γ,β (σ ) =  2−|❚  (n) |  (n) Zγ,β  ❊(n) ∏ e  √  β σx ψ˚ x  .  x∈❚(n)  The grand partition function satisfies Zγ,β = ❊(n) (n)  ∏  eV (ψ˚ x ) ,  x∈❚(n)  where V (ζ ) = log cosh( β ζ ). The rewriting of the system in terms of the mean field ψ˚ is what we call the Kac-Siegert transformation. The spin-spin interactions have been replaced by an interaction between spins σx and Gaussian field averages ψ˚ x . The variance of a Gaussian average ψ˚ x of i.i.d. standard normal Gaussians is γ d/2 . Hence, the variables ψ˚ x are expected to concentrate around minimizer(s) ξˆ with small fluctuations of order γ d/2 . Claim 2.14. As γ → 0 the field averages ψ˚ x are going to concentrate near the minimizers of Φ :  ❘ → ❘ with Φ(ζ ) =  ζ2 −V (ζ ), 2 22  (2.21)  where V (ζ ) = log cosh( β ζ ). The function Φ is studied in Proposition C.2. It is shown that for β ≤ 1 the function Φ has a unique minimizer ξˆ = 0. For β > 1 the function Φ has a double-well shape with two symmetrically located minimizers ±ξˆ . Hence, a phase transition is to be expected for β > 1. We finish the section on the Ising model by proving Lemma 2.13. Furthermore, we give a heuristic argument for the claim that the Gaussian averages settle near the minimizers of Φ. Proof of Lemma 2.13. Consider the identity  ∑  x∈❚(n)  m˚ x (σ )ξx =  ∑  x∈❚(n)  σx ψ˚ x .  Recall the definition of the Hamiltonian for the Ising model by (2.8). (n)  e−β Hγ  (σ )  √ √ β 2 ˚ = e 2 ∑x∈❚(n) m˚ x (σ ) = ❊(n) [e β ∑x∈❚(n) m˚ x (σ )ξx ] = ❊(n) [e β ∑x∈❚(n) σx ψx ].  (2.22)  where the expectation is w.r.t. the Gaussian field ξ . Formula (2.22) shows that the finite volume measure (2.9) can be equivalently defined by  Pn,γ,β (σ ) =  2−|❚  (n) |  (n) Zγ,β  ❊(n) ∏ e  √  β σx ψ˚ x  .  (2.23)  x∈❚(n)  (n)  The normalization constant Zγ,β is obtained by summing over all possible spin configurations on  ❚(n) .  ∑  √ √ √ e β σx ψ˚ x e− β ψ˚ x + e+ β ψ˚ x = ∏ exp[log cosh( β ψ˚ x )] ∏ 2 = ∏ 2 (n) (n) (n) ❚ x∈❚ x∈❚ x∈❚(n)  σ ∈{−1,+1}  Motivation for Claim 2.14. Let N  (n)  be the normalization constant of the Gaussian field on  ❚(n)  with covariance matrix I. The grand partition function in Lemma 2.13 satisfies (n)  Zγ,β =  1 N (n)  d ❚ (ξ )e− 2 ∑ ξx +V (ψ˚ x ) . (n)  1  2  ˚ x [ξ 2 ] ≥ ∑ ψ˚ x2 . The first equality holds because the Jensen’s inequality yields the bound ∑ ξx2 = ∑ Av sum runs over the torus. ψ˚ 2 ξx2 −V (ψ˚ x ) ≥ ∑ x −V (ψx ) = ∑ Φ(ψ˚ x ) 2 2 x∈❚(n) x∈❚(n) x∈❚(n)  ∑  It is expected that the field averages ψ˚ will settle near the minima of Φ(ζ ). 23  2.4.2  Widom-Rowlinson model  Let ξ = {ξx , x ∈ ❚(n) } be i.i.d. standard Gaussian random variables. Gaussian averages are defined •  •  in terms of coefficients w(x, y) = wn (x, y). Let o be the origin, then w(x, y) = ✶{y∈C(1) } , •  •  w=  ∑  •  (2.24)  w(o, y).  y∈❚(n)  v(x) ˇ  •  The normalization constant w is the number of lattice points in a cube C(1) . Define the averaging operator •  Avx [ξ ] = •  1 ξy w(x, y), w y∈∑ ❚(n) •  •  •  and set ψ x = Avx [ξ ]. The averages are constant on cubes at scale  •  •  : ψx = ψxˇ for all x ∈ Cxˇ .  −  Define the energy of a spin configuration σ immersed in the auxiliary field ξ to be •  Hn,γ,υ,r,β (σ , ψ) = −υβ  ∑  kxˇ −  •  β |C|  ˇ (n) x∈ ˇ ❚  ∑  σxˇ ψxˇ +  ˇ (n) x∈ ˇ ❚  kβ |C| 2  ∑  ρxˇ (k)2 .  (2.25)  ˇ (n) x∈ ˇ ❚  The first result shows that the measures (2.15) can be expressed as a Gibbs measure with the Hamiltonian Hn,γ,υ,r,β (σ , ξ ). Lemma 2.15. The finite volume measures Pn,γ,υ,r,β satisfy  Pn,γ,υ,r,β (σ ) =  1 (n)  Zγ,υ,r,β  |C|kxˇ  •  e−Hn,γ,υ,r,β (σ ,ψ) . ❊(n) ∏ kxˇ · k ! 2 xˇ ˇ (n) x∈ ˇ ❚ •  The rewriting of the system in terms of the mean field ψ is what we call the Kac-Siegert transformation. The second term in (2.25) shows that the spin-spin interactions have been replaced by an •  interaction between σxˇ and ψxˇ . ˇ (n) ). The grand partition function Zγ,υ,r,β is obtained by summing over all possible σ ∈ Σσ (❚ (n)  After applying the Kac-Siegert transformation the sum over σ will factor over cubes at scale Furthermore, it is possible to integrate out the signs  σxˇi  .  −  of the spin configurations. Let k ∈ ◆ repre-  sent the number of particles in a single cube and pick ζ ∈ ❘. Define the Hamiltonian in auxiliary field ζ by Hγ,υ,r,β (k, ζ ) = −υβ k − k|C|V (ζ ) +  rβ |C|ρ(k)2 , 2  where the function V : ❘ → ❘ is V (ζ ) = log cosh( β ζ ) and ρ(k) =  k |C| .  (C) Lemma 2.16. Let the cube partition function Zγ,υ,r,β : ❘ → ❘ be defined by  |C|k −Hγ,υ,r,β (k,ζ ) e , ∑ k=0 k! ∞  (C) Zγ,υ,r,β (ζ ) =  24  (2.26)  then Zγ,υ,r,β = ❊(n) (n)  •  ∏  ˇ (n) ) x∈( ˇ ❚  (C) Zγ,υ,r,β (ψxˇ )  (2.27)  •  The Gaussian averages ψ of i.i.d. standard normal Gaussian averages is γ d/2 . As before the averages ψ are expected to concentrate around minimizer(s) ξˆ . •  ˆ are the Claim 2.17. As γ → 0 the field averages ψ will concentrate around ±ξˆ where (±ξˆ , ρ) •  minimizers of rβ 2 log ρ ρ + ρ(log ρ − 1) + . 2 2|C|  F(ζ , ρ) = ζ 2 − υβ ρ −V (ζ )ρ + and V (ζ ) = log cosh( β ζ ).  The function F(ζ , ρ) is studied in Section C.2. It is shown that there exists υ0 = υ0 (k, β ) such ˆ Hence, that for υ ≥ υ0 and r > 1 the function F has two symmetrically located minimizers (±ξˆ , ρ). a phase transition is to be expected in this range of the parameters υ and r. We finish the section by providing proofs for Lemma 2.15 and Lemma 2.16. An heuristic argument is given as to why the averages settle near ±ξˆ . Proof of Lemma 2.15. Consider the useful identity  ∑  x∈❚(n)  2 ξx mv(x) ˇ (σ ) = •  1 |C(1) | x∈∑ ❚(n)  ∑  ˇ (n) y∈ ˇ ❚  ξx σyˇ ✶{y∈C = (1) ˇ } v(x) ˇ  ∑  σyˇ  ˇ (n) y∈ ˇ ❚  1 ξx ✶{x∈C(1) } |C(1) | x∈∑ yˇ ❚(n)  •  On the right-hand side we find the field average ψxˇ . The two-body interaction in the WR model satisfies exp  β 2  ∑  ˇ (n) x∈ ˇ ❚  mxˇ (σ )2 = ❊(n) exp •  β  ∑  x∈❚(n)  = ❊(n) exp  •  mxˇ (σ )ξx  β |C|  •  ∑  σxˇ ψxˇ  (2.28)  ˇ (n) x∈ ˇ ❚  where the expectation is with respect to the Gaussian field ξ . Formula (2.28) shows that the finite volume measure (2.15) can be equivalently defined by  Pn,γ,υ,r,β (σ ) =  |C|kxˇ  1 (n)  Zγ,υ,r,β  •  ❊(n) ∏ kx e−Hn,γ,υ,r,β (σ ,ψ) , 2 · k ! xˇ ˇ (n) x∈ ˇ ❚  (2.29)  (n)  where Zγ,υ,r,β is the grand partition function. (n)  Proof of Lemma 2.16. The grand partition function Zγ,υ,r,β is the normalization constant of (2.29) Zγ,υ,r,β = ❊(n) (n)  |C|kxˇ −Hn,γ,υ,r,β (σ ,ψ) e . · kxˇ ! •  ∑  ∏  ˇ (n) ˇ ❚ ❚ˇ (n) ) x∈  σ ∈Σσ (  25  2−kxˇ  (2.30)  (n)  •  Let Zγ,υ,r,β (ψ) denote the integrand on the right-hand side of (2.30). After the Kac-Siegert transfor•  mation all the terms in Hn,γ,υ,r,β (σ , ψ) factor over the cubes at scale  .  −  √ i |C|k β υk− rβ |C|ρ(k)2 k eσ β ψxˇ 2 e ∑ ∑ ∏ 2 , i=1 k≥0 σ ∈{−1,+1}k k! •  • (n) Zγ,υ,r,β (ψ)  =  ∏  ˇ (n) ) x∈( ˇ ❚  (2.31)  Fix x, ˇ then for a fixed k the sum over σ ∈ {−1, +1}k is readily computed. k  ∑  ∏  eσ  i  √ 2  σ ∈{−1,+1}k j=1  √  •  β ψxˇ  =  e  •  β ψxˇ  √ + e− β ψxˇ 2 •  k  •  = ekV (ψxˇ )  Substitute the identity back into (2.31) to obtain (2.27).  Variational problem In the final part of this section we give an heuristic argument for the statement of Claim 2.17. Be(n)  •  sides the Gaussian averages ψ the particle densities ρxˆ also appear in the partition function Zγ,υ,r,β . The particle density averages the number of particles in a cube C over the volume of |C|. Like the Gaussian averages, the particle averages are expected to concentrate around some minimizer ρˆ with small fluctuations of order  − d2  . The symmetry of the model and the concentration of both Gaussian ˆ and particle averages point in the direction of minimizers of the form (±ξˆ , ρ). −  The minimizers are found in a two-tier approach. First assume that the Gaussian averages concentrate around some ζ ∈ ❘. The particle averages are expected to concentrate around a minimizer ˜ ) which is a function of ζ . The second step is to minimize over all possible ζ in order to locate ρ(ζ ˜ ξˆ ). ξˆ and set ρˆ = ρ( Suppose all Gaussian averages ψ concentrate at ζ ∈ ❘. Following in the footsteps of Lebowitz and Penrose [42] we will show that the particle averages ρxˇ will concentrate at ρ˜ which is a mini•  mizer to a specific variational problem. Lemma 2.18. There exist constants a0 , a1 , independent of γ, such that the cube partition function satisfies (C) ˜ ˜ a0 e−|C|Fζ (ρ) ≤ Zγ,υ,r,β (ζ ) ≤ a1 e−|C|Fζ (ρ) ,  ˜ minimizes where Fζ (ρ) Fζ (ρ) = −υβ ρ −V (ζ )ρ +  log ρ rβ 2 ρ + ρ(log ρ − 1) + . 2 2|C|  26  R EMARK . Observe that in the limit γ → 0 on recovers a Lebowitz-Penrose limit lim  (C) (ζ ) log Zγ,υ,r,β  |C|  γ→0  = min −Fζ (ρ).  ♦  ρ≥|C|−1  The proof of Lemma 2.18 is deferred until Chapter 5. The first three terms in Fζ (ρ) correspond to the Hamiltonian (2.14) at fixed density ρ. The last two terms in Fζ (ρ) are the entropy of an ideal gas at fixed density ρ. Adding the Gaussian field to Fζ results in F(ζ , ρ) =  ζ2 rβ 2 log ρ − υβ ρ −V (ζ )ρ + ρ + ρ(log ρ − 1) + 2 2 2|C|  The claim about the concentration of the Gaussian averages can be stated in terms of F(ζ , ρ) and ˜ ) of Fζ (ρ) for ρ > |C|−1 . In particular, the two-tier approach suggests to slightly the minimizer ρ(ζ alter Claim 2.17. •  Claim 2.17 . The field averages ψx are going to concentrate near the minimizers of ˜ )). Φ(ζ ) = F(ζ , ρ(ζ  Motivation for Claim 2.17 . Let N  (n)  be the normalization constant of the Gaussian field on ❚(n)  with covariance matrix I. The grand partition function satisfies (n)  Zγ,υ,r,β =  1 N (n)  d ❚ (ξ ) (n)  exp −  ∏  x∈ ˇ Vˇ (❚(n) ) •  1 (C) ξx2 + log Zγ,υ,r,β (ψxˇ ) . ∑ 2 x∈Cxˇ •  •  Jensen’s inequality yields the bound ∑ ξx2 = ∑ Avx [ξ 2 ] ≥ ∑ ψx2 . The first equality holds because the sum runs over the torus. (n)  Zγ,υ,r,β ≤  1 N (n)  d ❚ (ξ ) (n)  ∏  exp −  x∈ ˇ Vˇ (❚(n) ) •  1 (C) (ψxˇ ) . ∑ ψx2 + log Zγ,υ,r,β 2 x∈C xˇ •  •  •  Apply Lemma 2.18 and note that ψx = ψxˇ for all x ∈ Cxˇ . −  1 (C) (ψxˇ ) ≈ |C|Φ(ψxˇ ) ∑ ψx2 + log Zγ,υ,r,β 2 x∈C xˇ •  •  27  •  Chapter 3  A cluster expansion for Gaussian fields In Chapter 2 we defined two particle models and used the Kac-Siegert transformation to rewrite the Gibbs measures as Gaussian integrals. We introduce the principal tool to analyze Gaussian integrals: a cluster expansion for Gaussian fields with a finite range decomposition. This form of the cluster expansion was developed by Guadagni [31]. The review of the elegant proof is included for two reasons. Firstly, the work of Guadagni is slightly modified to fit it to our needs. Secondly, we correct a minor error which occurred in Guadagni’s original work. In this work the cluster expansion is applied in Chapter 4 to show convergence of finite volume measures and to prove the concentration of Gaussian averages. However, the results in this chapter are of interest in their own right and apply to a wider range of problems. Therefore, the reader can go through this technical chapter as a standalone result. The other option is to read this chapter in unison with Chapter 4. Progressing through Chapter 4 the reader is referred back to the related sections in this chapter. Cluster expansions are formal power series for the logarithm of a grand partition function Z (n) . Section 3.1 defines the type of grand partition functions Z (n) which can be handled by the cluster expansion of Guadagni. This requires the definition of finite range decompositions of Gaussian fields. Section 3.2 derives the formal expression for log Z (n) . Moreover, it states the conditions under which the cluster expansion converges uniformly in n. Section 3.4 provides the proof of uniform convergence. The cluster expansion has a long history in statistical mechanics. A brief description of its history is included in the notes at the end of the chapter.  3.1  Finite range decompositions and polymer models  Consider the length-scale  +  . The torus ❚(n) with n ∈ ◆ is the graph with vertices [−n + , n + ]d ∩ ❩d  where opposite faces are identified. Vertices x, y in ❚(n) are connected by an edge if the supremum norm x − y  ∞=  1. The diameter of the torus is L(n) = 2n + . The notation ❚(n) may refer to either  the graph itself or the vertex set of ❚(n) . Based on the context it will be clear which interpretation 28  to use. In the next paragraphs we consider a Gaussian field ξ on ❩d and define what it means for ξ to admit a finite range decomposition. The Gaussian field ξ is used to construct finite volume Gaussian fields on ❚(n) . The grand partition function Z (n) will be a Gaussian integral with respect to the finite  volume Gaussian field on ❚(n) . The integrand is a polymer model which is defined in the final part of this section. Infinite volume measure. Consider a Gaussian field ξ on  ❩d with a covariance matrix W . The  covariance matrix W is assumed to be invariant under translations at scale  +  W (x, y) = W (x + z + , y + z + ), for all z ∈ ❩d . The distance dist(x, y) on ❩d is measured in the supremum norm. A Gaussian field  ζ on ❩d has range L if its covariance matrix has range L, i.e., ❊[ζx ζy ] = 0 if dist(x, y) > L. A given Gaussian field ξ with covariance matrix W∞ is said to have a finite range decomposition if there exist independent Gaussian fields {ζs } with covariance matrices {Ws } such that ξ=  ∑ λs ζs ,  (3.1)  s≥0  where ζs has range Ls and the scales λs and matrices Ws satisfy  ∑  y∈❩d  |Ws (x, y)| ≤ 1  and  ∑ λs < ∞.  (3.2)  s≥0  for all x ∈ ❩d . Let λ∞ = ∑s≥0 λs be the sum over scales. Gaussian measures are completely determined by their covariance matrices. Hence, a finite range decomposition of the Gaussian field is equivalent to the statement W∞ =  ∑ λ2s ·Ws .  (3.3)  s≥0  Examples of finite range decompositions for Gaussian fields are provided in the notes at the end of this chapter. R EMARK . The elements of the covariance matrix satisfy the diagonal bound |Ws (x, y)| ≤ Wmax where Wmax is the largest diagonal element of Ws . The finite range of Ws implies that the condition (2Ls )d Wmax ≤ 1, is sufficient for the first inequality in (3.2) to hold. 29  (3.4) ♦  The main aim of this chapter will be to study Gaussian fields on the torus  Finite volume measures.  ❚  (n)  as n → ∞. Therefore, it is necessary to define the finite volume Gaussian measures. Gaussian (n)  measures are completely determined by their covariance matrix. Let ζs  ❚  (n)  be the Gaussian field on  with covariance matrix (n)  Ws (x, y) =  ∑ Ws (x, y + 2zL(n) ),  z∈❩d  (n)  and define ξ (n) = ∑ ζs . The Gaussian field ξ (n) has covariance matrix W (n) =  (n)  ∑ λsWs  .  (3.5)  s≥0 (n)  The latter definitions are only valid if Ws  and W (n) are well-defined positive definite matrices. This  fact will be shown in the next lemma. (n)  Lemma 3.1. The matrices Ws  satisfy  ∑  (n)  |Ws (x, y)| ≤ 1.  y∈❚(n) (n)  Moreover, the matrices Ws  are symmetric and positive definite.  R EMARK . The lemma immediately implies that W (n) is a symmetric positive definite matrix which satisfies  ∑  y∈❚(n)  |W (n) (x, y)| ≤ λ∞ ,  for all x ∈ ❚(n) .  ♦  Proof. The assumptions on Ws show that (n)  |Ws (x, y)| ≤  ∑  y∈❚(n)  ∑ ∑ |Ws (x, y + 2zL(n) )| = ∑  y∈❚(n) z∈❩d  y∈❩d  Symmetry immediately follows by the symmetry of Ws . (n)  To show that Ws  is positive definite one must show that  ∑  (n)  ϕ(x)Ws (x, y)ϕ(y) ≥ 0,  x,y∈❚(n)  30  |Ws (x, y)| ≤ 1.  for all ϕ in ❘❚ . Let ϕ˜ be the periodic extension of ϕ. (n)  (n)  ϕ(x)Ws (x, y)ϕ(y) =  ∑  x,y∈❚(n)  ∑ ∑ ϕ(x)Ws (x, y + 2zL(n) )ϕ(y)  x,y∈❚(n) z∈❩d  =  ∑ ∑  y∈❩d x∈❚(n)  ˜ ˜ ϕ(x)W s (x, y)ϕ(y)  The right-hand side is invariant under a translation of x by 2zL(n) for any z ∈ ❩d . Given N define the set ZN = [−N, N]d ∩ ❩d .  ∑  (n)  ϕ(x)Ws (x, y)ϕ(y) =  x,y∈❚(n)  1 (2N)d  ∑ ∑  ∑  y∈❩d z∈ZN x∈❚(n) +2zL(n)  ˜ ˜ ϕ(x)W s (x, y)ϕ(y)  The equality is thus also true in the limit N → ∞. Given N define the set Z˜ N = {❚(n) +2zL(n) : z ∈ ZN } ˜ ✶{z∈Z˜N } . and the vector ϕN (z) = ϕ(z) (n)  ϕ(x)Ws (x, y)ϕ(y) = lim  ∑  N→∞  x,y∈❚  (n)  1 (2N)d  ˜ ϕN (x)Ws (x, y)ϕ(y)  ∑  x,y∈❩  d  ˜ The claim is that in the above expression ϕ(y) can be replaced by ϕN (y). Compute an upper-bound ˜ − ϕN (y). on the difference ϕ(y) 1 N→∞ (2N)d lim  1 N→∞ (2N)d  ˜ ˜ ≤ max ϕ(x)2 lim ϕ(x)W s (x, y)ϕ(y)  ∑  x∈❚  x∈Z˜ N ,y∈ / Z˜ N  (n)  ∑  |Ws (x, y)|  (3.6)  x∈Z˜ N ,y∈ / Z˜ N  For any fixed x the sum over |Ws (x, y)| is convergent. For a given L define the set YL = {y ∈ / Z˜ N : dist(y, Z˜ N ) ≤ L}. Pick any ε > 0 and pick L large enough such that ∑|w|≥L |Ws (o, w)| ≤ ε where o is the origin  ∑ ∑ |Ws (x, y)| ≤ ε|Z˜N | ≤ ε(2N)d .  / L x∈Z˜ N y∈Y  For the sum over YL use that ∑x |Ws (x, y)| ≤ 1.  ∑ ∑ |Ws (x, y)| ≤ |YL |.  y∈YL x∈Z˜ N  Perform the sum over x.  ∑  |Ws (x, y)| ≤ ε(2N)d + |YL | ≤ ε(2N)d + (2N)d−1 L  x∈Z˜ N ,y∈ / Z˜ N  31  Dividing by (2N)d shows that 1 (2N)d  ˜ ˜ ϕ(x)W ≤ε+ s (x, y)ϕ(y)  ∑  x∈Z˜ N ,y∈ / Z˜ N  L . 2N  The bound holds for any ε > 0. This shows that the limit (3.6) tends to zero as N → ∞. Hence, we have proved that  ∑  1 N→∞ (2N)d  (n)  ϕ(x)Ws (x, y)ϕ(y) = lim  x,y∈❚(n)  A matrix W on  ∑  ˜ ˜ ϕ(x)W s (x, y)ϕ(y)  (3.7)  x,y∈Z˜ N  ❩d is defined to be positive definite if WΛ = (W (x, y))x,y∈Λ is positive definite  for all finite Λ ⊂ ❩d . The terms in the sum on the right-hand side of (3.7) are all positive for any finite N. The limit must be non-negative.  Polymer models.  Let Cn+ be the partition of  ❚(n) into cubes C+ of side-length  +  such that one  cube is centered at the origin. A set X is spatially connected if for any pair x, y in X there exists a walk in X from x to y. The walk is allowed to make diagonal steps. A polymer X is a spatially connected set which is a union of cubes at scale  +  . Let P (n) be the set of all polymers on  ❚(n) .  Henceforth, any set mentioned in this chapter will be a polymer. Recall that the torus is the set [−L(n) , L(n) ]d ∩ ❩d with opposite faces identified. A polymer X ∈ P (n) is a subset of ❩d which is  connected on ❚(n) . However, X is not necessarily connected in ❩d . Being a subset of ❩d , a polymer X can be an element of both P (n) and P (m) for n = m. For example consider the cube at the origin. To each polymer X we assign a polymer activity K (n) (X, ξ ).  Definition 3.2. A polymer activity K (n) (X, ξ ) is a measurable function which only depends on {ξx }x∈X and is (i) consistent: K (n) (X, ξ ) = K (m) (X, ξ ) if X is a polymer in both P (n) and P (m) , (ii) translation invariant: K(X + a, ξ ) = K(X, θa ξ ) where a ∈ ❩d and θa ξx = ξa+x . Given a polymer activity K (n) and a field ξ , define the partition function Z˜ (n) (ξ ) by m  Z˜ (n) (ξ ) =  ∑  m≥0  ∑  ∏ K (n) (Xi , ξ ),  (3.8)  {X1 ,...,Xm } i=1 Xi ∩X j =0/ if i= j  where the sum runs over sets of polymers Xi ∈ P (n) . This is said to be the partition function of a polymer system in a fixed auxiliary field ξ . The Gaussian polymer model with polymer activities  32  K (n) (X, ξ ) is defined by m  Z (n) = ❊(n) [Z˜ (n) (ξ )] = ❊(n)  ∑  m≥0  ∏ K (n) (Xi , ξ ) ,  ∑  (3.9)  {X1 ,...,Xm } i=1 Xi ∩X j =0/ if i= j  where the expectation is with respect to a Gaussian field ξ (n) . The Gaussian field allows for the finite range decomposition (3.5).  3.2  Convergence of the pressure  Define the functions Pr(n) =  log Z (n) , ˆ ❚(n) ) N(  (3.10)  ˆ ❚(n) ) = (2n)d is the number of C+ cubes contained in the torus ❚(n) . The pressure Pr of where N( the polymer gas is defined to be the limit of Pr(n) as n → ∞. The aim is to find a sufficient condition on the polymer activities and the scales λs in the finite range decomposition such that Pr(n) admits an expansion in powers of K which is convergent uniformly in n. Take a Gaussian field ξ which admits a finite range decomposition and for convenience define ζ−1 ≡ 0 with λ−1 = 1. Define the cumulative fields ξs for s ≥ −1 by s  ξs =  ∑  λ jζ j.  (3.11)  j=−1  The cumulative fields ξs (x) and ξs (y) are independent provided that the distance between x and y is at least Ls ∧ Ls . s  s  s∧s  ❊(n) [ξs (x)ξs (y)] = ∑ ∑ λi λ j ❊(n) [ζi (x)ζ j (y)] = ∑ λi2 ❊(n) [ζi (x)ζi (y)]. i=0 j=0  i=0  The above expression is zero if distn (x, y) > Ls ∧ Ls and for a Gaussian measure this is equivalent to the variables being independent. ˜ = ◆ ∪ {−1, 0} then a scaled polymer U = (X, s) is a tuple consisting of a polymer X and Let ◆ ˜ . Define the collection of scaled polymers Ps(n) = P (n) × ◆ ˜ . A scaled polymer (X, s) a scale s ∈ ◆ has a scaled polymer activity defined by  K (n) (X, 0) (n) δ K (X, s) = K (n) (X, ξ ) − K (n) (X, ξ  s−1 )  s  if s = −1,  (3.12)  if s ≥ 0.  The scaled polymer activities are differences of polymer activities K (n) (X, ξ ). For a polymer X let ˆ N(X) be the number of C+ cubes which are contained in X. 33  Definition 3.3. The polymer activities K (n) (X, ξ ) are called semi-local with positive constants a1 and a2 < 81 (λ∞2 + 1)−1 if the differences satisfy ˆ N(X)  |δ K (n) (X, s)| ≤ λs a1  exp a2  2 (x) + ζs2 (x)) , ∑ (ξs−1  (3.13)  x∈X  for all s ≥ 0. R EMARK . The condition on a2 is an integrability constraint which allows to take the expectation ♦  of the right-hand side of (3.13).  The final ingredient to state the main result is defined in terms of distance. Recall that the distance dist(x, y) on ❩d is measured in the supremum norm. For a given set X let Vˆ (X) ⊂ ❩d be the set of all cube-centers of cubes C+ which are contained in X. Let o be the origin and define Nˆ s = {xˆ ∈ Vˆ (❩d ) : dist(o, x) ˆ ≤ Ls } , to be the number of C+ -cubes in ❚(n) which are within distance Ls of the origin. Theorem 3.4. Let the scaled polymer activities be semi-local with constants a1 and a2 . If there exists some a3 > 1 such that a1 and a2 satisfy 2  4ea1 ea2 (λ∞ +1)|C  +|  3d a3+ + ∑ λs aL3 s Nˆ s < 1, s≥0  then Pr(n) converges uniformly in n. Moreover, the pressure Pr = limn→∞ Pr(n) exists. The above theorem is a consequence of the stronger Theorem 3.7 whose proof is given in Section 3.4. R EMARK . The first minor error in Guadagni [31] is that the factor 3d is missing in his convergence condition. The second error is a technical detail buried in the derivation of the factor ∑ λs Nˆ s . In his proof Guadagni constructs graphs on the vertex set Vˆ = Vˆ (❚(n) ). Given such a graph Guadagni puts weights λs on the edges of the graph. However, the weight λs should be on the vertices of the graph ♦  instead.  3.3  The formal power series for log Z (n)  The cluster expansion is a Taylor expansion for log Z (n) in powers of K. By this we mean that when K is replaced by zK the expansion is the Taylor expansion for log Z (n) in powers of z. We find conditions on K such that as n → ∞ this series converges for z = 1. The formal series can be  34  readily obtained once Z˜ (n) (ξ ) has been expressed in convenient form. The main challenge is to find constraints on the model such that the formal series converges uniformly as n → ∞. The main problem with expression (3.9) is that the covariance of the Gaussian field need never vanish and the expectation does not factor over polymers. To overcome this problem the scaled polymers (X, s) were introduced. The latter objects have finite range Ls . The scaled polymer activity δ K (n) (X, s) is measurable w.r.t. {ζt (x)} x∈X . Therefore, define the scaled activities δ K (n) (X, s) and t≤s  δ K (n) (X , s ) to be independent if distn (X, X ) > Ls ∧ Ls . Scaled polymers U = (XU , sU ) and U = (XU , sU ) are defined to be independent if distn (XU , XU ) > LsU ∧ LsU . Otherwise, the scaled polymers are dependent. The generic notation A is used to denote sets of scaled polymers. Two sets Ai and A j are independent if U and U are independent for all possible scaled polymers U ∈ Ai and U ∈ A j . Let  ✶{Ai ,A j indep.} equal one when sets Ai and A j are independent and equal zero otherwise. A set A is Gaussian connected if there does not exist a partition of A into independent sets. The weight of A is defined to be  ❑(n) (A) = ✶{A Gs. conn.} ∏ δ K (n) (X, s),  (3.14)  (X,s)∈A  where  ✶{A Gs. conn.} is one when A is Gaussian connected and is zero otherwise. The support of a  scaled polymer U = (X, s) is sp(U) = X. The support for a set A is defined by sp(A) =  sp(U). U∈A  Proposition 3.5. The partition function satisfies m  Z (n) =  ∑  ∑  m≥0 {A1 ,...,Am }  ∏ ❊(n) [❑(n) (A j )] j=1  ∏ ✶{A ,A i  j  indep.} ,  1≤i< j≤m  (n)  where the sum runs over sets of Ai which themselves are sets of scaled polymers in Ps . Proof. The polymer activities satisfy ∞  K (n) (X, ξ ) =  ∑  δ K (n) (X, s).  s=−1  Substitution into (3.8) yields m  Z˜ (n) (ξ ) =  ∑  m≥0  ∑  ∏ ∑  {X1 ,...,Xm }  Xi ∩X j =0/ if i= j  35  i=1  s≥−1  δ K (n) (X, s) .  The expression expands into a sum of products over δ K (n) (X, s)’s for any family of disjoint poly˜ mers in P (n) and a family of scales s ∈ ◆ m  Z˜ (n) (ξ ) =  ∑ ∏ δ K (n) (Ui ).  ∑  (3.15)  m≥0 {U1 ,...,Um } i=1 (n)  The sum runs over sequences Ui = (Xi , si ) ∈ Ps  such that Xi ∩ X j = 0/ when i = j.  Given {U1 , . . . ,Um } construct the graph G = (V, E) with V = {U1 , . . . ,Um } and {Ui ,U j } ∈ E if and only if Ui and U j are dependent. Let {G1 , . . . , Gk } be the connected components of G. Let f map {U1 , . . . ,Um } to {G1 , . . . , Gk }. The map f is a bijection from {{U1 , . . . ,Um } : m ≥ 0} to {{A1 , . . . , Ak } : k ≥ 0}. The bijection f is used to change the order of summation k  Z˜ (n) (ξ ) =  ∑ ∑  k≥0 {A1 ,...,Ak }  ∏ ✶{A  δ K (n) (U)  ∏  U∈ f −1 (π(A)) k  =  ∑ ∑ ∏ ✶{A  j  conn.}  k≥0 {A1 ,...,Ak } j=1  j  ∏ ✶{A ,A  Gs. conn.}  i  j=1  ∏ δ K (n) (U)  U∈A j  j  indep.} ,  1≤i< j≤k  ∏ ✶{A ,A i  j  indep.} ,  1≤i< j≤k  where π(A) = {A1 , . . . , Ak }. The expectation of Z˜ (n) (ξ ) factors over the sets {A1 , . . . , Ak }. The key to obtain a formal expansion for log Z (n) is to express the product over indicator functions in terms of graphs.  ∏ ✶{A ,A i  j  indep}  ∏ (✶{A ,A  =  i  1≤i< j≤m  j  indep} − 1 + 1) =  1≤i< j≤m  ∏ (✶{A ,A  ∑  i  j  indep} − 1).  (3.16)  G on {1,...,m} {i, j}∈G  The sum runs over all graphs G and the product runs over edges in G. It is a standard exercise in cluster expansion theory that the net effect of taking the logarithm is to cancel all contributions of non-connected graphs e.g., see [13, 50]. Faris [21] describes how enumerative combinatorics can be used to prove this cancellation. Therefore, define the coefficient Jc (A1 , . . . , Am ) =  ∏ (✶{A ,A  ∑  i  G on {1,...,m}  j  indep} − 1),  (3.17)  {i, j}∈G  G connected  where the sum runs over all possible connected graphs on {1, . . . , m}. Theorem 3.6. For any n > 0 the logarithm of the partition function formally satisfies F  log Z (n) =  m  1 m! m≥1  ∑  ∏ ❊(n) [❑(n) (A j )]  ∑ sp(Ai  (A1 ,...,Am ) )⊂ (n) ∀1≤i≤m  ❚  Jc (A1 , . . . , Am ),  j=1  where the sum runs over sequences of Ai which themselves are sets of scaled polymers. 36  (3.18)  R EMARK . The phrase “formally satisfies” means that the sum over connected graphs is the Taylor expansion of log Z (n) in powers of z when K is replaced by zK. Functions are not necessarily equal to their Taylor series. In this case Z (n) is a polynomial 1 + a1 z + . . . because only finitely many polymers can fit into ❚(n) . Therefore, log Z (n) has a convergent expansion in powers of z for z small. Power series expansions are unique and given by the Taylor expansions. The power series expansion equals the Taylor expansion and the power series expansion ♦  converges wherever the Taylor expansion converges.  3.4  Convergence of the cluster expansion  We are interested in convergence of the formal series (3.18) for sets {A1 , . . . , Am } which are rooted at polymers. For given polymers X,Y define the sets Am (X) = {(A1 , . . . , Am )|sp(Ai ) ∩ X = 0/ for some 1 ≤ i ≤ m}, Am (X,Y ) = {(A1 , . . . , Am )|sp(Ai ) ∩ X = 0/ and sp(A j ) ∩Y = 0/ for some 1 ≤ i, j ≤ m}, We show convergence of the series P(n) (X) =  m  1 1 ∑ ∑ ˆ m! N(X) m≥1 (A1 ,...,Am )∈Am (X)  1 1 P (X,Y ) = ∑ ˆ N(X) m≥1 m! (n)  ∏ ❊(n) [❑(n) (A j )]  Jc (A1 , . . . , Am ),  j=1  (3.19)  m  ∏ ❊ [❑ (n)  ∑ (A1 ,...,Am )∈Am (X,Y )  (n)  (A j )] Jc (A1 , . . . , Am ),  j=1  for fixed non-empty polymers X,Y as n → ∞. The sum runs over sets Ai which themselves are sets (n)  of scaled polymers in Ps . R EMARK . Theorem 3.6 implies that Pr(n) = P(n) (❚(n) ). Hence, by definition Pr is the limit of  P(n) (❚(n) ) as n → ∞.  ♦  The proof of the convergence of the series P(n) (X) follows a classical combinatorial approach which is simplified through the use of tree-graph bounds as was done in [23]. A tree is a graph without loops where any pair of vertices are connected by a path. Theorem 3.7. Let the scaled polymer activities be semi-local with constants a1 and a2 . Fix any non-empty polymers X,Y ∈  P (n) with dist(X,Y ) >  +  .  (i) If a1 and a2 satisfy 2  ν := 4ea1 ea2 (λ∞ +1)|C  +|  3d + ∑ λs Nˆ s < 1, s≥0  37  then the bound 1 1 ˆ m! N(X)  m  ∏ ❊(n) [|❑(n) (A j )|] |Jc (A1 , . . . , Am )| ≤ λ∞  ∑  j=1  (A1 ,...,Am )∈Am (X)  νm , 1−ν  holds uniformly in n. (ii) If there exists some a3 > 1 such that a1 and a2 satisfy 2  ν˜ := 4ea1 ea2 (λ∞ +1)|C  +|  3d a3+ + ∑ λs aL3 s Nˆ s < 1, s≥0  then the bound 1 1 ∑ ˆ N(X) m! (A1 ,...,Am )∈Am (X,Y )  m  ∏ ❊(n) [|❑(n) (A j )|]  |Jc (A1 , . . . , Am )| ≤ λ∞  j=1  ν˜ m −dist(X,Y ) a , 1 − ν˜ 3  holds uniformly in n. We first show the uniform convergence of P(n) (X) in n. The remainder of the section is used to prove Theorem 3.7. Let ❊∞ denote the expectation with respect to the Gaussian field ξ on ❩d with  covariance matrix W∞ . Partition the lattice ❩d into cubes C+ and define P and Ps to be the set of polymers and scaled polymers in ❩d , respectively. For X ∈ P define K(X, ξ ) = lim K (n) (X, ξ ). n→∞  The limit is well-defined since by the consistency assumption, Definition 3.2 part (i), there exists some n0 such that X ∈ P (n) for all n ≥ n0 . Corollary 3.8. Fix any non-empty polymer X ∈ P. If the scaled polymer activities are semi-local with constants a1 , a2 and there exists a3 > 1 such that 2  ν := 4ea1 ea2 (λ∞ +1)|C  +|  3d a3+ + ∑ λs aL3 s Nˆ s < 1, s≥0  then P(n) (X) converges as n → ∞ to 1 1 ∑ ∑ ˆ m! N(X) m≥1 (A1 ,...,Am )∈A m (X)  m  ∏ ❊∞ [❑(A j )]  Jc (A1 , . . . , Am ).  j=1  Furthermore, the pressure Pr = limn→∞ Pr(n) exists. Proof of Corollary 3.8. Since translations act on the torus we can without loss of generality assume that X is centred at the origin. Let Bn be the ball of radius  38  L(n) 2  centered at the origin. Split the sum  P(n) (X) up in two parts P(n) (X) =  1 1 ∑ ˆ N(X) m≥1 m!  m  ∏ ❊(n) [❑(n) (A j )]  ∑  (A1 ,...,Am )∈Am (X)−Am (X,Bcn )  Jc (A1 , . . . , Am )  j=1  (3.20)  m  1 1 + ∑ ∑ ˆ N(X) m≥1 m! (A1 ,...,Am )∈Am (X,Bc )  ∏ ❊ [❑ (n)  (n)  (A j )] Jc (A1 , . . . , Am ).  j=1  n  The first sum runs over sets of scaled polymers which are contained in Bn . The second sum runs over sets of scaled polymers which venture into Bcn . The second sum can be bounded using part (ii) of Theorem 3.7. 1 1 ∑ ∑ ˆ N(X) m≥1 m! (A1 ,...,Am )∈Am (X,Bc ) n  m  ∏ ❊(n) [|❑(n) (A j )|]  |Jc (A1 , . . . , Am )|  j=1  −dist(X,Bcn )  ≤ λ∞ a3 The distance dist(X, Bcn ) is at least contribution of sets in  Am (X, Bcn )  L(n) 2  νm λ∞ −dist(X,Bcn ) a ≤ (1 − ν)2 3 m≥1 1 − ν  ∑  − diam(X) where diam(X) is the diameter of X. Hence, the  tends to zero as n → ∞.  Consider the first term on the right-hand side of (3.20). By part (i) of Theorem 3.7 the summands are O(ν m ) which holds uniformly in n. By the Dominated Convergence Theorem the limit and sums can be interchanged. It suffices to show lim ❊(n) [❑(n) (A j )] = ❊∞ [❑(A j )].  n→∞  (3.21)  Fix (A1 , . . . , Am ) which has a maximal scale smax . The functions ❑(n) (A j ) only depend on the (n)  Gaussian variables {ζs (x), s ≤ smax , x ∈  sp(Ai )}. Gaussian measures are completely determined  by their covariance matrix. In this case it suffices to consider the covariance matrix W˜ (n) (x, y) =  (n)  λsWs (x, y),  ∑  s≤smax  with x, y ∈  (n)  sp(Ai ). Recall the definition of Ws (x, y) as the periodic extension of Ws . Let o be the  origin and note that (n)  lim Ws (x, y) = Ws (x, y) + lim  n→∞  n→∞  (n)  ∑ Ws ❩  (x, y + zL(n) ) = Ws (x, y).  z∈  z=o  In the last step the finite range property of ζs was used. In the limit n → ∞ the covariance matrix W˜ (n) converges to the covariance matrix of {ζs (x), s ≤ smax , x ∈ sp(Ai )}. The very definition of K(X, ξ ) shows that  ❑(n) (A j ) = ❑(A j ) for n finite but sufficiently large. Formula (3.21) has been  proved.  39  Finally, consider the statement about the pressure. The latter statement can proved by translation invariance and a simple identity. The equation 1 ✶{x∈sp(A = 1, ˆ i )} |sp(A1 )| ∑ xˆ holds for any given set of scaled polymers A1 . Definition 3.2 part (ii) states that the polymer activities are translation invariant. Hence, the finite volume pressure satisfies Pr(n) = =  1  ✶{sp(A1 )  ∑∑ ∑  ˆ ❚(n) ) m≥1 N(  xˆ (A1 ,...,Am )  1  Cxˇ }  |sp(A1 )|  ✶{sp(A1 )  m  ∏ ❊(n) [❑(n) (A j )]  |Jc (A1 , . . . , Am )|  j=1  Co }  ∑ ∑ ∑ |sp(A1 )| ˆ ❚(n) ) m≥1 N( xˆ (A1 ,...,Am )  m  ∏ ❊(n) [❑(n) (A j )]  |Jc (A1 , . . . , Am )|,  j=1  ˆ ❚(n) ) which thus cancels out. It suffices to where o is the origin. The sum over xˇ gives a factor N( show convergence of the power series  ∑ ∑  ✶{sp(A1 )  m  ∏ ❊(n) [|❑(n) (A j )|]  Co }  |Jc (A1 , . . . , Am )|  j=1  m≥1 (A1 ,...,Am )  m  =  ∑  ∑  m≥1 (A1 ,...,Am )∈Am (Co )  ∏ ❊(n) [|❑(n) (A j )|]  |Jc (A1 , . . . , Am )|  j=1  Hence, convergence follows from the previous results.  3.4.1  Proof of Theorem 3.7.  The proofs of parts (i) and (ii) are almost identical. We first prove part (i) and then show where the proof of part (ii) differs. Fix m and define the sum bm (X) =  1 1 ˆ m! N(X)  k  ∑  ∏ ❊(n) [|❑(n) (A j )|] · |Jc (A1 , . . . , Am )|.  (A1 ,...,Am )∈Am (X) j=1  Separate estimates for ❊(n) [|❑(n) (A j )|] and the coefficients |Jc (A1 , . . . , An )| are obtained. Fix A = {U1 , . . . ,Uk } where Ui = (Xi , si ) and consider  ❊(n) [|❑(n) (A)|]. The set Xi has to be a  spatially connected set. Let ✶{Xi sp. conn.} be 1 when Xi is a spatially connected and zero otherwise. Use the conditions of Theorem 3.7. k  k  i=1  i=1  k  ˆ  i) ❊(n) ∏ |δ K (n) (Ui )| ≤ ∏ λsi aN(X ✶{Xi sp. conn.} ❊(n) ∏ exp a2 ∑ (ξs2i−1 (x) + ζs2i (x)) . 1  i=1  x∈Xi  Let I be the Gaussian integral on the left-hand side and continue with bounding the integral.  40  1/2 1/2  Use Cauchy-Schwarz to bound the square of the cumulative field ξsi−1 (x) = ∑ λ j λ j ζ j (x). si−1  ξs2i−1 (x) ≤ λ∞ ∑ λ j ζ j2 (x) j=1  The bound on I becomes k  I ≤ ❊(n)  ∏ exp a2 i=1  ζs2i (x) + λ∞  ∑  x∈Xi  si −1  ∑ λs ζs2 (x)  s=0  For x ∈ Xi define sx = si and let smax be the maximum of s1 , . . . , sk . Use the independence of the fields {ζs } followed by Cauchy-Schwarz. smax  I ≤ ∏ ❊(n) exp a2 s=0 smax  ∑ ζs (x)2  λ∞ λs ✶{s<sx } + ✶{s=sx }  x∈X  ≤ ∏ ❊(n) exp 2a2 λ∞ λs s=0  ∑ ζs (x)2 ✶{s<s }  1/2  x  x∈X  ❊(n) exp 2a2 ∑ ζs (x)2 ✶{s=sx }  1/2  x∈X  Use Lemma A.5 to bound the expectation. Let κs−1 = 4a2 λ∞ λs , the integrability condition in Definition 3.3 shows κs−1 ≤ 4a2 λ∞2 ≤  1 2  Recall the condition (3.2) on the covariance matrices Ws and apply Lemma A.5 with ν = 1. Compute the exponent 1 1 1 κs−1 1 = ≤ κs−1 = 2a2 λ∞ λs , −1 4 κs − 1 4 1 − κs 2 and conclude that smax  ∏ ❊(n)  exp 2a2 λ∞ λs  ∑ ζs (x)2 ✶{s<s }  1/2  x  x∈X  s=0  2  ≤ exp a2 λ∞ |X| ∑ λs = ea2 λ∞ |X| s≥0  A similar computation with κ −1 = 4a2 shows that smax  ∏ ❊(n)  exp 2a2  ∑ ζs (x)2 ✶{s=s } x  1/2  ≤ exp a2 |X| .  x∈X  s=0  We have proved the first partial necessary to bound ❊[|❑(n) (A)|]. k  k  i=1  i=1  k  ˆ  ˆ j) ❊(n) ∏ |δ K (n) (Ui )| ≤ ∏ λsi a1N(Xi ) ✶{Xi sp. conn.} exp a2 (λ∞2 + 1)|C+ | ∑ N(X j=1  41  (3.22)  The second partial result for the bound on ❊[|❑(n) (A)|] is obtained by using a tree-graph bound  for ✶{A Gs. conn.} . Let D(Ui ,U j ) equal 1 when Ui and U j are dependent and 0 otherwise.  ✶{A Gs. conn.} = sup  D(U,V ) ≤  ∏  ∑  T on A {U,V }∈T  ∏  D(U,V )  (3.23)  T on A {U,V }∈T  The sum runs over trees on A while the product runs over all edges in T . Combining bounds (3.22) and (3.23) into the final bound  ❊(n) [|❑(n) (A)|] ≤ ∑  λs α N(X) ✶{Xi sp. conn.} ˆ  ∏  ∏  D(U,V ) ,  (3.24)  {U,V }∈T  T on A (X,s)∈A  where α = a1 exp(a2 (λ∞2 + 1)|C+ |). The quantity Jc (A1 , . . . , Am ) will be bounded using the Penrose identity [49]. Apply the generalized version of the Penrose identity [23, Proposition 5]. |Jc (A1 , . . . , Am )| =  ∏ (✶{A ,A  ∑  i  G on {1,...,m}  j  indep} − 1)  ≤  {i, j}∈G  ∏ |✶{A ,A  ∑  i  j  indep} − 1|  T on {1,...,m} {i, j}∈T  G connected  where the sum runs over trees T on {1, . . . , m}. The factor |✶{Ai ,A j indep} − 1| is zero unless there exist U ∈ Ai and V ∈ A j which are dependent. |✶{Ai ,A j indep} − 1| ≤  ∑ ∑  D(U,V ).  U∈Ai V ∈A j  The two bounds combined yield |Jc (A1 , . . . , Am )| ≤  ∑  ∏  T on {1,...,m} {i, j}∈T  ∑ ∑  D(U,V ) .  U∈Ai V ∈A j  Expanding the product yields a sum over trees on (A1 , . . . , Am ) where an edge between Ai and A j is indexed by {U,V } with U ∈ Ai while V ∈ A j . Let T (A, m) be the collection of all those trees. The bound on Jc becomes |Jc (A1 , . . . , Am )| ≤  ∑  ∏  D(U,V ).  (3.25)  T ∈T (A,m) {U,V }∈T  Given a scaled polymer U let sU be the scale of U. Substituting the upper-bounds (3.24) and  42  (3.25) into the expression bm (X) yields bm (X) ≤  1 1 1 ∑ ∑ ˆ m! k! N(X) (A1 ,...,Am ) k≥m  k  ✶{  ∑  Ai ={U1 ,...,Uk }}  ∏ λs α N(X ) ✶{X sp. conn.} ˆ  i  i  i  i=1  (U1 ,...,Uk ) sp(Ui )∩X=0/  (3.26)  m  ∏ ∑  D(U,V )  ∏  i=1 Ti on Ai {U,V }∈Ti  ∑  ∏  D(U,V )  T ∈T (A,n) {U,V }∈T  Each set Ai has a tree Ti connecting its building blocks U j while the tree T connects the sets Ai . Colour the edges in the trees Ti blue while the edges in T are coloured red as is done in Figure 3.1. The trees Ti and T can be combined into one coloured tree T on {U1 , . . . ,Uk } with red and blue edges. Fix a set {U1 , . . . ,Uk } and any tree T on this set. Given A1 , . . . , Am we say that U ∼A V when U,V are both contained in the same set Ai while U ∼A if U,V are contained in different sets Ai and A j . The colourings of T in red and blue edges are in bijection with partitions (A1 , . . . , Am ) of (U1 , . . . ,Uk ) by the relation ∼A and ∼A . 1 m! (A  ∑ ✶{  Ai ={U1 ,...,Uk }}  1 ,...,Am )  ∏  D(U,V )  {U,V }∈T U∼AV  ∏  D(U,V ) = 2k−1  ∏  D(U,V )  (3.27)  {U,V }∈T  {U,V }∈T  U∼AV  Each colouring has the same weight. Moreover, the tree T has k vertices and k − 1 edges. The factor 2k−1 counts the number of colourings of T with two colours. In (3.26) interchange the sum over (A1 , . . . , Am ) with the sum over (U1 , . . . ,Uk ) and then use the identity (3.27). There must be at least m scaled polymers. bm (X) ≤  2k 1 ∑ ˆ N(X) k≥m k!  k  ∑  ∏ λs α N(X ) ✶{X sp. conn.} ∑ ˆ  i  i  i  ∏  D(U,V )  T on {U,V }∈T (U1 ,...,Uk )  i=1 (U1 ,...,Uk ) sp(Ui )∩X=0/  Figure 3.1: Blue (dashed) edges form a tree on Ui ∈ A j , red edges form a tree on {A j }  43  Consider scaled polymers U and V . D(U,V ) ≤  ∑  ∑  D((Cx+ˆ , sU ), (Cy+ˆ , sV )).  x∈ ˆ Vˆ (U) y∈ ˆ Vˆ (V )  Use the generic notation ∆i = ∆xˆi = (Cx+ˆi , sxˆi ) for a scaled polymer consisting of a single cube. Define the function fBB fBB ((X, s)) = {(Cx+ˆ , s) : xˆ ∈ Vˆ (X)}, which maps U down to its building blocks. bm (X) ≤  1 (2α) 1 ∑ ∑ ∑ ˆ N(X) k≥m k! (U1 ,...,Uk ) ≥k !  ∑ ✶{{∆1 ,...,∆ }=  k fBB (Ui )}  ∏ λs T (Xi ) ∑ i  ∏  T on {U,V }∈T (U1 ,...,Uk )  i=1  (∆1 ,...,∆ ) ∃ j: xˆ j ∈X  ∑  ∑  D(∆xˆ , ∆yˆ )  x∈ ˆ Vˆ (U) y∈ ˆ Vˆ (V )  In the above inequality the trivial fact 2k ≤ 2 was used as well. Expanding the product yields a sum over trees on {U1 , . . . ,Uk } where an edge between Ui and U j is indexed by {∆xˆ , ∆yˆ } with xˆ ∈ Vˆ (Ui ) while yˆ ∈ Vˆ (U j ). Let T (U, k) be the collection of all those trees. The condition {∆1 , . . . , ∆m } = 1 k!  ∑  ∏  T on {U,V }∈T (U1 ,...,Uk )  fBB (Ui ) indicates that (Cx+ˆ , sU ) = (Cx+ˆ , sxˆ ) and  ∑  ∑  D(∆xˆ , ∆yˆ ) =  ∑  ∏  D(∆xˆ , ∆yˆ )  (3.28)  ˆ y}∈T ˆ T ∈T (U,k) {x,  x∈ ˆ Vˆ (U) y∈ ˆ Vˆ (V )  ✶{Xi sp. conn.} . The condition for a set X to be spatially connected can be interpreted in terms of graphs on Vˆ (X). For x, ˆ yˆ ∈ Vˆ (❚(n) ) let xˆ ∼ yˆ if and only if Finally apply another tree bound for  the cubes Cx+ˆ and Cy+ˆ are contiguous. The set of edges is {{x, ˆ y} ˆ : xˆ ∼ y}. ˆ If Xi is connected, then it must be possible to construct a tree graph on Vˆ (Xi ). The condition for Xi to be connected is bounded by  ✶{Xi sp. conn.} ≤  ∑  ˆ y} ˆ , ∏ ✶{x∼  (3.29)  ˆ y}∈T ˆ T on Vˆ (Xi ) {x,  where the sum runs over all trees on Vˆ (Xi ). Apply the tree-graph identity (3.28) and tree-graph bound (3.29). The sets ∆i are indistinguishable and conditioning on {∃i : xˆi ∈ X} is equivalent to conditioning on {xˆ1 ∈ X}. Also use the fact  44  that because there are at least m scaled polymers, it follows that ≥ m. bm (X) ≤  1 ˆ N(X)  (2α) ! ≥m  ∑  1 ∑ ∑ k! ∑ ✶{{∆ ,...,∆ }= 1  (∆1 ,...,∆ ) k≥m xˆ1 ∈X  fBB (Ui )}  (U1 ,...,Uk )  (3.30)  ˆ y} ˆ ∏ ✶{x∼  ∑  ∑  ˆ y}∈T ˆ Ti on Vˆ (Ui ) {x, i  ∏ λs ∏ xˆ  D(∆xˆ , ∆yˆ )  {x, ˆ y}∈T ˆ  ˆ T ∈T (U,k) x∈T  Each set Ui has a tree Ti connecting its centers xˆi while the tree T connects the sets Ui . Colour the edges in the trees Ti blue while the edges in T are coloured red as is done in Figure 3.2. The trees Ti and T can be combined into one coloured tree T on {1, . . . , } with red and blue edges. Fix a set {1, . . . , } and any tree T on this set. Observe that there is a bijection between the colouring of T and the choice of sets {U1 , . . . ,Uk }. Define the relation ∼U by xˆ ∼U yˆ if and only if xˆ and yˆ are in the same component Ui for some i. Given T let F(T ) be the set of all colourings of T in red and blue edges. Given a colouring f ∈ F(T ) let R(T ) be the set of vertices which are adjacent to a red edge. 1 ∑ k! ∑ ✶{{∆ ,...,∆ }= 1  k≥m  k  fBB (Ui )}  ˆ y} ˆ ∏ λs ∏ ∏ ∏ ✶{x∼ xˆ  ˆ i=1 {x,ˆ y}∈T i x∼ ˆ U yˆ  (U1 ,...,Uk )  =  x∈T ˆ  D(∆xˆ , ∆yˆ )  {x, ˆ y}∈T ˆ  x∼ ˆ U yˆ  ∑ F(T )  ˆ y} ˆ ∏ λsxˆ ∏ D(∆i , ∆ j ) ∏ ✶{x∼ {i, j}∈T  {i, j} blue  x∈R(T ˆ )  (3.31)  {i, j}∈T  {i, j} red  In (3.30) interchange the sum over (U1 , . . . ,Uk ) with the sum over (∆1 , . . . , ∆ ). Use the tree identity (3.31) to get the final bound. bm (X) ≤  1 ˆ N(X)  (2α) ! ≥m  ∑  ∑  ˆ y} ˆ ∏ ∑ ∑ ∏ ✶{x∼  T on {1,..., } (∆1 ,...,∆ ) F(T ) {i, j}∈T {i, j} blue xˆ1 ∈Xˆ  x∈R(T ˆ )  λsxˆ  ∏  D(∆i , ∆ j )  {i, j}∈T  {i, j} red  Figure 3.2: Blue (dashed) edges form a tree on xˆi , red edges form a tree on {U j }.  45  (3.32)  Continue by trimming down the coloured trees of (3.32). For ∆ = (Cx+ˆ , s) and  Pruning the tree.  ∆ = (Cx+ˆ , s ) define the function g(∆, ∆ ) = ✶{x∼ ˆ xˆ } + λs D(∆xˆ , ∆xˆ )  (3.33)  and fix a tree T on {1, . . . , }. By pruning the tree T leaf by leaf  Consider the bound (3.32), fix one obtains the bound 1 ∑ ˆ N(X) (∆ ,...,∆ ) 1  ∑ ∏ ✶{xˆ ∼xˆ } ∏ i  F(T )  xˆ1 ∈Xˆ  ∏  λsxˆ  j  x∈R(T ˆ )  {i, j}∈T  {i, j} blue  D(∆i , ∆ j ) ≤ λ∞ sup  ∑  −1  g(∆, ∆ )  . (3.34)  ∆ ∆=(x,s) ˆ  {i, j}∈T  {i, j} red  Proof of (3.34). The proof goes by induction. For = 1 the identity is trivial. There is one cube which has to be placed in the set X and comes with some factor λs for s ≥ 0.  ∑  λs ✶{x∈X} = ˆ  (Cx+ ˆ ,s)  ˆ ∑ ∑ λs = N(X) ∑ λs .  x∈ ˆ Vˆ (X) s≥0  s≥0  Take > 1 and without loss of generality assume the vertex with label is a leaf with the single edge e = {k, }. Let T = T − e be the trimmed tree. The edge e could either be coloured red or blue. By summing over these two possibilities one is left with a unique colouring on T .  ∑ ∑ ∏  λsxˆ  (∆1 ,...,∆ ) F(T ) x∈R(T ˆ ) xˆ1 ∈X  =  ∑ ∑  i  {i, j}∈T  {i, j} red  {i, j} blue  ∑ ∏  (∆1 ,...,∆ −1 ) F(T xˆ1 ∈X  ≤  ∏ ✶{xˆ ∼xˆ }  D(∆i , ∆ j )  ∏ {i, j}∈T  ∑ ∏  (∆1 ,...,∆m−1 ) F(T xˆ1 ∈X  ∏ ✶{xˆ ∼xˆ } ∑  D(∆i , ∆ j )  ∏  λsxˆ  ˆ ) ) x∈R(T  j  i  {i, j}∈T  {i, j}∈T  {i, j} red  {i, j} blue  ∏  λsxˆ  ˆ ) ) x∈R(T  j  ∏ ✶{xˆ ∼xˆ }  D(∆i , ∆ j )  g(∆, ∆k )  ∆=(x,s) ˆ  i  {i, j}∈T  {i, j}∈T  {i, j} red  {i, j} blue  j  sup  ∑  g(∆, ∆ )  ∆ ∆=(C+ ,s) xˆ  This completes the proof for the inductive step. Substitute the upper-bound (3.34) into (3.32) and also use Cayley’s theorem which states that there are  −2  tree on vertices. −2  bm (X) ≤ λ∞  ∑  ≥m  !  Stirling’s formula implies the upper-bound bm (X) ≤ λ∞  (2α) · sup  ∑  g(∆, ∆ )  −1  ∆ ∆=(C+ ,s) xˆ −2  !  ∑ (2eα)  ≤e . · sup  ∑  ∆ ∆=(C+ ,s) xˆ  ≥m  46  g(∆, ∆ )  −1  Recall the definition (3.33) of g(∆, ∆ ) and fix a scaled cube ∆ = (Cx+ˆ , s ).  ∑  ˆ xˆ } + ∑ λs ∑ ✶{dist (C ∑ ✶{s=s } ✶{x∼  g(∆, ∆ ) =  ∆=(Cx+ ˆ ,s)  + + xˆ ,Cxˆ )≤Ls ∧Ls  n  xˆ  s≥0  (x,s) ˆ  }  ≤ 3d + ∑ λs ∑ ✶{distn (C+ ,C+ )≤Ls } xˆ  xˆ  s≥0  xˆ  ≤ 3d + ∑ λs Nˆ s s≥0  The bound is uniform in ∆ and also holds for the supremum over ∆ . The bound on bm (X) becomes bm (X) ≤ λ∞  3d + ∑ λs Nˆ s  ∑ (4eα) ≥m  Proof of part (ii).  −1  s≥0  Let bm (X,Y ) be equivalents to bm (X) when Am (X) is replaced by Am (X,Y ).  Repeat the exact same steps as before until reaching equation (3.32) which changes into bm (X,Y ) ≤  1 ˆ N(X)  (2α) ! ≥m  ∑  ∑  T on {1,..., }  ˆ y} ˆ ∏ ∑ ∏ ✶{x∼  ∑ (∆1 ,...,∆ )  F(T )  ˆ xˆ2 ∈Yˆ xˆ1 ∈X,  λsxˆ  x∈R(T ˆ )  {i, j}∈T  {i, j} blue  D(∆i , ∆ j )  ∏ {i, j}∈T  {i, j} red  Multiply and divide by factors a3 , the right-hand side becomes 1 ˆ N(X)  (2α) ! ≥m  ∑  ∑  T on {1,..., }  ∑ (∆1 ,...,∆ )  ˆ xˆ2 ∈Yˆ xˆ1 ∈X,  ˆ y} ˆ a3 ∑ ∏ ✶{x∼  +  F(T )  a3−  +  ∏  −Lsxˆ  Ls  λsxˆ a3 xˆ a3  x∈R(T ˆ )  {i, j}∈T  {i, j} blue  ∏  D(∆i , ∆ j )  {i, j}∈T  {i, j} red  The tree must have a path p connecting xˆ1 with xˆ2 . The distance dist(xˆ1 , xˆ2 ) is at least dist(X,Y ). Each blue edge corresponds to a jump of size − ˆ y} ˆ a3 ∏ ✶{x∼ {i, j}∈p  {i, j} blue  +  +  , a red edge corresponds to a jump of size Lsxˆ .  ∏  −Lsxˆ  a3  −dist(X,Y )  ≤ a3  .  x∈R(T ˆ )  After taking out the power of a3 simply drop the constraint that the polymers have to intersect with Y. −dist(X,Y )  bm (X,Y ) ≤  a3 ˆ N(X) (2α) ∑ ! ≥m  ∑  ˆ y} ˆ a3 ∑ ∑ ∏ ✶{x∼ ∏ +  T on {1,..., } (∆1 ,...,∆e ll F(T ) {i, j}∈T {i, j} blue xˆ1 ∈Xˆ  x∈R(T ˆ )  Ls  λsxˆ a3 xˆ  ∏  D(∆i , ∆ j )  {i, j}∈T  {i, j} red  Repeat the exact same steps as was done for bm (X) but now carrying the extra powers of a3 .  47  Notes In these notes we provide some background information on the history of the cluster expansion technique. Moreover, we discuss examples of Gaussian fields which admit a finite range decomposition. Cluster expansion. The cluster expansion is one of the oldest tools in statistical mechanics. It was originally developed to express thermodynamic potentials as power series in the activities. The cluster expansion was introduced by Ursell [58], Yvon and Mayer and collaborators [45–47]. The theory of cluster expansion techniques has roughly split into three branches. We summarize each of the branches. In this chapter we studied polymer models on the torus ❚(n) . However, the standard setting in  the literature is to consider polymer models Z(Λ) on Λ ⊂ ❩d where Λ is a finite subset of  ❩d . In  other words, the faces of Λ are not identified with one another. The main aim is to find convergence conditions for the formal power series of log Z(Λ) to converge as Λ  ❩d .  In the first and main branch of cluster expansion techniques, the grand partition function is defined in terms of polymers, polymer activities and a relationship ι between polymers. Let Λ be a finite set and let the set of polymers P(Λ) be the set of subsets of Λ. The polymer activity is assigned by a function K : P(Λ) → ❘. Let ι ⊂ P × P be a reflexive symmetric relationship. Two polymers X and Y are incompatible if (X,Y ) ∈ ι, otherwise they are compatible. The grand partition function is defined by Z(Λ) =  1 m≥0 m! X  ∑  ∑  K(X1 ) · · · K(Xm )  (3.35)  1 ,...,Xm ⊂Pn (Λ)  (Xi ,X j )∈ι∀i= j  Gruber and Kunz [30] were the first to study convergence conditions for log Z(Λ). They considered the special case (X,Y ) ∈ ι if X ∩ Y = 0. / An important step forward was the work of Koteck´y and Preiss [39]. They introduced the simplified setting with the relationship ι and provided an elegant condition for convergence. Fern´andez and Procacci [23] improved the convergence criteria of Koteck´y and Preiss by using a classical combinatorial approach with tree-graph bounds. The second branch of cluster expansion techniques, of which a particular case is studied in this chapter, concerns cluster expansions for Gaussian fields. Here the partition function takes the form Z(Λ) =  1 m≥0 m! X  ∑  ∑  dµΛ (ξ )K(X1 , ξ ) · · · K(Xm , ξ ),  (3.36)  1 ,...,Xm ⊂Pn (Λ) Xi ∩X j =0∀i= / j  where dµΛ is a normalized Gaussian measure on variables ξ = (ξx , x ∈ Λ). The earliest work was initiated by Glimm, Jaffe and Spencer [25, 26]. Since then it has been simplified and improved by many authors (see the most current publication [1] and references therein). The method used in  48  these papers is to first evaluate the Gaussian integral using tree-graph formulas and then to apply the cluster expansion. Guadagni’s method [31] takes a different approach. Using the finite range decomposition of the Gaussian measure it is possible to apply the cluster expansion without evaluating the Gaussian integrals. The third branch studies a more general setting than the polymer system (3.35). For example, it includes the case where the space is not discrete. Ueltschi [57] and Faris [20] study cluster expansion techniques which unify the theory for continuous and discrete systems. Finite range decompositions.  We conclude the chapter by discussing Gaussian fields with finite  range decompositions. The existence of finite range decompositions for a wide range of Gaussian measures is discussed in [2, 3, 11, 12, 15]. In this thesis we are interested in the following finite range decomposition. Let D be a diagonal matrix and A a matrix which is symmetric, positive definite and has range L. The most important example for this work is the finite range decomposition for the Gaussian field with covariance matrix W = (D − A)−1 . If Dxx , ∑ |Axy | ≤ λ min x x  for all x, then the inverse W = (D − A)−1 admits a particularly simple finite range decomposition. Details are provided in Section A.1.  49  Chapter 4  Perturbations of a Gaussian measure In this chapter we study finite volume measures νn which are absolutely continuous with respect to the standard Gaussian field. The standard Gaussian field is the collection of random variables {ξx } where ξx are i.i.d. standard Gaussian variables. We list four conditions on the Radon-Nikodym derivative of νn . It will be shown that if these conditions are satisfied, then the measures νn converge to a Gibbs state ν∞ as n → ∞. The measure ν∞ satisfies ν∞ = 12 ν + + 21 ν − where ν ± are ergodic, translation invariant measures. It is proved that the field averages in the measures ν ± concentrate around ±ξˆ . Hence, a coexistence of phases is achieved. The conditions on the Radon-Nikodym derivative ensure that the measures νn can be expressed as a small perturbation of a more complicated Gaussian measure. The main tool to study the small perturbations of the Gaussian measure is the cluster expansion introduced in Chapter 3. When the terminology and results of the cluster expansion are needed the reader is referred back to the appropriate section in Chapter 3. As a final remark we place this chapter in the context of the Ising and WR model of Chapter 2. The Kac-Siegert transformation was used to rewrite the Gibbs measures of these models as measures which are absolutely continuous with respect to the standard Gaussian field. In Chapter 5 it is shown that the Ising and WR measures satisfy the four conditions listed in this chapter. Then it will be shown that a coexistence of phases in the Gaussian model implies a coexistence of phases in the particle models.  4.1  Setup and conditions  We define the finite volume measures νn which are absolutely continuous with respect to the standard Gaussian field. The section starts with some geometric definitions which are necessary to introduce the Radon-Nikodym derivative. The section finishes with listing the conditions for the Radon-Nikodym derivative.  50  Geometric definitions. Given a positive number x define the numbers x  3  = max{3k : x ≥ 3k },  x  k∈◆  3  = min{3k : x ≤ 3k }, k∈◆  which are the largest lower-bound and smallest upper-bound of x in terms of powers of 3. There are three length-scales  −  <  1  <  constant. The length-scale  +  1  . The length-scales are  −  =  1−ε 3 1  and  +  1+ε 3 1  =  is γ −1 where γ is the Kac parameter. Hence,  1  for ε a small  should be thought  of as large but finite. The parameter ε is determined by forthcoming estimates and will have to be small. All the constants in this chapter, including ε, will be independent of γ. The torus ❚(n) for n ∈ ◆ is the graph with vertices [−L(n) , L(n) ]d ∩ ❩d where opposite faces are  ❚(n) are connected by an edge if the supremum norm x − y ∞ = 1. The diameter of the torus is L(n) = 2n + . The notation ❚(n) may refer to either the graph itself or the vertex set of ❚(n) . Based on the context it will be clear which identified. The length L(n) is equal to n + . Vertices x, y in  interpretation to use.  ❚(n) into cubes of side-length − such that one cube is centered at the origin. Let Cn+ be the coarser partition of ❚(n) into cubes with side-length + . Unless explicitly ˆ mentioned, all spatial sets are unions of cubes in Cn+ . For a subset X ⊂ ❚(n) define N(X) to be the Let Cn− be the partition of  number of C+ cubes contained in X. Given x ∈ ❚(n) let Cx+ be the unique cube in Cn+ such that x ∈ Cx+ . For a given set X let Vˆ (X) ⊂  ❚(n) be the set of all cube-centers of cubes C+ which are contained in X. Similarly, define Cx and Vˇ (X) at the smaller scale  . The notation Cx is a short form for what might naturally be called Cx− .  −  By convention the notation xˆ and xˇ is used to denote the cube-centers at scale Distances dist(x, y) for x, y ∈ ❚  (n)  +  and  are measured in the graph distance norm on ❚  , respectively.  −  (n) , i.e., dist(x, y)  is the number of edges in a shortest path connecting x and y. Any set X ⊂ ❚(n) has an interior and exterior boundary. For a given length define ∂ int (X) := {x ∈ X : dist(x, X c ) ≤ }, ∂ ext (X) := {x ∈ X c : dist(x, X) ≤ }. For any given set X let ∂ (X) = ∂ ext (X) 1 /2 Given a region X ⊂ ❚(n) let ·, ·  X  and  X = X ∪ ∂ (X).  denote the standard inner-product. v, w  X  =  ∑ vx wx  x∈X  51  For the cubes Cx+ˆ and Cxˇ a special notation is used. v, w xˆ = v, w  v, w xˇ = v, w  Cx+ ˆ  Cxˇ  We consider two different averages. The so-called site average and the  Two different averages. block average. For x ∈ ❚  (n)  let Cx(1) be the cube of side-length  centered at x  1  Cx(1) = {y ∈ ❚(n) : dist(x, y) ≤  1  2  }.  Let o be the origin, then w(x, ˚ y) = ✶{y∈C(1) } ,  w˚ =  x  ∑  w(o, ˚ y).  z∈❚(n)  Define the averaging operator ˚ x [ξ ] = 1 ∑ ξy w(x, Av ˚ y), w˚ y∈❚(n) ˚ x [ξ ]. This is the site average. Let Av ˚ be the matrix corresponding to the linear and let ψ˚ x = Av transformation (ξx , x ∈ ❚(n) ) → (ψ˚ x , x ∈ ❚(n) ). For a given connected set X ⊂ ❚(n) , not necessarily a union of C+ cubes, and x ∈ X let α˚ x (X) =  1 w˚  ∑  α˚ min (X) = min α˚ x (X).  w(x, ˚ y) and  (4.1)  x∈X  (1) y∈Cx ∩X  If the set X is a connected set of C+ cubes, then α˚ min (X) ≥ 2−d . The lower-bound on α˚ min (X) only depends on the dimension d. Block averages are defined in terms of weights w. Given x ∈ ❚(n) let v(x) ˇ denote the center of •  the cube in Cn− which contains x. Let o be the origin, then w(x, y) = ✶{y∈C(1) } , •  •  w=  ∑  •  (4.2)  w(o, y).  y∈❚(n)  v(x) ˇ  Define the averaging operator •  Avx [ξ ] = •  •  1 ξy w(x, y), w y∈∑ (n) ❚ •  •  •  and let ψ x = Avx [ξ ]. The averages ψx are constant on cubes at scale  •  •  , ψx = ψxˇ for all x ∈ Cxˇ .  −  •  Hence, these averages are called the block averages. Let Av be the matrix corresponding to the linear transformation (ξx , x ∈ ❚(n) ) → (ψx , x ∈ ❚(n) ). •  52  For a given connected set X ⊂ ❚(n) , not necessarily a union of C+ cubes, and x ∈ X let •  αx (X) =  1 w •  ∑  •  w(x, y) and  •  •  αmin (X) = min αx (X).  (1)  x∈X  (4.3)  y∈Cx ∩X  If the set X is a union of C+ cubes, then αmin (X) ≥ 2−d . R EMARK . All the statements in this chapter hold for both site and block averages. Only while deriving estimates in the appendix Section A.3.1 do we need to make the distinction between site and block averages. Therefore, we adopt the generic notation which simultaneously refers to both ♦  averages: ψ, αmin , etc.  Finite volume measure. Let ξ = {ξx , x ∈ ❚(n) } be i.i.d. standard Gaussian random variables. Let µn (dξ ) be the corresponding Gaussian measure. The cube Co is centered at the origin. Label the lattice sites in Co by x1 , . . . , xM such that x1 is the origin. The input for the measure νn is a function Z (C) : ❘M → ❘ which we call the cube partition function. The function Z (C) will have to obey some conditions which are not listed until Section 4.2. The probability measure νn studied in this chapter is defined by νn (dξ ) =  1 ∏ Z (C) (ψx1 +xˇ, . . . , ψxM +xˇ)µn (dξ ). ˜ Z (n) x∈ ˇ Vˇ (❚(n) )  (4.4)  The grand partition function Z˜ (n) is the normalization constant. Z (n) =  µn (dξ )  ∏  x∈ ˇ Vˇ (❚(n) )  Z (C) (ψx1 +xˇ , . . . , ψxM +xˇ )  (4.5)  •  R EMARK . If the block averages ψ are used then the cube partition function is only a function of •  ψxˇ . In this case the cube partition function Z (C) (ψx1 +xˇ , . . . , ψxM +xˇ ) is Z (C) (ψxˇ ).  Alternative definition.  ♦  The reformulation of the measures νn is needed for the analysis in this  chapter. Given ξ0 ∈ ❘ define the constant S(ξ0 ) = |C| and the function S S(x, ˇ ξ) =  ξ02 − log Z (C) (ξ0 ), 2  Avx [ξ 2 ] − log Z (C) (x, ˇ ξ ) − S(ξ0 ). 2 x∈Cxˇ  ∑  53  (4.6)  (4.7)  (a) Double-well function  (b) Parabolic lower-bounds  Figure 4.1: A double-well function with parabolic lower-bounds The function is normalized such that S = 0 if the field ξx = ξ0 for all x. For a region X ⊂ ❚(n) define S(X; ξ ) =  ∑  S(x; ˇ ξ ).  (4.8)  x∈ ˇ Vˇ (X)  Define the measure νn via dνn (ξ ) =  (n) 1 ❚(n) d (ξ )e−S(❚ ;ξ ) , (n) Z  (4.9)  where Z (n) =  d ❚ (ξ )e−S(❚ (n)  (n) ;ξ )  (4.10)  Two changes are made in the definition of the measures νn . The first change is that both numerator and denominator are multiplied by a factor N  ˆ ❚(n) ) (n) · e−S(ξ0 )N(  where N  (n)  is the normalization  constant of the measure µn . The factors cancel out. The second change is that ∑ ξx2 is replaced by ∑ Avx [ξ 2 ]. On the torus these sums are identical.  4.2  Phase coexistence  In this section we list four conditions for a cube partition function Z (C) to be well-adapted to a double-well function Φ. See Figure 4.1(a) for an example of a double-well function. The main goal is to show that when Z (C) is well-adapted function, then the field averages ψx are going to settle near one of the minimizers ±ξˆ of the function Φ. Furthermore, it must be shown that this in turn implies the coexistence of two phases. We give a heuristic argument as to why a phase coexistence is expected when field averages are forced to settle near either ±ξˆ . Let o be the origin and suppose ψo chooses the well +ξˆ . Field averages {ψx , x = o} can still choose to settle near either well ±ξˆ . However, a choice for the well −ξˆ causes a surplus of energy in the system because the field is forced to travel from one well to the other. Therefore, a vast majority of the spins is expected to choose the well +ξˆ . By symmetry the same picture holds when ψo chooses the well −ξˆ . A coexistence of two phases is expected. In  54  one phase a vast majority of the field averages settle near −ξˆ . The second phase has a vast majority of the field averages settling near +ξˆ .  4.2.1  Conditions on the Radon-Nikodym derivative  The first requirement for Z (C) to be well-adapted is that Z (C) is a symmetric function. Besides symmetry there are two main conditions. One condition is the close relationship of Z (C) with a double-well function Φ with two global minima located at ±ξˆ . The relationship ensures that the Gaussian averages ψx remain close to the minimizers ±ξˆ . Consider the Hessian HC of log Z (C) near ξˆ . Let DC be the block matrix with the block HC repeated along the diagonal. The Gaussian measure µn has the identity matrix I as its covariance matrix. Construct the matrix (I − AvDC Av)−1 . The other main condition ensures that the latter matrix is positive definite such that the Gaussian measure with covariance matrix (I − AvDC Av)−1 is well-defined. Definition 4.1. A function Φ : ❘ → ❘ is called a symmetric double-well function with minimizers ±ξˆ if (i) Φ(ζ ) = Φ(−ζ ), (ii) Φ has two global minimizers ±ξˆ , (iii) Φ admits parabolic lower-bounds underneath each of its wells, i.e., there exists κ > 0 such that for ζ > −ξˆ /2 κ Φ(ζ ) − Φ(ξˆ ) ≥ (ζ − ξˆ )2 , 2 and for ζ < ξˆ /2 κ Φ(ζ ) − Φ(ξˆ ) ≥ (ζ + ξˆ )2 . 2 The requirements for a Φ to be a symmetric double-well function are depicted in Figure 4.1(b). For a constant ξ0 ∈ ❘ define Z (C) (ξ0 ) = Z (C) (ξ0 , . . . , ξ0 ). Having defined a symmetric doublewell function we state the conditions Z (C) has to satisfy to be well-adapted. Throughout the thesis the conditions listed in Definition 4.2 are referred to as Condition 0, Condition 1, etc. Definition 4.2. The partition function Z (C) is called well-adapted to a symmetric double-well function Φ with minimizers ±ξˆ if 0. (Symmetry) The cube partition function satisfies Z (C) (ζ1 , . . . , ζM ) = Z (C) (−ζ1 , . . . , −ζM )  55  1. (Taylor) There are positive constants a0 , ξ0 ∈ ❘ with |ξ0 − ξˆ | ≤ a0 (log |C|)2 |C|−1 and a positive definite symmetric M × M matrix HC such that M  log Z (C) (ζ ) = log Z (C) (ξ0 ) + ξ0 ∑ (ζi − ξ0 ) + i=1  1 (ζ − ξ0 ), HC (ζ − ξ0 ) + R(ζ ), 2  for ζ ∈ ❘M . Furthermore, there exists a constant a1 such that remainder satisfies |R(ζ )| ≤ a1 (log |C|)2 |C| max |ζi − ξ0 |3 . 1≤i≤M  2. (Finite range) Define the matrix A via the relation ζ , Aζ  Co  = Av[ζ ], HC Av[ζ ]  Co ,  where o is the origin. There exists λ < 1 such that the matrix A satisfies  ∑ |Axy | ≤ λ αx (Co ),  (4.11)  y∈C0  for all x ∈ C0 . 3. (Double-well) Suppose |ξ0 − ξˆ | ≤ a0 (log |C|)2 |C|−1 for some constant a0 . The cube partition function must satisfy 1  M  2  e− 2 ∑i=1 ζi Z (C) (ζ ) e  − 12 |C|ξ02  · Z (C) (ξ0 )  M  ≤ a1 exp − ∑ Φ(ζi ) − Φ(ξˆ ) , i=1  for some constant a1 . R EMARKS . • In Condition 1 one would expect the Taylor expansion of log Z (C) around ξˆ . However, ξˆ is the minimizer of the mean field theory which arises in the limit γ → 0. Taking γ positive we have to work with ξ0 which is a small perturbation of the minimizer ξˆ . • Condition 2 combined with Proposition A.2 guarantees that (I − A)−1 is a positive definite matrix. • We give an heuristic sketch why the field averages are expected to concentrate near ±ξˆ . The length scale  −  was chosen such that  −  is large for small γ but still satisfies  −  γ −1 . When γ  is very small the average field is essentially constant in a cube C or ψx1 ≈ . . . ≈ ψxM . Condition  56  3 shows that a small deviation δ from ±ξˆ is heavily punished M  2  exp − ∑ (Φ(ψxi ) − Φ(ξˆ )) ≈ exp − |C|(Φ(ψx1 ) − Φ(ξˆ )) ≈ e− 2 |C|δ . κ  i=1  In the final approximation we applied the parabolic lower-bound of the double-well func♦  tion.  4.2.2  Main result  The main result is the existence of a phase transition for the field ξ described by the finite volume measures νn . The terminology used in the statement of the theorem was introduced in Section 2.2. The constants ξˆ and λ appear in Definition 4.2 for a partition function to be well-adapted. Theorem 4.3 (Phase coexistence). If the cube partition function Z (C) is well-adapted, then there exists γ0 = γ0 (ξˆ , λ , d) such that for all 0 < γ < γ0 (i) the measures νn converge locally to ν∞ = 21 ν + + 21 ν − . (ii) The measures ν ± are ergodic, translation invariant with an exponential decay of correlations. (iii) Let δ =  d 12 ,  then for any cube C 1 δ ν ± ({ξ : sup |ψx ∓ ξˆ | < 2γ d/2−δ }) > 1 − e− 2 1 .  x∈C  4.3  Contours and proof strategy  As was mentioned before two phases are expected to coexist. Either the vast majority of the Gaussian field averages settle near the well +ξˆ or they choose to settle near −ξˆ . We refer to these two phases as the plus and minus phase. Consider the plus phase and partition space into cubes. In the plus phase one expects an “ocean” of cubes where the field averages are all close to +ξˆ . Inside this ocean of pluses there are rare “islands” of cubes where the field settles near −ξˆ . A similar ocean-island picture with an ocean of cubes where the field averages settle near −ξˆ describes the minus phase. Contours are the boundaries of the islands i.e., the regions where the field transfers from one well to the other. The contours are the regions in the system where there is an excess of energy. Therefore, they should be rare objects. We adopt the following strategy to prove a phase transition. First, show that the probability of a contour decays exponentially with the size of the contour. In the literature this is called a Peierls estimate. If contours are exponentially unlikely, then there must be a large region (which would correspond to the ocean) where the field can choose to either settle 57  near ±ξˆ . By symmetry each of these scenarios has equal probability 21 . Heuristically the measures ν ± are defined by forcing the field averages in the ocean to settle near the well ±ξˆ . Strategy. In this section we formally define contours in the finite volume ❚(n) as the regions where there are large fluctuations from either minimizer ±ξˆ . Contours are classified into two types: the restricted and unrestricted contours. For restricted contours there will be a unique ocean where the averages settle either near +ξˆ or −ξˆ . The unrestricted contours might not have such an ocean. The immediate goal is to prove the Peierls estimate which will hold for any contour, restricted or unrestricted. The Peierls estimate can be applied to show that unrestricted contours vanish in the limit n → ∞. With the unrestricted contours eliminated it is possible to study the local convergence of the measure νn . For restricted contours two measures νn± can be defined by forcing the field averages to settle near ±ξˆ in the aforementioned ocean. The measures ν ± are the local limits of the finite volume measures νn± . Outline. We sketch the outline for the remainder of this chapter. Section 4.4 shows that the grand partition function Z (n) can be re-expressed as a Gaussian integral with respect to a different Gaussian measure than the original standard Gaussian field. In the terminology of Chapter 3 the new expression for Z (n) is a Gaussian polymer model. Section 4.5 uncovers the benefits of the new expression for Z (n) . It is shown that by choosing the new Gaussian measure properly the integrand is a small perturbation. Section 4.6 and Section 4.7 make use of the small perturbations to prove the Peierls estimate and local convergence, respectively.  4.3.1  Definition and Peierls estimate  For xˇ ∈ Vˇ (❚(n) ) define the indicator functions χ +1 (x, ˇ ξ ) = ✶ |ψx − ξ0 | ≤ χ −1 (x, ˇ ξ ) = ✶ |ψx + ξ0 | ≤  −d/2+δ 1 −d/2+δ 1  for all x ∈ Cxˇ , for all x ∈ Cxˇ ,  (4.12)  χ 0 (x, ˇ ξ ) = 1 − χ −1 (x, ˇ ξ ) − χ +1 (x, ˇ ξ ), with δ = d/12. Three different cases are considered, either the field averages are “close” to ±ξ0 or the averages are “not close” to either ±ξ0 . R EMARK . Condition 1 shows that the difference |ξˆ − ξ0 | is negligible in comparison with the cut−d/2+δ off . The choice to center the fluctuations at ξ0 instead of ξˆ makes for slightly neater 1  ♦  computations.  58  Define the set of spin configurations η by Σ(n) = {−1, 0, +1}V (❚ ˇ  (n) )  .  Spin configurations allow to formally define contours and contour ensembles. Definition 4.4 (Contours). (i) Given η ∈ Σ(n) a cube C+ ∈ C + is said to be +-correct if ηxˇ = +1 for all xˇ ∈ Vˇ (C+ ∪ ∂2ext+ (C+ )). Similarly, a cube can be −-correct. An incorrect cube is a cube which is neither ±-correct. (ii) Given η ∈ Σ(n) the contour ensemble ω consists of the union of all incorrect cubes. A contour Γ is a connected component of ω. Let sp(ω) be the union of the contours Γ contained in ω. (iii) The set of all contour ensembles is Ω(n) = {ω(η) : η ∈ Σ(n) } while the set of contours is defined by Ψ(n) = {Γ : Γ ∈ ω for some ω ∈ Ω(n) }. (iv) For a contour ensemble ω define Σ(n) (ω) = {η ∈ Σ(n) : ω(η) = ω}.  R EMARK . Contour ensembles and contours are purely defined in terms of spatial sets. No infor♦  mation about the spin configuration η is retained.  Peierls estimate. The spin configurations are in correspondence with the indicator functions χ by defining χ η (ξ )X =  ∏  χ ηxˇ (x, ˇ ξ)  χ+(ξ )X =  and  x∈ ˇ Vˇ (X)  ∏  χ+(x, ˇ ξ ),  (4.13)  x∈ ˇ Vˇ (X)  for any region X ⊂ ❚(n) . The probability of coming across a specific contour Γ ∈ Ψ(n) is νn (Γ) =  νn (χ η (ξ )❚ ). (n)  ∑  ∑  ω∈Ω(n)  η∈Σ(n) (ω)  ω Γ  Let Ψ =  Ψ(n) be the set of all possible contours. Recall that the constant λ appeared in Condition  2. Theorem 4.5 (Peierls estimate). Let δ = for  1  d 12  and take Γ ∈ Ψ. There exist ε = ε(ξˆ , λ , d) such that  large enough νn (Γ) ≤ e−  δ ·N(Γ) ˆ 1  The bound holds uniformly for those n with Γ ∈ Ψ(n) .  59  .  4.3.2  Contour classification  Suppose the diameter of a given contour Γ is smaller than  ❚  (n) − Γ  whose volume exceeds  |❚(n) | 2 .  L(n) 10 ,  then there must be a component in  This component is the obvious candidate for the “ocean”  mentioned in the introduction to this section. The contours satisfying this constraint are classified as restricted contours. Let unrestricted contours be contours which are not restricted and might be missing a clear candidate for the “ocean”. Definition 4.6 (Contour classification). (i) A contour Γ ∈ Ψ(n) is called unrestricted if diam(Γ) >  L(n) 10 ,  otherwise a contour is called  restricted. A contour ensemble ω is called restricted if all of its contours Γ ∈ ω are restricted. (n)  (n)  (ii) Let Ωr be the set of all restricted contour ensembles in Ω(n) while Ψr contains all possible restricted contours.  The diameter of an unrestricted contour is proportional to diameter of the torus. The size of the unrestricted contours make them improbable objects. In fact the Peierls estimate can be used to show that, in the limit n → ∞, the unrestricted contours do not contribute to the measure νn . Claim 4.7. For  1  large enough, the probability that there exists an unrestricted contour satisfies νn ({∃ unrestricted contour}) ≤ (2n)d e−n .  The bound on the right-hand side converges to 0 as n tends to ∞. Proof. An unrestricted contour Γ can be rooted at any of the (2n)d points in Vˆ (❚(n) ). Furthermore, Γ needs to consist of at least  −1 L(n) + 10  =  n 10  cubes. Apply the Peierls estimate Theorem 4.5 followed  by the combinatorial result Lemma B.1.  ∑  Γ: unrestricted  νn (Γ) ≤  ∑  x∈ ˆ Vˆ (❚(n) )  ∑  e−  δ N(Γ) ˆ 1  ≤ (2n)d e−n  Γ xˆ n ˆ N(Γ)> 10  The right-hand side tends to zero as n → ∞.  The unrestricted decomposition.  The appearance of unrestricted contours makes for a more com-  plicated analysis. In particular, unrestricted contours cannot be analyzed in a unified manner. Each distinct set of unrestricted contours has to be studied separately. This is achieved by decomposing the grand partition function Z (n) into a restricted partition function combined with a sum over unrestricted partition functions. 60  (n)  Define the restricted partition function Zr  by only summing over spin configurations η with  restricted contours. (n)  Zr =  d ❚ (ξ )e−S(❚ (n)  ∑  ∑  (n) ;ξ )  χ η (ξ )❚(n)  (n) (n) ω∈Ωr η∈Σ (ω)  The definition of unrestricted partition functions calls for more definitions. Definition 4.8 (Unrestricted configurations). (i) Given η ∈ Σ(n) let ωu (η) be the subset of the unrestricted contours contained in ω(η). The set of all non-empty unrestricted contour ensembles is (n)  Ωu = {ωu (η) : η ∈ Σ(n) and ωu (η) = 0}. / (n)  (n)  (ii) Take ωu ∈ Ωu and define Σu (ωu ) to be the set of all spin configurations which have ωu as their set of unrestricted contours (n)  Σu (ωu ) = {η : η ∈ Σ(n) and ωu (η) = ωu } Let η X be the restriction of η to X. The complete set of restrictions for ωu is (n) Σ˜ u (ωu ) = {η sp(ωu ) : η ∈ Σu (ωu )}  (n)  (iii) For ωu ∈ Ωu define the unrestricted partition function by (n)  Zu (ωu ) =  d ❚ (ξ )e−S(❚ (n)  ∑  (n) ;ξ )  χ η (ξ )❚(n) .  (n)  η∈Σu (ωu )  The grand partition function Z (n) has the following unrestricted decomposition (n)  Z (n) = Zr +  (n)  Zu (ωu ).  ∑  (4.14)  (n)  ωu ∈Ωu  (n)  The decomposition arises by conditioning on all the possible ωu ∈ Ωu one could encounter. The (n)  term Zr handles the case when the set of unrestricted contours is empty.  4.4  Gaussian polymer models  In this section we study Gaussian polymer models and a Gaussian field with a finite range decomposition (FRD). These concepts were introduced in Section 3.1. Importantly, here we achieve our 61  goal to rewrite the grand partition function Z (n) as a small perturbation of a Gaussian measure which admits a FRD. Ideally we would like to express the entire partition function Z (n) as one single Gaussian polymer model. Unfortunately, the unrestricted contours ruin such a unified approach. As a workaround use the decomposition (4.14) into sub-partition functions. Each of the sub-partition functions can be expressed as a Gaussian polymer model where the polymer activities are small perturbations. The set of polymers P (n) contains all connected sets which are of unions of cubes C+ in Cn+ .  Let ❊H and NH (n)  (n)  denote the expectation and normalization constant of the Gaussian measure with  covariance matrix H −1 . (n)  Lemma 4.9. There exist a covariance matrix Hn−1 and polymer activities Kr (X, ξ ) such that the (n)  (n)  (n)  restricted partition function satisfies Zr = Z+ + Z− and (n)  (n)  Z± = NH ·  m  ∑  m≥0  (n) ❊(n) H ∏ K± (Xi , ξ ) .  ∑  (4.15)  i=1  {X1 ,...,Xm } Xi ∩X j =0/ if i= j  (n)  (n)  The sum runs over sets of polymers in P (n) . For ωu ∈ Ωu and η ∈ Σ˜ u (ωu ), there exist Ku (X, ξ |η) such that (n)  (n)  Zu (ωu ) = NH ·  ∑ ∑  η∈Σ˜ u (ωu ) m≥0  ✶{  ∑  m  (n) (n) Xi ⊃sp(ωu )} ❊H ∏ Ku (Xi , ξ |η) .  (4.16)  i=1  {X1 ,...,Xm } Xi ∩X j =0/ if i= j  Furthermore, the covariance matrix Hn−1 allows for a FRD with λs = λ s/2 and λ < 1. R EMARK . For unrestricted partition functions there is a different Gaussian polymer model for every spin configuration η in Σ˜ u (ωu ). ♦ The definition of Hn is deferred until Section 4.4.1. The technical definition of the polymer activities is deferred until the end of this section. The polymer activities for restricted and unrestricted contour ensembles are presented in Definition 4.14 and Definition 4.17, respectively. The proofs for (n)  (n)  the restricted partition function Zr and the unrestricted partition functions Zu (ωu ) are provided in (n)  Section 4.4.2 and Section 4.4.3, respectively. The proof for Zr needed to prove (4.16). Hence, the proof for  (n) Zr  already contains all the techniques  is described in full detail while the proof of (4.16)  omits some details where the reader is referred to Section 4.4.2.  4.4.1  Gaussian measure and FRD.  The Gaussian polymer model for each of the sub-partition functions in (4.14) has different polymer activities. However, the Gaussian measure is the same for all sub-partition functions. In fact, the Gaussian measure is extracted from the Taylor expansion for log Z (C) . 62  Let HC be the symmetric positive definite matrix of Condition 1. Define the matrix H in the region Cxˇ by ξ , Hξ  xˇ  =  ∑ Avx [ξ 2 ] −  Av[ξ ], HC Av[ξ ] xˇ ,  (4.17)  x∈Cxˇ  where ξ 2 is the vector with entries ξx2 . Define the matrix H = H(X) for any region X in ❚(n) via ξ , Hξ  X  =  ξ , Hξ  ∑  (4.18)  xˇ  x∈ ˆ Vˇ (X)  The matrix H(X) naturally arises as it is the Hessian of the function S(X; ξ ) near the constant field ξ0 . Let Hn be the matrix H(❚(n) ). On the torus ❚(n) the sums satisfy ∑ Avx [ξ 2 ] = ∑ ξx2 . The matrix Hn can be written as Hn = (I − A) with I the identity matrix. Condition 2 ensures that  ∑  y∈❚(n)  |Axy | ≤ λ < 1.  Proposition A.2 shows that the covariance matrix Hn−1 exists and admits a finite range decomposition. The finite range decomposition for the Gaussian field ξ with covariance matrix Hn−1 becomes ξ=  ∑ λs ζs  Hn−1 =  and  s≥0  ∑ λ2s ·Cs  (4.19)  s≥0  with λs = λ s/2 and Cs has range Ls = s( 1 + −). There is a little technical detail which has to be dealt with. For the cluster expansion technique in Section 3.1 the input was a covariance matrix W on ❩d . The finite volume covariance matrices W (n) were then defined as the periodic extensions of W W (n) (x, y) =  ∑ W (x, y + 2zL(n) ).  z∈❩d  In this chapter we start with matrices Hn whose inverses define covariance matrices W˜ (n) . Furthermore, the matrices HC define the matrix H∞ on ❩d by ξ , H∞ ξ =  ∑  x∈Cx+ ˆ  Avx [ξ 2 ] −  ∑  x∈ ˆ Vˇ (❩d )  Av[ξ ], HC Av[ξ ]  xˇ  Hence, the matrix W = H∞−1 and the matrices W (n) are the periodic extensions of H∞−1 . It is not immediate that W (n) = W˜ (n) or equivalently whether the periodic extension commutes with the inverse. We verify that they do indeed commute such that the results of Chapter 3 can be applied. Claim 4.10. Let A be a matrix on ❩d which commutes with translations at scale  63  +  . Let A(n) be the  periodic extension of A restricted to ❚(n) A(n) (x, y) =  ∑ A(x, y + 2zL(n) ).  z∈❩d  If A has inverse B, then the inverse of the matrix A(n) is B(n) (x, y) =  ∑ B(x, y + 2zL(n) )  z∈❩d  Proof. Let θz denote translation by 2zL(n) . The matrix B commutes with translations because Bθz ξ = Bθz (ABξ ) = θz Bξ . Check whether A(n) · B(n) is the identity matrix by checking the matrix elements. (A(n) · B(n) )(x, y) =  ∑  A(n) (x, z)B(n) (z, y)  z∈❚(n)  =  ∑  z∈❚  (n)  A(x, z + 2vL(n) )  ∑  B(z, y + 2wL(n) )  ∑  w∈❩  v∈❩  d  d  Use the fact that B commutes with translations and make the change of variables w → w − v. (A(n) · B(n) )(x, y) =  ∑ ∑ ∑  A(x, z + 2vL(n) )B(z + 2vL(n) , y + 2wL(n) + 2vL(n) )  ∑ ∑ ∑  A(x, z + 2vL(n) )B(z + 2vL(n) , y + 2wL(n) )  v∈❩d w∈❩d z∈❚(n)  =  v∈❩ w∈❩ z∈❚ d  d  (n)  Combine the sum over v ∈ ❩d with z ∈ ❚(n) . (A(n) · B(n) )(x, y) =  ∑ ∑ A(x, z)B(z, y + 2wL(n) )  w∈❩d z∈❩d  =  ∑ ∑ δ (x, y + 2wL(n) )  w∈❩d z∈❩d  When restricted to x, y ∈ ❚(n) the right hand side is the identity.  4.4.2  Restricted contour ensembles  A contour ensemble ω ∈ Ω(n) partitions ❚(n) − ω into connected components. Let I0 be one of those components and suppose both contours Γ and Γ are adjacent to I0 . Let ∂ (Γ), ∂ (Γ ) be union of C+ cubes which are adjacent to I0 and contained in Γ, Γ respectively. The sign of the spin configuration in ∂ (Γ) and ∂ (Γ ) must be the same. This global condition between the contours is known as the compatibility condition. The global compatibility condition makes the analysis of the measures νn difficult. However, for restricted contour ensembles symmetry allows to remove the compatibility 64  constraint. The proof (4.15) goes in two steps. First symmetry is used to to remove the compatibility condition between contours. Then the polymer activities are defined based on the notion that the Gaussian averages will concentrate around ±ξ0 . Symmetry reduction.  For restricted contour ensembles it is possible to define exterior and interior  components. The exterior component is the clear candidate for the “ocean” as mentioned in the introduction to Section 4.3. Definition 4.11 (Components and signs). (i) Let Γ be a restricted contour. The set Γc has components I0 (Γ), . . . , Im (Γ) with m = m(Γ). The component I0 (Γ) is the unique component which has volume at least  L(n) 2  and is called the  exterior component. Components I1 (Γ), . . . , Im (Γ) are the interior components. (ii) Let ω be a restricted contour ensemble. A contour Γ ∈ ω is called an exterior contour if Γ is not contained in any of the interior components {Ii (Γ ) : i = 1, . . . , m(Γ ) with Γ ∈ ω − {Γ}}. Let E (ω) be the set of exterior contours in ω. (iii) Let ω be a restricted contour ensemble. The set ω c has components I0 (ω), . . . , Im (ω) with m = m(ω). The exterior component I0 (ω) is defined to be intersection of the sets I0 (Γ)  Γ∈ω  .  (n)  (iv) Consider a tuple (η, Γ) where η ∈ Σr and Γ ∈ ω(η). Define sgni (Γ) to be the sign of η in ∂ ext (Γ) ∩ Ii (Γ) with i = 0, . . . , m. If sgn0 (Γ) is ±1, then Γ is called a ±-contour. + (n)  (v) Take Γ ∈ Ψr and define the contour configurations (n)  Γ Σ± c (Γ) := η : η ∈ Σr with Γ ∈ ω(η) and Γ is a ±-contour ,  where η Γ denotes the restriction of η to Γ.  R EMARKS . • The benefit of defining exterior contours is that the contour ensemble ω can be viewed as a set of nested contours as depicted in Figure 4.3(a). This notion fails for sets of unrestricted contour ensembles, e.g., see Figure 4.3(b). • Observe that part (iii) of Definition 4.11 is only valid in dimensions d ≥ 2 where I0 (ω) is a connected set. If d = 1 and a contour ensemble ω has k components, then I0 (ω) necessarily consists of k connected sets. Figure 4.2 shows an example where ω has two contours.  65  Figure 4.2: Example in d = 1 where I0 (ω) consists of two connected sets Proposition 4.12. Let  +1 if η = 0, +1 xˇ σx = −1 if η = −1, xˇ (n)  (n)  (n)  where xˇ is the center of Cx . The partition function Zr = Z+ + Z− and d ❚ (ξ )e−S(ω (n)  (n)  Z± =  ∑  c ;ξ +ξ ) 0  χ+(ξ + ξ0 )ω c  (n)  ∏ Γ∈ω  ω∈Ωr  ∑  e−S(Γ;σ (ξ +ξ0 )) χ η (σ (ξ + ξ0 ))Γ .  η∈Σ± c (Γ)  (n)  (n)  Proof. For a contour ω ∈ Ωr , either the sign of η is positive or negative in I0 (ω). Let Z+ and (n)  Z− correspond to η being positive and negative in I0 (ω), respectively. By symmetry it suffices to (n)  study Z+ . (n)  Define Σ+ (ω) to be the subset of Σ(n) containing those η such that ω(η) = ω and η ≡ +1 in  (a) Two families of nested contours  (b) Unrestricted contours  Figure 4.3: Unrestricted contours versus restricted contours  66  I0 (ω). The restricted grand partition function satisfies (n)  Z+ =  ∑  (n)  Z+ (ω),  (n)  ω∈Ωr  where d ❚ (ξ )e−S(❚ (n)  (n)  Z+ (ω) =  ∑  (n) ;ξ )  χ η (ξ )❚(n) .  η∈Σ+ n (ω) (n)  For the rest of the proof fix ω ∈ Ωr . Make the change of variables ξx → σx ξx . m  d ❚ (ξ ) ∏ e−S(Ii (ω);σ ξ ) χ η (σ ξ )Ii (ω) (n)  (n)  Z+ (ω) =  ∑  i=0  (n)  η∈Σ+ (ω)  ∏ e−S(Γ;σ ξ ) χ η (σ ξ )Γ Γ∈ω  For each of the components Ii = Ii (ω) the variable σ is constant, either ±1, in Ii ∪ ∂ ext (Ii ). This 1 /2 implies that Avx [σ ξ ] = σx Avx [ξ ] for all x ∈ Ii . Use the symmetry of S and the identity χ η (σ ξ )Ii = χ+(ξ )Ii . (n)  Z+ (ω) =  d ❚ (ξ )e−S(ω (n)  c ;ξ )  χ+(ξ )ω c  ∑ ∏ e−S(Γ;σ ξ ) χ η (σ ξ )Γ  (4.20)  (n)  η∈Σ+ (ω) Γ∈ω  The sum over η can be performed in two steps. First, choose the sign of η in each of the components I0 (ω), . . . , Im (ω). Then sum η inside each contour Γ ∈ ω with the restriction that sgni (Γ) with i = 0, . . . , m(Γ) match up with the chosen signs in step 1. Some notation is needed. The set A+ (ω) records all possible ways to assign signs ±1 to the components of ω A+ (ω) = {(α0 , . . . , αm(ω) ) : α0 = +1 and αi = ±1 for i = 1, . . . , m(ω)}. Given α ∈ {−1, +1}m(Γ)+1 and a contour Γ define the set of spin configurations (n)  Σ(Γ, α ) := η Γ : η ∈ Σr with Γ ∈ ω(η) and sgni (Γ) = αi for i = 0, . . . , m , and define the weight WΓ (α0 , . . . , αm ) =  ∑  e−S(Γ;σ ξ ) χ η (σ ξ )Γ .  η∈Σ(Γ,α )  Given α ∈ A+ (ω), a contour Γ and i = 0, . . . , m(Γ) set αiΓ = α j where j satisfies I j (ω) ⊆ Ii (Γ). Define W (ω) by W (ω) =  ∑ ∏ e−S(Γ;σ ξ ) χ η (σ ξ )Γ = ∑ ∏ W (α0Γ , . . . , αmΓ ). α∈A+ (ω) Γ∈ω  (n)  η∈Σ+ (ω) Γ∈ω  67  Employ the symmetry of the model. Consider WΓ (α0 , . . . , αm ) for a contour Γ. By symmetry of S it follows that S(Γ; σ ξ ) = S(Γ; α0 σ ξ ). Moreover, the product of indicator functions satisfies χ η (σ ξ )Γ = χ α η (α0 σ ξ )Γ . Substituting these equalities into the weight WΓ yields 0  WΓ (α0 , . . . , αm ) =  e−S(Γ;α0 σ ξ ) χ α0 η (α0 σ ξ )Γ = WΓ (+1, α0 α1 , . . . , α0 αm ).  ∑  (4.21)  α0 η∈Σ(Γ,α0 α )  The identity (4.21) can be applied inductively to drop the compatibility condition between contours. In particular this is achieved by extracting the innermost contours and then working up to the outermost contours. Inductively partition the set ω Em = E (ω), Ei = E ω − (Em ∪ . . . ∪ Ei+1 )  for i = m − 1, . . . , 1  where m = m (ω) is the smallest integer such that ω − E1 − . . . − Em = 0. / Define the corresponding contour ensembles Ei .  ωi = i≤ j≤m  By (4.21) the weight W (ω) satisfies m  W (ω) =  ∑  m  ∏ ∏ WΓ (α0Γ , . . . , αmΓ ) =  α∈A+ (ω1 ) i=1 Γ∈Ei  ∑ ∏ ∏ WΓ (+1, α0Γ α1Γ , . . . , α0Γ αmΓ ).  (4.22)  α∈A+ (ω1 ) i=1 Γ∈Ei  Define the weight W + (Γ) by W + (Γ) =  ∑  e−S(Γ;σ ξ ) χ η (σ ξ )Γ .  η∈Σ+ c (Γ)  Pick any contour Γ ∈ E1 , then there are no contours contained in the interior components of Γ. Hence, in the sum of α over A+ (ω1 ) one is free to pick αiΓ to be either −1 or +1 for i = 1, . . . , m. Regardless whether α0Γ = ±1 one obtains the identity  ∑  WΓ (+1, α0Γ α1Γ , . . . , α0Γ αmΓ ) = W + (Γ).  (4.23)  α1Γ ,...,αmΓ ∈{−1,+1}m  Substituting (4.23) into (4.22) yields m  W (ω) =  ∏ W + (Γ) Γ∈E1  ∑ ∏ ∏ WΓ (+1, α0Γ α1Γ , . . . , α0Γ αmΓ )  .  (4.24)  α∈A+ (ω2 ) i=2 Γ∈Ei  In the new expression (4.24) repeat the argument used for E1 to extract the contours in E2 .  68  The extraction argument can be continued until ω is exhausted. Hence, the expression for W (ω) becomes W (ω) =  ∏ W + (Γ). Γ∈ω  Substitute the identity for W (ω) into (4.20) and finish the proof by making the change of variables ξ → ξ + ξ0 .  Cube- and contour-weights. Pick a cube Cx+ˆ which is located outside the contours. All the field averages ψx in x ∈ Cx+ˆ are close to ξ0 . Proposition 4.12 suggests that S(Cx+ˆ , ξ + ξ0 ) should be well approximated by the Hessian of S(Cx+ˆ , ξ ) evaluated at ξ0 . Condition 1 states that the Hessian of S(Cx+ˆ ; ξ + ξ0 ) near ξ0 is given by (4.18). Therefore, define the cube-weight ξ ,Hξ xˆ −S(Cx+ ˆ ;ξ +ξ0 ) χ (ξ +  1  F(Cx+ˆ , ξ ) = e 2  +  + ξ0 )Cxˆ − 1.  (4.25)  Fix a contour Γ and suppose the field in ∂ (Γ) is fixed at ξ ∈ ❘∂ (Γ) . For ξ ∈ ❘Γ let ξ ⊕ξ ∈ ❘Γ be the vector whose restrictions to Γ and ∂ (Γ) are ξ and ξ , respectively. For a contour Γ the fluctuations of the field averages are too large to hope that the Hessian will be a good approximation. However, the weight of a contour is expected to be exponentially small in comparison with the contribution when η ≡ +1 in Γ. In the latter case the contribution should be well-approximated by the Gaussian integral d Γ (ξ )e− (ξ ⊕ξ ),H(ξ ⊕ξ ) Γ dξ .  NH (Γ|ξ ) =  (4.26)  For a spin configuration η on Vˇ (Γ) define the contour-weight Wη (Γ|ξ ) as the ratio Wη (Γ|ξ ) =  1  d Γ (ξ )e−S(Γ;σ (ξ ⊕ξ +ξ0 )) χ η (σ (ξ ⊕ ξ + ξ0 ))Γ .  (n) NH (Γ|ξ )  (4.27)  The cumulative contour-weight W± (Γ|ξ ) is defined by W± (Γ|ξ ) =  Wη (Γ|ξ ).  ∑  (4.28)  η∈Σ± c (Γ)  Substituting the cube- and contour-weights into the partition function yields (n)  Z± =  d ❚ (ξ )e− 2  1  (n)  ∑  ξ ,Hξ  ❚(n)  ∏  (F(Cx+ˆ , ξ ) + 1) ∏ W± (Γ|ξ ).  xˆ∈ /Vˆ (sp(ω))  (n)  ω∈Ωr  (n)  Divide and multiply by NH  (n)  Γ∈ω  and expand the product over cubes located outside the contours. (n)  Z± = NH ·  ∑  (n) ∑ ❊H [F(ξ )Y W± (ξ )ω ],  c (n) ω∈Ωr Y ⊂sp(ω)  69  (4.29)  Figure 4.4: The cluster rule with three clusters: X1 , X2 and X3 where by notation F(ξ )Y =  ∏  F(Cx+ˆ , ξ ),  and W± (ξ )ω =  x∈ ˆ Vˆ (Y )  ∏ W± (Γ|ξ ).  (4.30)  Γ∈ω  The product over the empty set equals 1. For forthcoming estimates it will be necessary that the single cubes in the set Y in (4.29) are in the deep interior of a polymer. Let Y˜ ∪ ω˜ be the enlarged set obtained from a given Polymer activities.  (Y, ω) by taking the union of Y and sp(ω) and including all cubes which are contiguous to either Y or sp(ω). A polymer is a connected component of the enlarged set. The single cubes inside a polymer have an insulation layer and are at least distance  +  away from the boundary of a polymer. We call  this procedure the cluster rule. The cluster rule is depicted in Figure 4.4 where after clustering a total of three connected sets of C+ -cubes remain. In this example the polymer X3 wraps around the torus. We show that the polymer activities can be defined as conditional expectations. Consider Figure 4.4 where Y is the union of Y j with j = 1, . . . , 5 and ω = {Γ}. Let ξ i be the restriction of ξ to ∂ int (Xi ). Let Y (i) ⊂ Y be the union of those Y j such that Y j ⊂ Xi . Define the 1 /2 contour ensembles ω1 = {Γ} while ω2 = ω3 = 0. / The contribution of the configuration in Figure 4.4 is  3  (n) (n) Y ω Y ωi ❊(n) H [F(ξ ) W± (ξ ) ] = ❊H ❊H ∏ F(ξ ) W± (ξ ) |ξ 1 , ξ 2 , ξ 3 . i=1  70  (i)  The conditional expectation factors over the sets Xi . 3  (n) (n) Y ω ❊(n) ∏ ❊H [F(ξ )Y W± (ξ )ωi |ξ i . H [F(ξ ) W± (ξ ) ] = ❊H (i)  i=1  The above computation suggests to define polymer-activities as conditional expectations. The polymer activity of X is obtained by reverse engineering the cluster rule. For later reference the definition is broken up in two parts. Definition 4.13 (Minimal fillings). (i) Let Ω = {{Γ1 , . . . , Γm } : m ∈ ◆, Γi ∈ P (n) } be a given set and for ω ∈ Ω let sp(ω) =  ωΓ  where the union runs over all Γ ∈ ω. The set Ω is referred to as the family of compatible sets. (ii) A polymer filling of X is a tuple (ω,Y ) where ω ∈ Ω and Y is a union of C+ -cubes such that sp(ω),Y ⊂ X and Y ∩ sp(ω) = 0. / (iii) The support of a polymer filling is (ω,Y ) is defined to be sp(ω,Y ) = sp(ω) ∪Y . (iv) A polymer filling is minimally admissible if • dist(sp(ω,Y ), X c ) =  +  ,  • sp(ω,Y ) ∪ ∂ ext (sp(ω,Y )) = X. + (v) Given a polymer X and a family of compatible sets Ω, let Fm (X, Ω) be the set of minimally admissible polymer fillings of a polymer X. Given a polymer X sum over the weights of all possible tuples (Y, ω) such that after applying the cluster rule one obtains the set X. Definition 4.14 describes the technical details. (n)  Definition 4.14 (Restricted polymer activities). Let Fr (X) = Fm (X, Ωr ), the restricted polymer activity is (n)  K± (X, ξ ) =  Y ω ❊(n) H F(ξ ) W± (ξ ) ξ ,  ∑ (ω,Y )∈Fr (X)  where ξ = ξ  ∂ int/2 (X)  . The sum over the empty set is zero.  1  (n)  All that remains is to check that (4.15) indeed holds with the above choice of K± (X, ξ ). Let  f denotes the cluster rule, i.e., the map from F (❚(n) , Ωr ) to {X1 , . . . , Xk } where k depends on (n)  (ω,Y ). (n) ❊(n) ∑ F(ξ )Y W± (ξ )ω = ❊H H (ω,Y )  k  ∑  ∑  ∏ ❊H  {X1 ,...,Xk } (ω,Y )∈ f ({X1 ,...,Xk }) i=1  71  (n)  [F(ξ )Yi W± (ξ )ωi |ξ i ]  The constraint that (ω,Y ) is in the inverse image of {X1 , . . . , Xk } is easily checked. Let (ωi ,Yi ) denote the restriction of (ω,Y ) to Xi . Each restriction (ωi ,Yi ) must be contained in Fr (Xi ) while sp(ω,Y ) ⊂  Xi . The right-hand side becomes  ∑  ∑ ✶{sp(ω,Y )⊂  k  Xi }  {X1 ,...,Xk } (ω,Y )  (n) Yi ωi ❊(n) H ∏ ✶{(ωi ,Yi )∈Fr (Xi )} ❊H [F(ξ ) W± (ξ ) |ξ i ] . i=1  Hence, it has been shown that  ❊(n) ∑ F(ξ )Y W± (ξ )ω = H (ω,Y )  k  (n) ∑ ❊H ∏  {X1 ,...,Xk }  Yi ωi ❊(n) H [F(ξ ) W± (ξ ) |ξ i ] .  ∑  i=1 (ω,Y )∈Fr (Xi )  This concludes the proof for the restricted partition function.  4.4.3  Unrestricted contour ensembles  In this section we derive the Gaussian polymer model for unrestricted contour ensembles. The proof is very similar to the proof for restricted contours ensembles. Hence, some details are omitted and when needed the reader is referred to Section 4.4.2. Partial symmetry reduction. For restricted contour ensembles symmetry reduction was used to remove the compatibility condition between restricted contours. The argument fails for contour ensembles which contain unrestricted contours. Suppose ω contains the unrestricted contour ensemble ωu . The region ❚(n) − ωu is divided into connected components I0 (ωu ), . . . , Im (ωu ). In each of these connected components ω can only have restricted contours. The symmetry technique of Section 4.4.2 can thus be applied in each of these sub-regions. Stating the partial symmetry reduction result requires some extra definitions of restricted spin configurations in sub-regions X of ❚(n) . The set X should be thought of as ❚(n) − ωu . Definition 4.15 (Restricted configurations in sub-regions). Let X be a strict subset of ❚(n) . (i) Let η X denote the restriction of η to X. (n) Σ˜ u (X) = η X : η ∈ Σu (X)  (ii) Given ωu define the set of restricted contours (n)  Ωr (ωu ) = {ω(η) − ωu : η ∈ Σu (ωu )} The partial symmetry result can now be stated. Some details in the proof of Proposition 4.16 are omitted and the reader is referred to the proof of Proposition 4.12. 72  Proposition 4.16. Given η define the spin configuration  +1 if η = 0, +1, xˇ σx = −1 if η = −1, xˇ (n)  (n)  where xˇ is the center of the cube Cx . Let ωu ∈ Ωu , then the partition function Zu (ωu ) satisfies (n)  Zu (ωu ) =  d ❚ (ξ ) (n)  c  c  e−S(ωu ;ξ ) χ+(ξ + ξ0 )ωu  ∑  ∏ e−S(Γ;σ (ξ +ξ )) χ η (σ (ξ + ξ0 ))Γ 0  u  ηu ∈Σ˜ u (ωu )  Γ∈ωu  c  c  e−S(ωr ;ξ ) χ+(ξ + ξ0 )ωr  ∑  ωr ∈Ωr (ωu )  e−S(Γ;σ (ξ +ξ0 )) χ ηr (σ (ξ + ξ0 ))Γ .  ∑  ∏  ηr ∈Σ+ c (Γ)  Γ∈ω  (n)  The expression for Zu (ωu ) is lengthy. The interpretation is that the first sum runs over all possible ways to get the unrestricted contour ensemble ωu while the second sum enriches ωu with restricted contour ensembles in the complement of ωu . Proof. Make the change of variables ξ → σ ξ . c  (n)  Zu (ωu ) =  c  e−S(ω(η) ;ξ ) χ+(ξ )ω(η)  ∑ (n)  ∏  e−S(Γ;σ (ξ +ξ0 )) χ η (σ ξ )Γ  Γ∈ω(η)  η∈Σu (ωu )  The spin configuration η can be decomposed into a spin configuration ηu on ω u and spin configurations ηr on Γ for Γ ∈ ω(η) − ωu . The partition function is equal to (n)  Zu (ωu ) =  d ❚ (ξ ) (n)  c  ∑  c  e−S(ωu ;ξ ) χ+(ξ )ωu  ∏ e−S(Γ;σ ξ ) χ η (σ ξ )Γ  η∈Σ˜ u (ωu )  ∑  ωr ∈Ωr (ωu )  Γ∈ωu −S(ωrc ;ξ )  e  ωrc  χ+(ξ )  ∏ Γ∈ω  ∑  e−S(Γ;σ ξ ) χ ηr (σ ξ )Γ  ✶{compatible} ,  ηr ∈Σσc (Γ)  where ✶{compatible} is one if the spin configurations on ωu and ωr match up and zero otherwise. Perform the symmetry argument of Proposition 4.12 on the contour ensemble ωr . In this way all contours in ωr can be turned into +-contours. However, it stops there because for unrestricted contours there is no sensible way to define exterior and interior components. Apply the translation ξ → ξ + ξ0 to finish the proof.  Cube- and contour-weights. Proceed as in Section 4.4.2 with a minor difference in defining the contour-weights. For a cube Cx+ˆ located outside the contours define the cube-weight by (4.25). For a restricted contour Γ ∈ Ωr (ωu ) define the contour-weight by (4.28). The weight of an unrestricted contour Γ ∈ ωu depends on η and is thus defined by (4.27). The unrestricted partition function  73  satisfies (n)  (n)  Zu (ωu ) = NH  ∑  (n) Y ω ∪ω ] ˜ )=0} / ❊H [F(ξ ) Wu (ξ |η) ∑ ✶{Y ∩sp(ω∪ω u  ∑  r  u  η∈Σ˜ u (ωu ) ωr ∈Ωr (ωu ) Y ⊂❚(n)  where F(ξ )Y was defined by (4.30) and by notation Wu (ξ |η)ω =  ∏  Wη (Γ|ξ )  Γ∈ω Γ∈ωu (η)  Polymer activities.  ∏  W± (Γ|ξ ).  Γ∈ω Γ∈ω / u (η)  The polymer activities for unrestricted contour ensembles are defined to be  conditional expectations, see the paragraph on polymer activities in Section 4.4.2. There is one more condition for unrestricted polymer filings to be admissible: there can be no factors F(Cx+ˆ , ξ ) (n)  or W± (Γ|ξ ) with either Cx+ˆ or Γ overlapping with sp(ωu ). Given ωu ∈ Ωu define (n)  Ωu (ωu ) = {ω(η) : η ∈ Σ(n) and ωu (η) = ωu }.  (n)  Definition 4.17 (Unrestricted polymer activities). Let Fu (X, ωu ) ⊂ Fm (X, Ωu (ωu )) contain the polymer fillings which satisfy sp(ω,Y ) ∩ Γ = 0/  for some Γ ∈ ωu  ⇒  ω  Γ.  The unrestricted polymer activity is (n)  Ku (X, ξ |η) =  Y ω ❊(n) H [F(ξ ) Wu (ξ |η) |ξ ]  ∑ (ω,Y )∈Fu (X)  where ξ = ξ  ∂ int/2 (X)  . The sum over the empty set is zero.  1  4.5  Cluster expansion bounds for a generalized polymer model  While deriving the Peierls estimate for contours and proving a phase coexistence we will encounter a few different Gaussian polymer models. Each Gaussian polymer model uses the Gaussian measure with covariance matrix H −1 . However, the polymer activities will differ slightly. To accommodate all these different models we introduce a generalized Gaussian polymer model. In this section we list constraints under which the generalized Gaussian model satisfies the assumptions for the cluster expansion technique in Section 3.4. The section finishes by showing that the cube- and contour-weights introduced in Section 4.4 do satisfy these constraints. The proofs of the results in this section are given in Section 4.5.1–Section 4.5.3. However, the proofs can be 74  skipped and the reader can go directly to the proofs of the Peierls estimate and the phase coexistence in Section 4.6 and Section 4.7 Setup. Define the partition function m  (n)  Zg =  ∑  m≥0  ∑  (n) ❊(n) H ∏ Kg (Xi , ξ ) .  ❚  {X1 ,...,Xm }⊂ (n) Xi ∩X j =0/ if i= j  (4.31)  i=1  The two assumptions of the generalized model are the use of the Gaussian measure with covariance (n)  matrix H −1 and the specific form of the polymer activities Kg (X, ξ ). Recall Definition 4.13 and assume there exists some set of compatible elements Ω. Let Fg (X) ⊆ Fm (X, Ω) be a subset of the set of minimally admissible polymer fillings. Assume there exist cubefunctions Fg (Cx+ˆ , ξ ), which depend on ξx for x ∈ C+xˆ . There exists j different contours functions  Wg (Γ|ξ ) with i = 1, . . . , j and j ∈ ◆ with j finite. The contours function are functions of the field ξx with x ∈ Γ. The polymer activity is Kg (X, ξ ) =  Y ω ❊(n) H [Fg (ξ ) Wg (ξ ) ξ ],  ∑  (4.32)  (ω,Y )∈Fg (X)  where ξ = ξ  ∂ int/2 (X)  and  1  Fg (ξ )Y =  ∏  Fg (Cx+ˆ , ξ ),  and Wg (ξ )ω =  x∈ ˆ Vˆ (Y )  ∏ Wg (Γ|ξ ). Γ∈ω  Intermezzo. The next definitions were provided in Section 3.2 to Section 3.4. For convenience of the reader they are repeated in this short intermezzo. Scaled polymers are sets (X, s) where X is a ˜ = ◆ ∪ {−1, 0} a scale. The generic notation Ai is used to denote a set of scaled spatial set and s ∈ ◆ polymers, the support of Ai is sp(Ai ) =  X. (X,s)∈Ai  It was shown in Section 3.3 that, formally, the logarithm satisfies (n)  log Zg =  m  ∑ ∑  m≥1 (A1 ,...,Am )  (n) (n) ∏ ❊H [❑g (A j )]  Jc (A1 , . . . , Am ),  (4.33)  j=1  where the sum runs over sequences of scaled polymers Ai such that sp(Ai ) ⊂ ❚(n) . The weight  ❑(n) g (A j ) is defined by (3.14) which in itself depends on definition (3.12). The real coefficients  ♣  Jc (A1 , . . . , Am ) are defined by (3.17).  Recall that for given polymers X,Y the sets of rooted polymers Am (X), Am (X,Y ) are defined 75  by Am (X) = {(A1 , . . . , Am )|sp(Ai ) ∩ X = 0/ for some 1 ≤ i ≤ m}, Am (X,Y ) = {(A1 , . . . , Am )|sp(Ai ) ∩ X = 0/ and for some sp(A j ) ∩Y = 01 / ≤ i, j ≤ m}. In this section the major result will be to show uniform convergence in n of the series (n)  Pg (X) = (n)  Pg (X,Y ) =  m  1 1 ∑ ∑ ˆ N(X) m≥1 m! (A1 ,...,Am )∈Am (X) 1 1 ∑ m! ˆ N(X) m≥1  (n) (n) ∏ ❊H [❑g (A j )]  Jc (A1 , . . . , Am ),  j=1  (4.34)  m  (n) (n) ∏ ❊H [❑g (A j )] Jc (A1 , . . . , Am ),  ∑  j=1  (A1 ,...,Am )∈Am (X,Y )  for fixed non-empty polymers X and Y . Given a field ξ and x, y in ❚(n) define the differences ϕx (y) = ψy − ψx ,  ϕ(y) = ϕyˇ (y),  and  (4.35)  where yˇ is the center of the cube Cy . The first results states a sufficient condition on the weights F (n)  and W under which Pg (X) converges as n → ∞. Theorem 4.18. If the positive constants a0 = a0 (λ , d) and ε = ε(λ , d) are small enough and |Fg (x, ˆ ξ )| ≤  1−λ a0 ∑+ exp 4 |C+ | x∈C xˆ  ∑  ψy2 + ϕx (y)2 ,  y∈Cxˇ  and ˆ N(Γ)  |Wg (Γ|ξ )| ≤ a0  1 ε 1  exp λ 4  ∑ ξx2  ,  x  then (n)  (i) Pg (X) converges as n → ∞ for any X ∈ P. (n)  (ii) There exists a finite constant a0 such that |Pg (X)| ≤ a0 holds uniformly for all those n with X ∈ P (n) . (iii) There exists a finite constant a1 such that 1 ∑ ∑ ˆ N(X) m≥1 {A1 ,...,Am }∈Am (X,Y )  m  (n) (n) ∏ ❊H [|❑g (A j )|]  1 −1 + dist(X,Y )  |Jc (A1 , . . . , Am )| ≤ a1 λ 4  .  j=1  The second result shows that the cube- and contour-weights which were introduced in Section 4.4 do indeed satisfy the conditions imposed in Theorem 4.18 provided  76  1  is large enough.  Theorem 4.19. For ε small enough (depending on the dimension d) and  large enough the  1  weights F and Wη , defined by (4.25) and (4.28), satisfy −d/5 1  |F(Cx+ˆ , ξ )| ≤  |C |  1−λ 4  exp  ∑  +  x∈Cx+ ˆ  ∑  ψy2 + ϕx (y)2  (4.36)  y∈Cx  ˜ Let η ∈ Σ+ c (Γ) or η is the restriction to Γ of some spin configuration in Σu (ωu ). Wη (Γ|ξ ) ≤ e−  3δ 2 ˆ 1 N(Γ)  1 ε 1  exp λ 4  ∑ ξx2  ,  (4.37)  x∈Γ  The bound (4.37) is referred to as the pre-Peierls estimate. The pre-estimate already shows that the weight of a contour decays exponential with its size. In the upper-bound (4.36) the first sum runs over Cx+ˆ a cube at scale  4.5.1  +  while the second sum in the exponent runs over a cube Cx at scale  .  −  Proof of Theorem 4.18  Recall Definition 3.3 for the definition of semi-local polymer activities with parameters a1 , a2 . By (n)  Theorem 3.7 the statement of Theorem 4.18 holds when the polymer activities Kg (X, ξ ) are semilocal with parameters a1 , a2 small enough. The assumptions in Theorem 4.18 are used to show that d  (n)  1 ε  the polymer activities Kg (X, ξ ) are semi-local with a1 = 3ea−3 and a2 = λ 2 1 . 0 Proposition 4.20. Fix a polymer X and a scale s ≥ −1, then −d  (n)  |δ Kg (X, s)| ≤ 3ea30  ˆ N(X)  1 ε 1  λs exp λ 4  ξ s−1 (x)2 + ζ s (x)2  ∑  x∈∂ int (X) 1  We first show that if a0 is small enough, then Proposition 4.20 suffices to prove the conclusions of Theorem 4.18. After that is done we turn to the proof of the proposition. Proof of Theorem 4.18. Recall the finite range decomposition (4.19) of H −1 . Theorem 3.7 and Corollary 3.8 shows that it suffices to find a constant a3 > 1 such that 2 |C+ |  4ea1 e2a2 (λ∞ +1)  3d a3+ + ∑ λs aL3 s Nˆ s < 1, s≥0  −3d  with a1 = 3ea0  and a2 = λ  1 ε 2 1  1 −1 +  . Pick the constant a3 to be λ − 4  follows that 2 |C+ |  e2a2 (λ∞ +1)  1 ε (1+ε)d 1 (λ∞ +1)2 1  = e2λ 2  77  . By choosing  ≤ 2.  1  large enough it  Compute the sum over scales ∑ λs aL3 s Nˆ s where Nˆ s = {xˆ ∈ Vˆ (❩d ) : dist(0, x) ˆ ≤ s( 1 + −)} The sum is easily bounded by decomposing the scales into appropriate sets. Define the sets of scales Sm = {s : (m − 1)  +  ≤ Ls < m + }. ∞  ∞  ∑ λs Nˆ s = ∑ ∑ λs aL3 Nˆ s ≤ ∑ 3dm ∑ λs aL3 s  s≥0  m=1 s∈Sm  m=1  If Ls ≥ (m − 1) + , then s must exceed λs aL3 s ≤ λ s/4 .  ∞  s  ≥  m−1 ε 2 1.  ∞ s≥0  ∑ (3d λ λ  ∑ 3dm  m=1  s∈Sm  (m−1) + 1+ −  ∑ λs Nˆ s ≤ 3d λ∞  ≤  λs aL3 s  ∑  s:Ls ≥(m−1)  +  Recall that λs = λ s/2 which implies that  1 ε 8 1  m  ≤ 2 · 3d λ∞  m=0  The bound on the right-hand side holds provided  1  is large enough. The proof is completed by  choosing a0 small enough such that −d  ν := 24e2 a30 and setting a0 =  1  3d λ − 4 + 2 · 3d λ∞ < 1,  1 1−ν .  Proof of Proposition 4.20 Some notation is necessary to deal with the boundary condition in ∂ int (X). Define the sets 1 /2 ∂˚ (X) = ∂ int (X) 1 /2  X˚ = X − ∂˚ (X).  and  Fix a polymer filling f = (ω,Y ) ∈ F (X). The scaled polymer activities satisfy δ K f (X, s) = ❊H [F(ξ )Y W (ξ )ω ξ s ] − ❊H [F(ξ )Y W (ξ )ω ξ s−1 ], (n)  (n)  ˚ Given a boundary condition ξ ∈ ❘X˚ c where ξ¯s , ξ¯s−1 are the boundary conditions in X˚ c = ❚(n) − X. (n) ˚ and ξ ∈ ❘X let ξ ⊕ ξ ∈ ❘❚ be the vector whose restrictions to X˚ and X˚ c are ξ and ξ , respectively. The proof heavily relies on a study of the minimizer of (ξ ⊕ ξ ), H(ξ ⊕ ξ ) n . The definition (4.18) shows that H = (I − A) where I is the identity matrix. Condition 2 shows that, in the language of Section A.1, the matrices D and A are (1, λ ,  1+  )-compatible. Hence, Lemma A.3 can be used  −  and lists some properties of the minimizer. Throughout the proof all subscripts g and superscripts (n) are repressed. For v, w ∈ ❘X the notation v, w boundary condition in X˚ c . ˚  78  0  = v ⊕ 0, w ⊕ 0  n  is used to denote a zero  ˚ X ∗ and ∂˚ X Figure 4.5: The sets X, For t ∈ [0, 1] define the interpolated field ξ¯ (t) = t ξ¯s + (1 − t)ξ¯s−1 . 1  δ K f (X, s) =  dt 0  d (n) ❊ [F(ξ )Y W (ξ )ω |ξ¯ (t)] dt H  Before taking the derivative it is important to make a change of variables. Let gˆt ∈ ❘X be the minimizer of gt , Hgt with the boundary condition gt ≡ ξ (t) in X˚ c . Let X ∗ = sp(ω,Y ) ∪ ∂ ext (sp(ω,Y )) 1 /2  and define g˜t (x) = gˆt (x)✶X−X ∗ . In this way g˜t ≡ 0 in X ∗ and the weights with the translated field satisfy F(ξ + g˜t )Y = F(ξ )Y and W (ξ + g˜t )ω = W (ξ )ω . This is important because the weights do not depend on the variable t and are unaffected when taking the derivative with respect to t. The ˚ X ∗ and ∂˚ X are depicted in Figure 4.5. sets X, The conditional expectation is a fraction of two integrals. In the numerator make the change of variables ξ → ξ + g˜t while in the denominator ξ → ξ + gˆt . 1  ❊  (n) Y ω ¯ H [F(ξ ) W (ξ ) |ξ (t)]  e− 2  g˜t ,H g˜t  =  1  ˚  d X (ξ )e− 2 1  e− 2  gˆt ,H gˆt  ξ ,Hξ  0  e− ξ ⊕0,H g˜t F(ξ )Y W (ξ )ω  ˚  1  d X (ξ )e− 2  ξ ,Hξ  0  = e 2 Q(gˆt ) ❊H [eL(ξ ,gˆt ) F(ξ )Y W (ξ )ω ] 1  where the quadratic and linear term are defined by Q(gˆt ) = gˆt , H gˆt − g˜t , H g˜t ,  L(ξ , gˆt ) = − ξ ⊕ 0, H gˆt ✶❚(n) −X ∗ .  (4.38)  R EMARK . The linear term ξ ⊕ 0, H gˆt did not appear in the exponent of the denominator because ♦  Lemma A.3 shows that this term is zero.  79  The expectation has the right form to take the derivative. 1 d (n) L(ξ ,gˆt ) ❊ [F(ξ )Y W (ξ )ω |ξ¯ (t)] = e 2 Q(gˆt ) Q (gˆt )❊(n) F(ξ )Y W (ξ )ω ] H [e dt H  ❊  (n) H [L  (gˆt , ξ )e  L(ξ ,gˆt )  (4.39) Y  ω  F(ξ ) W (ξ ) ]  The work is to bound all the separate terms which occurred after taking the derivative. Let wt ∈ (n) ❘❚ which is zero on X˚ and wt = ξ (t) in X˚ c . Proposition A.4 shows that the minimizer gˆt = H −1 wt . Take the derivative d d gˆt (x) = H −1 ξ x (t)✶{x∈∂˚ (X)} = H −1 (ξ s (x) − ξ s−1 (x)✶{x∈∂˚ (X)} ) = λs H −1 ws , dt dt where ws ∈ ❘❚  (n)  is zero on X˚ and ws = ζ s (x) in X˚ c . Proposition A.4 then implies that d gˆt (x) = λs gˆs (x), dt  (4.40)  where gˆs is the minimizer of gs , Hgs with gˆs ≡ ζ s in ∂˚ (X). Proposition 4.21. The quadratic term satisfies Q(gˆt ) ≤ 0 and 1 ε  λs 2λs λ 3 1 1 d Q(gˆt ) ≤ gˆt (x)2 + gˆs (x)2 ≤ ∑ 2 dt (1 − λ ) x∈X ∗ (1 − λ )3  ∑  ξ s−1 (x)2 + ζ s (x)2  x∈∂˚ (X)  Proof. The fact that Q(gˆt ) ≤ 0 immediately follows because gˆt is the minimizer and as such satisfies g˜t , H g˜t ≥ gˆt , H gˆt . Lemma A.3 and symmetry of H also show that 0 = gˆt ✶X ∗ , H gˆt = gˆt , H gˆt ✶X ∗ . This observation shows that g˜t , H g˜t = gˆt (1 − ✶X ∗ ), H gˆt (1 − ✶X ∗ ) = gˆt , H gˆt + gˆt ✶X ∗ , H gˆt ✶X ∗ , and Q(gˆt ) = − gˆt ✶X ∗ , H gˆt ✶X ∗ . Take the derivative of Q(gˆt ), use (4.40) and symmetry d Q(gˆt ) = −2λs H 1/2 gˆs ✶X ∗ , H 1/2 gˆt ✶X ∗ . dt  80  Apply Cauchy-Schwarz followed by the Schur test [61, Corollary 3.7]. d 1/2 gˆt ✶X ∗ , H gˆt ✶X ∗ Q(gˆt ) ≤ 2λs gˆs ✶X ∗ , H gˆs ✶X ∗ dt 2λs 1/2 1/2 ≤ ∑∗ gˆs (x)2 ∑∗ gˆt (x)2 1 − λ x∈X x∈X The distance from X ∗ to ∂˚ (X) is at least − + 1  +  +  1/2  − 1 , see Figure 4.5. The relative distance is then  1  ≥  1+ε 1  −  2  −  1  1  ≥  ε 1  3  .  Apply Young’s inequality and then use Lemma A.3. 1 ε  d 2λs 2λs λ 3 1 Q(gˆt ) ≤ gˆt (x)2 + gˆs (x)2 ≤ ∑ dt (1 − λ ) x∈X ∗ (1 − λ )3  ∑  ξ x (t)2 + ζ s (x)2  x∈∂˚ (X)  Use Young’s inequality one more time on the term ξ x (t) = (ξ s−1 (x) + tζ s (x))2 . 1 ε  4λs λ 3 1 d Q(gˆt ) ≤ dt (1 − λ )3  ∑  ξ s−1 (x)2 + ζ s (x)2  x∈∂˚ (X)  Lemma A.3 shows that the linear term is L(ξ , gˆt ) = − ξ ⊕ 0, H gˆt + ξ ⊕ 0, H gˆt ✶X ∗ = ξ ⊕ 0, H gˆt ✶X ∗ ,  (4.41)  while (4.40) shows d L(ξ , gˆt ) = λs ξ ⊕ 0, H gˆs ✶X ∗ = L(ξ , gˆs ) dt Use the simple bound |x| ≤ cosh(x) and Proposition 4.21 to obtain that for  1  large enough  1 ε  d (n) 2λ 3 1 ❊H [F(ξ )Y W (ξ )ω |ξ¯ (t)] ≤ λs · exp ξ¯s−1 (y)2 + ζ¯s (y)2 dt (1 − λ )3 y∈∂∑ ext (X)  (4.42)  ❊  (n) L(ξ ,gˆt ) cosh(L(ξ , gˆs ))F(ξ )Y W (ξ )ω ] H [e  We focus our attention to the expectation on the right-hand side and set B± = The aim is to bound  L(ξ ,gˆt ) ±L(ξ ,gˆs ) ❊(n) e F(ξ )Y W (ξ )ω ] . H [e  (4.43)  B+ +B− . 2  Let µY denote the product measure of the independent random variables {Zyˆ : yˆ ∈ Vˆ (X)} where  81  Zyˆ uniformly chooses y ∈ Cy+ˆ . Throw in the assumptions of Theorem 4.18 on F and W . ˆ )+N(sp(ω)) ˆ N(Y  B± ≤ a0  µY  L(ξ ,gˆt )±L(ξ ,gˆs ) ❊(n) H e  exp  1−λ 4  ∑ ∑  1 ε 1  ψy2 + ϕZyˆ (y)2 + λ 4  y∈ ˆ Vˆ (Y ) y∈CZyˆ  ∑ ∑ ξx2 Γ∈ω x∈Γ  Define the matrix Rt by ξ , Rt ξ = ξ , Hξ − t  1 ε 1  ψy2 + ϕZyˆ (y)2 + λ 4  ∑ ∑  y∈ ˆ Vˆ (Y ) y∈CZyˆ  ∑ ∑ ξx2 Γ∈ω x∈Γ  Proposition A.2 can be applied because of Condition 2 and Proposition A.6. Hence, Rt−1 is a positive definite matrix with  ∑ |Rt−1 (x, y)| ≤ ∑  y∈X  −ε/2 1 − λ ) 1  λ + (1 +  2  s≥0  1 ε 1  +λ 4  s  ,  for all x ∈ X. Let r denote the finite quantity on the right-hand side r=  ∑  λ + (1 +  −ε/2 1 − λ ) 1  2  s≥0  1 ε 1  +λ 4  s  ≤  ∑  (1 +  −ε/2 1 + λ s ) 1  s≥0  2  <∞  The bound can be expressed in terms of Gaussian measures. Let NRt (X) and ❊Rt denote the normalization constant and expectation, respectively, of the Gaussian measure with covariance matrix Rt−1 and zero boundary condition. ˆ )+N(sp(ω)) ˆ N(Y  B± ≤ a0  NR1 (X) ❊R eL(ξ ,gˆt )±L(ξ ,gˆs ) ] NR0 (X) 1  µY  Recall the expression for the linear term (4.40) to conclude that  ❊R1 eL(ξ ,gˆt )+L(ξ ,gˆs ) ] = e 2  1  H(gˆt ✶X ∗ +gˆs ✶X ∗ ),R−1 1 H(gˆt ✶X ∗ +gˆs ✶X ∗ )  It is a given that ∑ |Hxy | ≤ 1 and ∑ |R1−1 (x, y)| ≤ r, apply the Schur test to find  ❊R1 eL(ξ ,gˆt )±L(ξ ,gˆs ) ≤ e 2 ∑x∈X (gˆt ✶X ∗ +gˆs ✶X ∗ ) ≤ exp r ∑ (gˆt (x)2 + gˆs (x)2 ) . ∗ r  2  x∈X  Finally apply Proposition 4.21 which implies that for  1  large enough  1 ε  ❊R1 eL(ξ ,gˆt )±L(ξ ,gˆs ) ≤ exp  2rλs λ 3 1 (1 − λ )3  82  ∑  x∈∂ int (X) 1  ξ s−1 (x)2 + ζ s (x)2  .  (4.44)  Consider the size of NR1 (X) in comparison with NR0 (X). log  ˚ NR1 (X) = NR0 (X)  1  dt 0  d log NRt (X) dt  ≤ ❊Rt  ∑ ∑  1 ε 1  ψy2 + ϕZyˆ (y)2 + λ 4  y∈ ˆ Vˆ (Y ) y∈CZyˆ  ∑ ∑ ξx2 Γ∈ω x∈Γ  The variances can be bounded using Proposition A.6. Let ey be the vector with all zeroes except at position y where it is 1.  ∑ ❊R [Avy [ξ ]2 ] = ∑ t  y∈CZyˆ  ey , AvRt−1 Avey =  y∈CZyˆ  ≤r  ∑  Av(ey ), Rt−1 Av(ey )  ∑  y∈CZyˆ  Av(ey ), Av(ey ) = r  2εd 1  y∈CZyˆ  To obtain the small factor  2εd 1  ≤  1 2  it was crucial here that there is only one small cube CZyˆ for every  +  cube Cyˆ ∈ Y . The same computations, in conjunction with Proposition A.6, show that for  1  large  enough  ❊Rt  ∑ ∑  1 ε 1  ϕZyˆ (y)2 + λ 4  y∈ ˆ Vˆ (Y ) y∈CZyˆ  ∑ ∑ ξx2  ≤r  2εd 1  ˆ ) + N(sp(ω)) ˆ N(Y  Γ∈ω x∈Γ  Therefore, it has been shown that  NR1 (X) ˆ ≤ eN(X) . NR0 (X)  (4.45)  Substituting (4.44) and (4.45) into (4.43) yields 1 ε  B+ + B− 2rλs λ 3 1 ˆ )+N(sp(ω)) ˆ ˆ N(Y ≤ a0 eN(X) exp 2 (1 − λ )3 By taking  1  ξ s−1 (x)2 + ζ s (x)2  ∑  x∈∂ int (X) 1  large enough the bound (4.42) becomes  d (n) ❊ [F(ξ )Y W (ξ )ω |ξ¯ (t)] ≤ dt H ˆ )+N(sp(ω)) ˆ ˆ N(Y N(X)  λs · a1  e  1 ε 1  exp λ 4  ∑  ξ s−1 (x)2 + ζ s (x)2  x∈∂ int (X) 1  Minimal polymer fillings were defined such that at least every set of 3d cubes contains at least one cube which either belongs to Y or to sp(ω). It has been proved that −d  |δ K f (X, s)| ≤ λs · ea30  ˆ N(X)  1 ε 1  exp λ 4  ∑  ξ s−1 (x)2 + ζ s (x)2 .  x∈∂ int (X) 1  Coming back to δ K(X, s) one has to sum over all polymer fillings f . Each cube in X has tree 83  ˆ  options: it can belong to Y , ω or to neither. Hence, for a polymer X there can be at most 3N(X) polymer fillings |δ K(X, s)| ≤  −d  |δ K f (X, s)| ≤ λs · 3ea30  ∑  ˆ N(X)  ξ s−1 (x)2 + ζ s (x)2 .  ∑  x∈∂ int (X)  f ∈F (X)  4.5.2  1 ε 1  exp λ 4  1  Proof of bound (4.36)  The main ingredient of the proof is to use Condition 1 which shows that S(Cx+ˆ , ξ + ξ0 ) is wellapproximated by its Hessian ξ , Hξ xˆ . Proposition 4.22. Take X ⊂ ❚(n) and suppose that |ψx − ξ0 | ≤ γ d/2−δ for all x ∈ X, then there exists a constant a such that S(X; ξ + ξ0 ) − ξ , Hξ  X  ≤ a(log |C|)2  −d( 41 −ε) ˆ N(X). 1  (4.46)  Proof. The term S(X; ξ + ξ0 ) is S(X; ξ + ξ0 ) =  ∑  x∈X  =  1 Avx [(ξ + ξ0 )2 ] − S(ξ0 ) − ∑ log ZC (x, ˇ ξ + ξ0 ) 2 x∈ ˇ Vˇ (X)  1 ξ , Hξ 2  X  −  ∑  1  log ZC (x, ˇ ξ + ξ0 ) − log ZC (ξ0 ) − ξ0  x∈ ˇ Vˇ (X)  ∑ ψx − 2  ψ, HC ψ  xˇ  x∈Cxˇ  The difference satisfies S(X; ξ + ξ0 ) −  1 ξ , Hξ 2  ≤  X  ∑  log ZC (x, ˇ ξ + ξ0 ) − log ZC (ξ0 ) − ξ0  x∈ ˇ Vˇ (X)  1  ∑ ψx − 2  ψ, HC ψ  xˇ  x∈Cxˇ  Apply the Taylor expansion provided by Condition 1. There exists some constant a such that the difference is bounded by a1 (log |C|)2 |C+ | max |ψx − ξ0 | x∈X  Recall that the scale  +  =  1+ε 1  3  ≤ a1 (log |C|)2 |C+ |  where ε is a small constant and δ =  −d/2+δ 3 ˇ N(X). 1 d 12 .  ˆ ξ ) be the indicator function of the event that the maxiReturn to the proof of (4.36). Let χ˜ +(x, −d/2+δ + . In other words it is the equivalent of χ + (x, ˇ ξ ) at the mum of |ψx − ξ0 | in Cxˆ is bounded by 1 scale  +  . Recall the definition of F(Cx+ˆ , ξ ) by (4.25). 1  |F(Cx+ˆ , ξ )| ≤ e 2  ξ ,Hξ xˆ −S(Cx+ ˆ ,ξ +ξ0 )  − 1 χ˜ +(x, ˆ ξ + ξ0 ) + (1 − χ˜ +(x, ˆ ξ + ξ0 ))  84  (4.47)  Consider the contribution of the first term in (4.47). The exponent is bounded by Proposition 4.22, then use the bound |ex − 1| ≤ 2|x| which holds for |x| < 12 . 1  e2  ξ ,Hξ xˆ −S(Cx+ ˆ ,ξ +ξ0 )  − 1 χ˜ +(x, ˆ ξ + ξ0 ) ≤ 2a(log |C|)2  −d( 41 −ε) 1  Consider the contribution of the second term in (4.47). 1 − χ˜ +(x, ˆ ξ + ξ0 ) = ✶{∃x∈C+ : |ψ |≥ x  Suppose |ψx | ≥ than  1 −d/2+δ . 2 1  −d/2+δ 1  x  ∑ ✶{|ψ |≥  ≤  −d/2+δ } 1  x  x∈Cx+  −d/2+δ } 1  then for each y ∈ Cx either |ψy | or the difference |ϕx (y)| must be greater  Recall that  −  =  1−ε 1 ,  then the sum satisfies |C| 4  ∑ ψy2 + ϕx (y)2 ≥  y∈Cxˇ  −d+2δ 1  1 4  =  2δ −εd . 1  Summing over all the possible x ∈ Cx+ , the bound becomes ˆ ≤ |Cx+ |e− 1 − χ +1 (ξ + ξ0 , x)  1−λ 2δ −εd 16 1  |Cx+ |−1  ∑  exp  x∈Cx+  ≤ e− 1 |Cx+ |−1 δ  ∑  exp  x∈Cx+  The second inequality holds provided |Cx+ |−1  ∑  x∈Cx+  1  1−λ 4  ∑  1−λ 4  ∑  ψy2 + ϕx (y)2  y∈Cx  ψy2 + ϕx (y)2 .  y∈Cxˇ  is large enough. Using the trivial inequality  exp  1−λ 4  ∑  ψy2 + ϕx (y)2  ≥ 1,  y∈Cx  the bounds on the first and second term of (4.47) can be combined to |F(Cx+ˆ , ξ )| ≤ a1 (log |C|)2  −d( 41 −ε) + e− 1  δ 1  |Cx+ |−1  ∑  x∈Cx+  Taking ε small enough and  4.5.3  1  exp  1−λ 4  ∑  ψy2 + ϕx (y)2 .  y∈Cxˇ  large enough one obtains the bound (4.36).  Proof of the pre-Peierls estimate  In this section we complete the proof of Theorem 4.19 by verifying the pre-Peierls estimate. We start the section by reminding the reader what pre-Peierls estimate is and then start the proof. The proof has a long buildup with many auxiliary results. These results are finally put together on page 99. Let Γ be a contour which can be either restricted or unrestricted and η a spin configuration on the contour. If the contour is restricted, then η ∈ Σ+ c (Γ). If the contour is unrestricted, then take  85  η ∈ Σ˜ u ({Γ}). Define  +1 if η = 0, +1, xˇ σx = −1 if η = −1, xˇ where xˇ is the center of the cube Cx . The contour-weight is Wη (Γ|ξ ) =  1 NH (Γ|ξ )  d Γ (ξ )e−S(Γ;σ (ξ ⊕ξ +ξ0 )) χ η (σ (ξ ⊕ ξ + ξ0 ))Γ .  The constant NH (Γ|ξ ) is defined by (4.26). The pre-Peierls estimate is Wη (Γ|ξ ) ≤ e−  3δ 2 ˆ 1 N(Γ)  1 ε 1  exp λ 4  ∑ ξx2  .  x∈Γ  In this section we consider functions χ˜ · (x, ˇ ξ ) which are refinements of the functions χ · (x, ˇ ξ ) by ˜ χ χ χ splitting 0 (x, ˇ ξ ) into two cases. Let ± (x, ˇ ξ ) = ± (x, ˇ ξ ) and define χ˜ 0± (x, ˇ ξ ) = ✶{±ψxˇ >0} χ˜ 0 (x, ˇ ξ ). The functions χ˜ 0± (x, ˇ ξ ) not only tell whether there is a large fluctuation in Cxˇ but also disclose whether the field-average at the center of the cube is positive or negative. ˇ Refined function χ˜ · call for refined spin configurations η˜ in {−1, 0− , 0+ , +1}V (Γ) . A refined spin configuration η˜ is mapped to a spin configuration η by mapping 0± to 0. The set of admissible ˜ ) is the inverse image of η under this mapping. The contour-weight refined spin configurations Σ(η ˜ ). Wη (Γ|ξ ) can be obtained by summing over all possible η ∈ Σ(η Wη (Γ|ξ ) =  ∑  Wη (Γ|ξ ),  η∈Σ(η )  where Wη (Γ|ξ ) =  1 (n) NH (Γ|ξ )  d Γ (ξ )e−S(Γ;σ (ξ ⊕ξ +ξ0 )) χ η (σ (ξ ⊕ ξ + ξ0 ))Γ  For convenience the tilde was dropped in the notation but η, Σ(η ) and χ η refer to the refined counterparts. Refined spin configurations η call for refined spin configuration σ . Hence, define  +1 if η = 0+ , +1 xˇ σx = −1 if η = 0− , −1. xˇ Undo the change of variables ξ → σ (ξ +ξ0 ) and then make the change of variables ξ → σ (ξ +ξ0 ).  86  Wη (Γ|ξ ) =  1 (n) NH (Γ|ξ )  d Γ (ξ )e−S(Γ;σ (ξ ⊕ξ +ξ0 )) χ η (σ (ξ ⊕ ξ + ξ0 ))Γ  (4.48)  For the majority of the proof we will focus on proving a bound for (4.48) with a fixed η ∈ Σ(η ). At the very end of the proof we sum over the possible η ∈ Σ(η ). Outline of the proof. The entire proof is based on two heuristic arguments which we refer to as the choose-well argument and the follow-suit argument. The follow-suit argument relies on the choice of scale  −  =  1−ε 1  with ε a small constant.  Figure 4.1 provides an intuitive idea for the choose-well argument. The field averages want to settle near either well ±ξˆ . For xˇ ∈ Vˇ (Γ) the refined spin ηxˇ selects which well the spin averages ψxˇ choose. If ηxˇ is positive, then ψxˇ must select the well +ξˆ . If ηxˇ is negative, then ψxˇ must select the well −ξˆ . The field average ψx runs over the field ξ in Cx(1) . The scale (1)  −  was chosen such that  −  1  (1)  and consequently Cx ≈ Cy for x, y ∈ Cxˇ which in turn implies ψx ≈ ψy . This we call the follow-suit argument. The choose-well and follow-suit argument provide an heuristic proof. In the contour Γ two bad events can occur. The first event is the large fluctuation event. Suppose ηxˇ = 0± , then there exist −d/2+δ ψx ∈ Cxˇ such that the displacement of ψx from either well ±ξˆ is larger than . The follow-suit argument then claims that all the displacements of ψy for y ∈ Cxˇ exceed  1 −d/2+δ . 1  This event has  exponential small probability because the standard deviation of Gaussian averages ψx is of the order |C(1) |−1/2 =  −d/2 . 1  The second event is the neighbour-clash event when two neighbouring cubes Cxˇ and Cyˇ have opposite signs σxˇ = σyˇ . By the choose-well argument ψxˇ chooses the well +ξˆ while ψyˇ chooses to be in the well −ξˆ . According to the follow-suit argument all field averages ψx for x ∈ Cxˇ choose the well +ξˆ . Similarly, the field averages ψy for y ∈ Cyˇ choose the well −ξˆ . Consider pairs ψx and ψy with x ∈ Cxˇ and y ∈ Cyˇ . The distance between x and y is at most 2  −  1  and still ψx ≈ ψy .  However, ψx and ψy are forced to choose a different well. This causes excessive friction in the system and the probability for this event is exponentially small as well.  A partition of Γ by σ -contours.  The region Γ is partitioned into three types of regions: the buffer  region, the large-fluctuation region and the neighbour-clash region. The buffer region BΓ = ∂ int (Γ) + ˆ is where the field averages still remain close to ±ξ . Its task is to make sure the dependence on the boundary condition ξ is exponentially small. The two remaining regions is where the corresponding bad events (see outline of the proof) occur. The large-fluctuation region and neighbour-clash regions require a more complicated description in terms of σ -contours, i.e., contours due to fluctuations in σ . Figure 4.6 depicts the partition for a 87  given contour Γ. Definition 4.23 (σ -contours). (i) Given η let Ξ be the union of all facets between cubes with a different sign of σ . Ξ = {Cxˇf ∩Cyˇf : x, ˇ yˇ ∈ Vˇ (Γ) and σxˇ = σyˇ }, where Cxˇf ⊂ ❘d is the closed cube of side-length  −  centered at x. ˇ  (ii) The set of σ -contours Fi , i = 1, . . . , mF are the connected closed surfaces in Ξ. The size of a contour N(Fi ) is measured in the number of facets it contains. (iii) Let ΞT be the union of the thickened contours ∂ ext (Fi ) for i = 1, . . . , mF . The sets ∆i , i = + 1, . . . , m∆ ≤ mF are the connected components of ΞT and ∆ =  ∆.  (iv) For 1 ≤ i ≤ m let Ii denote the ith bounded component of Γ − ΞT and I =  Ii .  The crucial step in the proof of (4.37) is make the right choice for the change of variables ξ → ξ + v. Different types of behaviour in the regions ∂ Γ, Ii and ∆¯ i determines the choice of v. The buffer region BΓ . In the buffer region BΓ no bad events occur because all averages are close to ±ξˆ . The main task in BΓ is to make a proper change of variables such that the dependence on the boundary condition ξ is exponentially small. Let gˆ ∈ ❘Γ be the minimizer of g, Hg tion A.2. Let ξ , Hξ  0 BΓ  = ξ ⊕ 0, H(ξ ⊕ 0)  Γ  with gˆ ≡ ξ in ∂ (Γ). The minimizer is studied in Sec-  BΓ  be the inner-product with a zero boundary condition  in ∂ (Γ).  Figure 4.6: A contour Γ with its σ -contours.  88  Proposition 4.24. There exist {vx }x∈BΓ with vx = ξ x for all x ∈ ∂ (Γ), vx = 0 for all x with dist(x, BΓ ) ∨ dist(x, I ∪ ∆) ≤ 2 1 . such that after translation ξ → ξ + v e−S(BΓ ;σ (ξ ⊕ξ +ξ0 )) χ η (σ (ξ ⊕ ξ + ξ0 ))BΓ ≤ e  −d/5 ˆ N(BΓ ) 1  ·e  ˆ gˆ Γ − 21 ξ ,Hξ − 12 g,H  0 + BΓ  ξ ⊕0,v˜  BΓ  .  Moreover, the sum of squares of v˜ is bounded by v, ˜ v˜  BΓ  1 ε 1  ≤λ4  ∑  ξ 2x .  x∈∂ (Γ)  R EMARK . The subset of x with dist(x, BΓ ) ∨ dist(x, I ∪ ∆) ≤ 2 tance 2  1  1  are those x which are within dis-  of the boundary between BΓ and the regions I ∪ ∆.  ♦  Proof. The buffer region is a disjoint union of connected sets, e.g. see Figure 4.6 for an example with two disjoint sets. On each of these connected sets the spin-variable η is constant either ±1. e−S(BΓ ;σ (ξ ⊕ξ +ξ0 )) χ η (σ (ξ ⊕ ξ + ξ0 ))BΓ = e−S(BΓ ;ξ ⊕ξ +ξ0 ) χ+(ξ ⊕ ξ + ξ0 )BΓ Proposition 4.22 shows that, for  1  large enough,  e−S(BΓ ;ξ ⊕ξ +ξ0 ) χ+(ξ ⊕ ξ + ξ0 )BΓ ≤ e  −d/5 ˆ N(BΓ ) 1  1  · e− 2  ξ ⊕ξ ,H(ξ ⊕ξ )  BΓ  .  Let gˆ = g(ξ ˆ ) be the minimizer among all g ∈ ❘Γ such that g(ξ ˆ ) = argmin g, Hg  Γ  g≡ξ in ∂ (Γ)  Define v for the region BΓ by  0 if dist(x, I ∪ ∆) ≤ 2 1 , vx = g(x) ˆ else.  (4.49)  Let g˜ = v on BΓ and extended to Γ by setting g(x) ˜ ≡ 0 in Γ − BΓ . Let Γ0 ⊂ Γ be the region where g˜ = 0. With the above choice of v it follows that (ξ + v), H(ξ + v  BΓ  = g, ˜ H g˜  Γ+  89  ξ , Hξ  0 BΓ  + 2 ξ ⊕ 0, v  BΓ  Observe that gˆ is the unique minimizer thus implying that g, ˜ H g˜  ≥ g, ˆ H gˆ Γ .  Γ  Pay attention to the linear term. Let ξ ∈ ❘ be the extension of ξ ⊕ 0 by setting ξx = 0 in Γ  Γ − BΓ . Lemma A.3 guarantees that ξ , H gˆ = 0. ξ ⊕ 0, v  ∂Γ  = ξ , H(gˆ✶Γ−Γ0 )  Γ  = ξ , H gˆ  ξ , H(gˆ✶Γ0 )  Γ−  Γ  = − ξ , H(gˆ✶Γ0 ) Γ .  Inverting the averages it follows that ξ , H(gˆ✶Γ0 )  Γ  =  ξx gˆx ✶Γ0 − λ Avx [Av[gˆ✶Γ0 ]]  ∑ x∈Γ0 ∩BΓ  The above equation defines v. ˜ By Young’s inequality and Jensen’s inequality the sum of squares satisfies  ∑ v˜2x ≤ 2 ∑ x∈BΓ  If dist(Γ0 , x) <  1,  gˆ2x ✶Γ0 + Avx [Av[gˆ2 ✶Γ0 ]] ≤ 4  x:dist(Γ0 ,x)<  x∈Γ0 ∩BΓ  gˆ2x  ∑ 1  then the relative distance to cross from x to Γc is at least dist(Γ0 , Γc ) − 1+ −  1  ≥  1+ε 1  −3 2 1  1  >  ε 1  3  Using Lemma A.3 the inner-product can be bounded H gˆ✶Γ0 , H gˆ✶Γ0  Γ  1 ε 1  ≤ a1 λ 3  ξ 2x ,  ∑  (4.50)  x∈∂ (Γ)  for some constant a1 . Taking  1  1 ε 1  large enough such that a1 λ 3  1 ε  ≤ λ 4 1.  Follow-suit. The follow-suit argument is the core principle to extract exponentially small factors from the regions I and ∆. It is the only point in the proof where a distinction between the averages ψ˚ •  and ψ is necessary. Given σ define signed averages ψxσ = Avx [σ ξ ], differences ϕxσ (y) = ψyσ − ψxσ and ξ0σ (x) = Avx [σ ξ0 ]. For the block averages ψ˚ it follows that ϕ˚ xσ (y) = 0 if x and y are in the same •  cube C. This is not the case for the site averages ψ. Lemma 4.25. Let Cxˇ be a cube with cube-center x. ˇ There exist positive constants κ, a0 such that ˆ (ξ +ξ0 )) e−S(x,σ ≤ E(x, ˇ ξ ) · exp −  1 ∑ Avy [(ξ + ξ0 )2 ] − (ψyσ + ξ0σ (y))2 + κ(ψyσ + ξ0σ (y) − σy ξ0 )2 2 y∈C xˇ  90  where the error term is 2  E(x, ˇ ξ ) = ea0 (log |C|) · exp  ∑ σy (ξˆ − ξ0 )ψ˚ σ (y) + 16κ ϕ˚ σ (y)2 ✶{σ ϕ˚ y  σ (y)<−ξ /4} 0  .  y∈Cxˇ  R EMARK . The many terms in Lemma 4.25 disguise why it is called the follow-suit lemma. However, observe that the quadratic term preceded by κ forces ψyσ into the well σy ξ0 = σxˇ ξ0 which is ♦  selected by ψxˇ .  The proof of Lemma 4.25 requires more information about the field average differences ϕxσ (y) when the distance between x and y is at most  . Sites x and y are not assumed to be in the same  −  cube C. Claim 4.26. Take x, y ∈ ❚(n) and suppose that dist(x, y) ≤  . If umax = max |ux | denotes the maxi-  −  mum, then Avx [u] − Avy [u] ≤ umax  − 21 ε . 1  Proof. The proof is a direct application of (A.6). Avx [u] − Avy [u] ≤ umax  ∑  z∈❚  |ϕy (x, z)|  (n)  Proof of Lemma 4.25. Take θ = ±1. As an intermediate step we claim that if θ ψxˆ > 0, then there exists positive constants κ and a1 , independent of Z (C) (x, ˇ ξ )e−S(ξ0 ) ≤ a1 exp  1,  such that  ψx2 κ ∑ 2 − 2 (ψx − θ ξˆ )2 + 4κϕ(x)2 ✶{θ ϕ(x)<−ξˆ /2} , x∈Cxˇ  (4.51)  By symmetry it suffices to consider the case with θ = +1. Recall the definition (4.6) of S(ξ0 ) and apply Condition 3. Z (C) (x, ˇ ξ )e−S(ξ0 ) ≤ a1 exp  ψx2 − Φ(ψx ) + Φ(ξˆ ) . x∈Cxˇ 2  ∑  Definition 4.1 shows that the function Φ allows for parabolic lower-bounds for some κ > 0. κ Φ(ψx ) − Φ(ξˆ ) ≥ (ψx − θ ξˆ )2 ✶{θ ψx ≥−ξˆ /2} + (ψx + θ ξˆ )2 ✶{θ ψx <−ξˆ /2} 2 κ = (ψx − θ ξˆ )2 + 4θ ξˆ ψx ✶{θ ψx <−ξˆ /2} 2  91  The additional information that θ ψxˇ > 0 yields 4θ ξˆ ψx ✶{θ ψx <−ξˆ /2} = 4θ ξˆ (ϕ(x) + ψxˇ )✶{θ ψx <−ξˆ /2} ≥ 4θ ξˆ ϕ(x)✶{θ ψx <−ξˆ /2} . The events {θ ψxˇ > 0} and {θ ψx < −ξˆ /2} imply that {θ ϕ(x) < −ξˆ /2}. 4θ ξˆ ψx ✶{θ ψx <−ξˆ /2} ≥ −8ϕ(x)2 ✶{θ ϕ(x)<−ξˆ /2} This completes the proof for the intermediate step (4.51). The spin configuration σ has been chosen such that {σxˇ (ψxσˇ + ξ0σ (x)) > 0} for all xˇ ∈ Vˇ (Γ). Apply the bound (4.51) with the variables σ (ξ + ξ0 ). ˇ (ξ +ξ0 )) e−S(x;σ = exp  1 Avy [(ξ + ξ0 )2 ] − log Z (C) (x, ˇ σ (ξ + ξ0 )) − S(ξ0 ) 2 y∈Cxˇ  ∑  ≤ a1 exp −  1 ∑ Avy [(ξ + ξ0 )2 ] − (ψyσ + ξ0σ (y))2 + κ(ψyσ + ξ0σ (y) − σy ξˆ )2 2 y∈C xˇ − 4κ(ϕ σ (y) + t(y))2 ✶{σy (ϕ σ (y)+t(y))<−ξˆ /2} ,  where t(y) = ξ0σ (y) − ξ0σ (x) ˇ Extract the linear term by replacing σy ξˆ by σy ξ0 . κ σ κ (ψ + ξ0σ (y) − σy ξˆ )2 = (ψyσ + ξ0σ (y) − σy ξ0 )2 + κσy (ξˆ − ξ0 )ψyσ 2 y 2 κ + (ξˆ − ξ0 )2 + κσy (ξˆ − ξ0 )(ξ0σ (y) − σy ξ0 ). 2 By Condition 1 the difference |ξˆ − ξ0 | is bounded by a (log |C|)2 |C|−1 . 0  κ κ σ (ψy + ξ0σ (y) − σy ξˆ )2 ≤ (ψyσ + ξ0σ (y) − σy ξ0 )2 + κσy (ξˆ − ξ0 )ψyσ + a0 (log |C|)|C|−1 2 2 Claim 4.26 shows that t(y) is small which implies that σy ϕ σ (y) < − ξ40 . (ϕ σ (y) + t(y))2 ✶{σy (ϕ σ (y)+t(y))<−ξˆ /2} ≤ 2ϕ σ (y)2 ✶{σ ϕ σ (y)<− ξ0 } y  4  Collecting all the terms finishes the proof.  The regions Ii . The regions Ii are the large-fluctuation regions: there are x ∈ Ii such that the fluctu−d/2+δ ation of ψx around ξˆ is at least . Therefore, we extract exponentially small factors. Let µX 1  92  denote the product measure of the independent random variables {Zxˇ } with xˇ ∈ Vˇ (X). The random variable Zxˇ uniformly chooses x ∈ Cxˇ . Let Nˇ LF (Ii ) = Nˇ LF (Ii , η) be the number of xˇ ∈ Vˇ (Ii ) such that ηxˇ = 0± . Proposition 4.27 (Large-fluctuation). There exist positive constants κ, a0 and a1 such that e  −S(Ii ;σ (ξ +ξ0 ) χ  η (σ (ξ  Ii  + ξ0 )) ≤ e  −  7δ 4 ˇ 1 NLF (Ii )  · e∑x∈Ii  2 − 12 Avx [ξ 2 ]+ 1−κ 2 ψx  ˜ i , ξ ), E(I  where the error term is ˇ i) ˜ i , ξ ) = ea0 (log |C|)2 N(I E(I µIi exp  ∑ ∑ a1 (ϕ(y)2 + ϕZ (y)2 ) + v˜x ψx xˇ  ,  x∈ ˇ Vˇ (Ii ) y∈Cxˇ  and |v˜x | ≤ a0 (log |C|)2 |C|−1 . R EMARK . In the large-fluctuation regions Ii no translation is needed, i.e., vx = 0 for x ∈ Ii .  ♦  Proof. The regions Ii were defined such that σ is constant on I i . e−S(Ii ;σ (ξ +ξ0 )) χ η (σ (ξ + ξ0 ))Ii = e−S(Ii ;ξ +ξ0 ) χ σ η (ξ + ξ0 )Ii The spin σxˇ ηxˇ is an element of {0+ , +1} for all xˇ ∈ Vˇ (Ii ). Hence, the field averages at the cubecenters all satisfy ψxˇ + ξ0 > 0. Apply Lemma 4.25 and let E(Ii , ξ ) = ∏ E(x, ˇ ξ ) where the product runs over xˇ ∈ Vˇ (Ii ). e−S(Ii ;ξ +ξ0 ) ≤ E(Ii , ξ ) exp  1  1  1  1−κ 2 ψx 2  κ  ∑ 2 Avx [(ξ + ξ0 )2 ] − 2 (ψx + ξ0 )2 + 2 ψx2  x∈Ii  ≤ E(Ii , ξ ) exp  ∑ 2 Avx [ξ 2 ] −  x∈Ii  (4.52)  In the last line we used that Avx [ξ ξ0 ] = ξ0 Avx [ξ ] = ξ0 ψx . Condition 1 ensures the upper-bound v˜y = (ξ0 − ξˆ ) ≤ a0 (log |C|)2 |C|−1 . Consider the large-fluctuation event which corresponds to σxˇ ϕxˇ = 0+ . χ 0+ (x, ˇ ξ + ξ0 ) ≤  ∑ ✶{|ψ +ξ −ξˆ |≥  x∈Cxˇ  x  0  −d/2+δ } 1  ≤  ∑ ✶{|ψ |≥  x∈Cxˇ  x  1 −d/2+δ } 2 1  In the last line we used Condition 1 by which |ξ0 − ξˆ | ≤ a0 log(|C|)2 |C|−1 . Suppose the field average 1 −d/2+δ 2 1 1 −d/2+δ . Recall 4 1  |ψx | ≥  then for each y ∈ Cx either |ψy | or the difference |ϕx (y)| must be greater than that  −  =  1−ε 1 ,  then the sum satisfies |C|  ∑ ψy2 + ϕx (y)2 ≥ 16  y∈Cxˇ  93  −d+2δ 1  =  1 16  2δ −εd . 1  Summing over all the possible x ∈ Cxˇ , the bound becomes  ∑ χ0  2δ −εd 1  κ  +  (x, ˇ ξ + ξ0 ) ≤ |Cx+ |e− 64  |Cx+ |−1  ∑  κ 4  exp  x∈Cx+  x∈Cxˇ  ≤ e−  7δ 4 1  |Cx+ |−1  ∑  exp  x∈Cx+  κ 4  ∑  ∑  ψy2 + ϕx (y)2  y∈Cx  ψy2 + ϕx (y)2 .  (4.53)  y∈Cxˇ  The last inequality holds for ε small enough and 1 large enough. The bound can be interpreted in terms of the random variables {Zxˇ }. Let VˇLF (Ii ) ⊂ Vˇ (Ii ) contain those cube-centers such that ηxˇ = 0± .  ∏  x∈ ˇ VˇLF (Ii )  |Cx+ |−1  ∑  exp  x∈Cx+  κ 4  ∑  ψy2 + ϕx (y)2  ≤ µIi exp  y∈Cxˇ  κ 4  ∑  ψy2 + ϕZxˇ (y)2  (4.54)  y∈Cxˇ  Consider the error term E(Ii , ξ ) which is easily bounded 2 N(I ˇ i)  E(Ii , ξ ) ≤ ea0 (log |C|)  exp  ˚ 2 ], ∑ v˜x ψ˚ x + 16κ ϕ(y)  (4.55)  y∈Ii  with v˜x = (ξˆ − ξ0 ). Substitute equations (4.53)–(4.55) into (4.52) and set κ = ˜ i , ξ ) and finishes the proof. error term E(I  κ 2  . This defines the  The region ∆i . The regions ∆i are the neighbour-clash regions: in ∆i there are neighbouring cubes Cxˇ and Cyˇ with opposite signs σxˇ = σyˇ . Proposition 4.28 shows that the neighbour-clash event causes enormous friction in the system. Proposition 4.28 (Neighbour-clash). There exists {vx }x∈∆i with vx = 0 for x ∈ ∂ (∆i )∪∂ int (∆i ) such 1 that after the translation ξ → ξ + v e−S(∆i ;σ (ξ +ξ0 ) χ η (σ (ξ + ξ0 ))∆i ≤ e−a0  d(1−2ε) F(∆i ) 1  1  · e∑x∈Ii − 2 Avx [ξ  2 ]+ 1−κ ψ σ (x)2 x 2  ˜ i , ξ ), E(∆  for some positive constants a0 and κ. Moreover, there exists a constant a1 such that the error term satisfies ˇ i) ˜ i , ξ ) = e3a0 (log |C|)2 N(∆ E(∆ µ∆i exp a  ∑ ∑ ϕ σ (y)2 + ∑ v˜y ξy  x∈ ˇ Vˇ (∆i ) y∈Cxˇ  ,  y∈∆i  and |v˜x | ≤ a0 (log |C|)2 |C|−1 . R EMARK . There might possibly be large-fluctuation events going on in ∆i . These events are not taken into account in the bound for the neighbour-clash region.  94  ♦  Apply Lemma 4.25 by setting E(∆i , ξ ) = ∏ E(x, ˇ ξ ) where the product runs over all xˇ ∈ Vˇ (∆i ). e−S(∆i ;ξ +ξ0 ) ≤ E(∆i , ξ )· 1 exp − ∑ Avy [(ξ + ξ0 )2 ] − (ψyσ + ξ0σ (y))2 + κ((ψyσ )2 + ξ0σ (y) − σy ξ0 )2 2 y∈Ii  (4.56)  As in the buffer region BΓ the key is to work with the minimizer of a positive definite function. Lemma 4.29. Let kˆ be the unique minimizer to Q(k) =  1 ∑ Avx [k2 ] − Avx [k]2 + κ(Avx [k] − σx ξ0 )2 . 2 x∈∆ i  The minimizer kˆ has the following properties: ˆ ≥ κξ 2 (i) Q(k) 0  d(1−2ε) F(∆i ) 1  (ii) Let k ∈ ❘∆i , then  ∑ Avx [kk]ˆ − Avx [k]Avx [k]ˆ + κAvx [k] · (Avx [k]ˆ − σx ξ0 ) = 0. x∈∆i  (iii) The minimizer satisfies |kˆ x | ≤ ξ0 for all x. (iv) For x ∈ ∂ (∆i ) ∪ ∂2int1 (∆i ) the minimizer satisfies 1 ε 1  |kˆ x − σx ξ0 | ≤ 2ξ0 (1 − κ) 2  Lemma 4.29 contains all the necessary ingredients to finish the proof of Proposition 4.28. We first show how to use the lemma to finish the proof in the neighbour-clash region ∆i and then return to the proof of Lemma 4.29. Define the vector v ∈ ❘∆i by  0 vx = σ kˆ − ξ x x 0  if x ∈ ∂ (∆i ) ∪ ∂ int (∆i ), 1  .  otherwise.  The choice of vx = 0 makes sure that v lines up with the choice of v in the regions BΓ and Ii . Make the change of variables ξ → ξ + v. Expand the quadratics and regroup the terms in a convenient way while setting k˜ x = vx + ξ0 . Define the linear term L(ξ + v) by L(ξ + v) =  ∑ Avx [ξ k]˜ − Avx [σ ξ ]Avx [σ k]˜ + κAvx [σ ξ ](Avx [σ k]˜ − σx ξ0 ). x∈∆i  95  (4.57)  After the translation ξ → ξ + v the bound (4.56) becomes 1 ˜ + L(ξ + v) − 1 ∑ Avy [ξ 2 ] − (1 − κ)(ψyσ )2 e−S(∆i ;ξ +ξ0 ) ≤ E(∆i , ξ + v) · exp − Q(σ k) 2 2 y∈Ii ˜ ≥ Q(k). ˆ Lemma 4.29 part (i) provides a lowerThe vector kˆ is the minimizer which implies Q(σ k) ˆ bound for Q(k). κ  e−S(∆i ;ξ +ξ0 ) ≤ E(∆i , ξ + v) · e− 2  ξ02  d(1−2ε) F(∆i ) 1  exp L(ξ + v) −  1 ∑ Avy [ξ 2 ] − (1 − κ)ψ σ (y)2 . 2 y∈I i  The linear term is closely related to the minimizer kˆ and will be negligible as will the error term E(∆i , ξ + v). Proposition 4.28 is a direct consequence of the next proposition by setting the vector v˜ = v(1) + v(2) . 1 ε  (1) Proposition 4.30. The linear term satisfies L(ξ , v) = ∑∆i ξx v(1) x and |vx | ≤ (1 − κ) 3 1 . The error  term satisfies ˇ i) ˜ i , ξ + v) ≤ ea0 (log |C|)2 N(∆ E(∆ µ∆i exp  ∑ aϕ σ (y)2 + vy ξy (2)  ,  y∈Cxˇ 2 −1 where |v(2) y | ≤ a0 (log |C|) |C| .  Proof of Proposition 4.30. Start with the linear term. The vector σ k˜ is defined by  σ ξ x 0 ˜ x= (σ k) kˆ x  if x ∈ ∂ (∆i ), if x ∈ ∆i − ∂ (∆i ),  where ∂ (∆i ) = ∂ (∆i ) ∪ ∂ int (∆i ). Equivalently, the vector is defined by σ k˜ = kˆ + u where the vector 1 ˆ ✶∂ (∆ ) . u = (σx ξ0 − k) i  Substitute the decomposition σ k˜ = kˆ + u into the linear term. It is convenient to use the trivial ˜ = Avx [(σ ξ )(σ k)]. ˜ identity Avx [ξ k] L(ξ , v) =  ∑ Avx [(σ ξ )u] − (1 − κ)Avx [σ ξ ]Avx [σ u] x∈∆i  +  ∑ Avx [(σ ξ )k]ˆ − Avx [σ ξ ]Avx [k]ˆ + κAvx [σ ξ ](Avx [k]ˆ − σx ξ0 ), x∈∆i  The second term is zero by Lemma 4.29 part (ii). The vector u is zero outside ∂ (∆i ). The sum can  96  be restricted to x in the region ∂ (∆i ) ∪ ∂2int1 (∆i ) where σ is constant. L(ξ , v) =  Avx [ξ (σ u)] − (1 − κ)Avx [ξ ]Avx [u]  ∑  x∈∂ (∆i )∪∂2int (∆i ) 1  1 ε  By Lemma 4.29 part (iv), the entries ux satisfy |ux | ≤ 2ξ0 (1 − κ) 2 1 . Invert the averages to obtain the expression for v(1) . This concludes the proof of the linear term. Consider the error term E(∆i , ξ + v). Set tx = Avx [σ v] − Avxˇ [σ v] and use Lemma 4.29 part (iii) to conclude that |vx | ≤ 2ξ0 . Claim 4.26 shows that |tx | is O(  ∑ (ϕ σ (y) + ty )2 · ✶{σ (ϕ y  σ (y)+t )<−ξ /4} y 0  ≤  − 21 ε ). 1  ∑ (ϕ σ (y) + ty )2 · ✶{σ ϕ y  σ (y)<−ξ  0 /5}  ≤4  y∈Cxˇ  y∈Cxˇ  y∈Cxˇ  ∑ ϕ σ (y)2 .  The linear term is easily dealt with because |v| ≤ 2ξ0 and Condition 1. ˆ i ) + ∑ σy (ξ − ξ0 )ψ σ (y) ∑ σy (ξ − ξ0 )(ψ σ (y) + Avy [σ v]) ≤ a0 (log |C|)2 N(∆ y∈∆i  y∈∆i 2  = a0 (log |C|) +  ∑v  (2)  (4.58)  ξy  y∈∆i  The exact expression for v(2) can be obtained by inverting the averages. To summarize the bound on the error term becomes 2 N(∆ ˇ  E(∆i , ξ + v) ≤ e2a0 (log |C|)  i)  · exp  ∑ 64κϕ(y)2 + ∑ v y∈∆i  (2)  ξy  y∈∆i  The proof of Proposition 4.30 is complete. Proof of Lemma 4.29. The proof is very similar to the proofs provided in Section A.2. In particular, we will extensively use the finite range decomposition provided by Proposition A.2. Define the matrix W by k,W k = k,W k  ∆i  :=  ∑ Avx [k2 ] − (1 − κ)Avx [k]2 . x∈∆i  ˜ Define the function Q(k) by 1 1 ˜ Q(k) = k,W k − κ Av[k], σ ξ0 = k,W k − κ k, ξ0σ (x) 2 2 The minimizer of Q˜ is the minimizer of Q because the difference Q − Q˜ = κ2 |∆i |ξ02 is constant. The proof of part (ii) is done via the directional derivative. The unique minimizer kˆ satisfies ˆ k = 0 = ∇Q(k),  d Q(kˆ + εk) dε  97  ε=0  ,  for all k ∈ ❘X . Compute the right-hand side. d Q(kˆ + εk) dε  ε=0  =  1 1 ˆ k,W kˆ + k,W k − κ k, ξ0σ (x) . 2 2  By symmetry of W it follows that d Q(kˆ + εk) dε  ε=0  = k, (W kˆ − κξ0σ ) = 0  (4.59)  for all k ∈ ❘∆i . Write out the definition of W to complete the proof of (ii). ˆ namely kˆ = κW −1 ξ σ . The existence of Equation (4.59) provides an explicit expression for k, 0 W −1 is ensured by Proposition A.2. Let D be the diagonal matrix with diagonal elements Dxx = αx , then W −1 = D−1 ∑ (1 − κ)s s≥0  Av2 D  where  s  s≥0  Av2 D  Cs =  = D−1 ∑ (1 − κ)sCs ,  s  ,  and ∑x∈∆i Cs (x, y) = 1 for all x ∈ ∆ and Cs has range s 1 . The finite range decomposition immediately proves parts (iii) and (iv). The vector kˆ satisfies |kˆ x | ≤ κ ∑ (1 − κ)s  ∑ Cs (x, y)  s≥0  y∈∆i  ξ0σ (y) ≤ κξ0 ∑ (1 − κ)s ∑ Cs (x, y) = ξ0 . Dyy s≥0 y∈∆i  Consider part (iv) and take x ∈ ∂ (∆i ) ∪ ∂2int1 (∆i ). |kˆ x − σx ξ0 | ≤ κ ∑ (1 − κ)s s≥0  ∑ Cs (x, y) y∈∆i  ξ0σ (y) − σx ξ0 Dyy  The fluctuations of σ occur deep in the interior of the set ∆i while x is near the boundary of ∆i . Hence, a site y with 2  + 1  =  ε 1  2  ξ0σ (y) Dyy  = σx ξ0 has to be at least distance  steps to cover the distance |kˆ x − σx ξ0 | ≤ κ  ∑  +  −3  1  away from x. It takes at least  − 3 1.  (1 − κ)s  s≥ ε1 /2  +  ∑ Cs (x, y) y∈∆i  ξ0σ (y) − σx ξ0 ≤ 2κξ0 ∑ (1 − κ)s Dyy s≥ ε /2 1  ˜ + κ |∆i |ξ 2 . We finally turn to the proof of (i). Use the fact that Q(k) = Q(k) 0 2 ˆ = Q(k)  κ 1 ˆ ˆ ˆ ξ0σ . |∆i |ξ02 + k,W k − κ k, 2 2  98  ˆ Apply the identity (4.59) with k = k. ˆ = Q(k)  κ 2  ∑ ξ02 −  ˆ ξ0σ k,  ≥  x∈∆i  κ 2 ξ ∑ (1 − |Avx [σ ]|) 2 0 x∈∆ i  In the last inequality we used that |kˆ x | ≤ ξ0 . Suppose x ∈ C and C is adjacent to a facet in F(∆i ), then σ varies in Cx(1) . To obtain an upperbound on (1 − Avx [σ ]) consider the case where σ ≡ +1 in Cx(1) −Cx while σ ≡ −1 on Cx . 1 − |Avx [σ ]| = 1 −  |C| (|C(1) | − |C|) − |C| = 2 (1) (1) |C | |C |  However, this bound holds for any x in which is in a cube directly adjacent to a facet in F(∆i ). ˆ ≥ κξ02 N(Fi ) Q(k)  Proof of pre-Peierls estimate.  |C|2 ≥ κξ02 N(Fi ) |C(1) |  d(1−2ε) 1  Propositions 4.24, 4.27 and 4.28 suffice to finish the bound. The  final part of the proof are tedious computations to show that the error terms are small and do not disturb the exponential decay obtained in Proposition 4.27 and Proposition 4.28. c Recall that Zx(i) ˇ was uniformly distributed in Cxˇ and let BΓ = Γ − BΓ . Define the matrix Qt by  ξ , Qt ξ  0  = ξ , Hξ  0 BΓ  +  ∑ ∑ Avy [ξ 2 ] − t(1 − κ)ψ σ (y)2 − ta1  ϕ σ (y)2 + ϕZσxˇ (y)2 .  x∈ ˇ Vˇ (BcΓ ) y∈Cxˇ  Hence, the matrix Qt is given by D − At with ξ , Dξ  0 Γ  =  ∑ Avx [(ξ ⊕ 0)2 ],  x∈Γ  and ξ , At ξ  0 Γ  =  ∑  ψx , HC ψx 0xˇ + t  x∈ ˇ Vˇ (BΓ )  ∑ ∑ (1 − κ)ψ σ (y)2 + a1  ϕ σ (y)2 + ϕZσxˇ (y)2  x∈ ˇ Vˇ (BcΓ ) y∈Cxˇ  The diagonal elements of D are αx (Γ). Condition 2 and Proposition A.6 ensure that  ∑ |At (x, y)| ≤  y∈Γ  1−  κ αx (Γ), 2  where κ was chosen small enough such that 1 − κ2 ≥ λ . Therefore, Proposition A.2 can be applied and Qt−1 is a positive definite matrix with  ∑ |Qt−1 (x, y)| ≤ ∑  x∈X  s≥0  99  1−  κ 2  s  = 2κ −1 .  (4.60)  Propositions 4.24, 4.27 and 4.28 show that, after a change of variables ξ → ξ + v, the contourweight Wη (Γ|ξ¯ ) satisfies 7 2 N(Γ)− 4δ ˇ ¯ ˆ 1 (NLF (I)+F(∆))  Wη (Γ|ξ¯ ) ≤ ea0 (log |C|)  1  e− 2  g,H ˆ gˆ  µΓ  Γ  NQ1 (Γ) NH (Γ|ξ )  · ❊ΓQ1 exp  ∑  y∈❚(n)  ξy v˜y  .  (4.61)  Combine the bound (4.60) with the Schur test [61, Corollary 3.7].  ❊ΓQ1 e∑x∈Γ ξx v˜x = e 2 1  v,Q ˜ −1 1 v˜  Γ  ≤ e2κ  −1  v, ˜ v˜  Γ  Recall that by definition |v˜y | is bounded by a0 (log |C|)2 |C|−1 in the region BcΓ . In the region BΓ there is an exponentially small dependence on ξ . 1 ˇ ❊ΓQ1 e∑x∈Γ ξx v˜x ≤ exp N(Γ) + a2 λ 4  −ε 1  ∑  ξ 2x  (4.62)  ∂ ext (Γ) 1  Consider NQ1 (Γ) and consider its size in comparison with NQ0 . log  NQ1 (Γ) = NQ0 (Γ) =  1  dt 0  d log NQt (Γ) dt  ∑ ∑ ❊ΓQ  t  (1 − κ)ψ σ (y)2 + a1 ϕ σ (y)2 + ϕZσxˇ (y)2  x∈ ˇ Vˇ (BcΓ ) y∈Cxˇ  The variances can be bounded using Proposition A.6. Let Avσ denote the linear transformation which maps {ξx } → {Avx [σ ξ ]} and ey the vector with all zeroes except at position x where it is 1.  ∑ ❊ΓQ [(1 − κ)Avy [σ ξ ]2 ] = ∑ t  y∈Cxˇ  ey , Avσ Qt−1 Avσ ey =  y∈Cxˇ  ≤  ∑  Avσ (ey ), Qt−1 Avσ (ey )  y∈Cxˇ  ∑ 2κ −1  Avσ (ey ), Avσ (ey ) = 2κ −1  y∈Cxˇ  The same computations, in conjunction with Proposition A.6, show that a1  ∑ ❊ΓQ [ϕ σ (y)2 + ϕZσ (y)2 ] ≤ 1 t  xˇ  y∈Cxˇ  Therefore, it has been shown that  N1 (Γ) ˇ ≤ e2N(Γ) . N0 (Γ)  100  |C| ≤1 |C(1) |  The matrix HC is positive definite which implies that  ∑ Avx [(ξ ⊕ 0)2 ] − ∑  0 xˇ  ψx , HC ψx  0 Γ.  ≤ ξ , Hξ  x∈ ˇ Vˇ (BΓ )  x∈Γ  Hence, we find that 1  e− 2  g,H ˆ gˆ  Γ  NQ1 (Γ) NH (Γ|ξ )  1  ˇ  ≤ e2N(Γ) e− 2  g,H ˆ gˆ  Γ  NQ0 (Γ) NH (Γ|ξ )  1  ˇ  ≤ e2N(Γ) e− 2  g,H ˆ gˆ  NH (Γ)  Γ  NH (Γ|ξ )  In the denominator make the change of variables ξ → ξ + gˆ which changes the boundary condition into zero boundary condition. ξ + g, ˆ Hξ + gˆ  Γ  = g, ˆ H gˆ  Γ+  ξ , Hξ  0 Γ  No linear terms appear because ξ has zero boundary condition and Lemma A.3 ensures that the linear term ξ ⊕ 0, H gˆ  Γ  = 0. The conclusion of this part of the computations is 1  e− 2  g,H ˆ gˆ  Γ  NQ1 (Γ)  ˇ  ≤ e2N(Γ)  NH (Γ|ξ )  (4.63)  Recall the definition of Wη (Γ|ξ ). Substitute (4.62) and (4.63) into (4.61). Wη (Γ|ξ ) ≤  2 N(Γ) ˇ  e2a0 (log |C|)  ∑  e−  7 4δ ˇ 1 (N(I)+F(∆))  1 ε 1  exp[λ 4  ∑  ξ 2x ]  x∈∂ ext (Γ) 1  η∈Σ(η ) 2 N(Γ) ˇ  ≤ e4a0 (log |C|)  e−  7 4δ ˇ 1 (N(I)+F(∆))  1 ε 1  exp[λ 4  ξ 2x ]  ∑  x∈∂ ext (Γ) 1 ˇ  In the final step we used that there are at most 2N(Γ) spin configuration in Σ(η ). Convert from counting cubes at scale  −  to counting cubes at scale  ˇ 4a0 (log |C|)2 N(Γ) = 4a0 (log |C|)2  +  .  |C+ | ˆ · N(Γ) = 4a0 (log |C|)2 |C|  2εd 1  ˆ · N(Γ) ≤  3εd ˆ 1 N(Γ)  ˆ By the very definition of contours there must be at least 5−d N(Γ) cubes in Γ where either a largefluctuation or neighbour-clash event occurs. Wη (Γ|ξ ) ≤ e(  7 3εd −5−d 4 δ )N(Γ) ˆ 1 1  1 ε 1  exp λ 4  ∑  ξ 2x  x∈∂ ext (Γ) 1  The proof of the pre-Peierls estimate (4.37) is completed if ε is chosen small enough (but only depending on dimension) while  1  must be large enough.  101  4.6  Proof of the Peierls estimate  All the of the necessary work to prove Theorem 4.5 has already been done in Section 4.4 and Section 4.5. The whole proof boils down to making the right definitions and applying the results of these sections in the proper way. The theorem we aim to prove is the Peierls estimate. Theorem 4.5. Let δ =  d 12  and take Γ ∈ Ψ. There exist ε = ε(λ , d) such that for δ ·N(Γ) ˆ 1  νn (Γ) ≤ e−  1  large enough  .  The bound holds uniformly for all those n such that Γ ∈ Ψ(n) . Along the way we will need a supplementary claim to bound the sum over all possible unrestricted contour ensembles. Claim 4.31. For n large enough  ∑  δ N(sp(ω ˆ u )) 1  e−  ≤ nd e−n  (n) ωu ∈Ωu  Proof. An ensemble ωu has m = m(ωu ) ≥ 1 contours. All the contours in ωu have a diameter exceeding  L(n) 10  and as such contain at least  −1 L(n) + 10  =  n 10  cubes at scale  +  . Release the constraint that  contours in ωu cannot be adjacent to obtain an upper-bound.  ∑ ∏ e− (n) ωu ∈Ωu Γ∈ωu  δ N(Γ) ˆ 1  ≤  ∑  m≥1  ∑  x∈ ˆ Vˆ (❚(n) )  ∑  e−  δ N(Γ) ˆ 1  m  Γ:Γ xˆ n ˆ N(Γ)≥ 10  The sum over xˆ ∈ Vˆ (❚(n) ) selects a root for the contours. The rooted sum can be bounded by the combinatorial result Lemma B.1. e−  ∑  δ N(sp(ω ˆ u )) 1  ≤  ∑ (nd e−2n )m ≤ nd e−n ,  m≥1  (n) ωu ∈Ωu  for n large enough. Proof of Theorem 4.5. Fix a contour Γ and define the set  {ω ∈ Ωu(n) : ω ∪ {Γ} ∈ Ω(n) } u u Ωu (Γ) = {ω ∈ Ωu(n) : ω Γ} u  u  if Γ is restricted, if Γ is unrestricted.  Use the Gaussian polymer models of Lemma 4.9 to find an exact expression for νn (Γ).  102  As an intermediate step define some auxiliary partition functions. If Γ is a restricted contour define (n)  m  (n)  Zr (Γ) = 2NH ·  ∑  m≥0  ∑  (n) ✶{∃i:Xi contains contour Γ} ❊(n) H ∏ K+ (Xi , ξ ) ,  ❚  i=1  {X1 ,...,Xm }⊂ (n) Xi ∩X j =0/ if i= j  (n)  while for unrestricted contours set Zr (Γ) = 0. For any contour, restricted or unrestricted, and η ∈ Σ˜ u (ωu ) define (n)  (n)  Zu (ωu , Γ|η) = NH ·  ∑  m≥0  ∑  ✶{∃i:Xi contains contour Γ} ✶{  ❚  m  (n) (n) Xi ⊃ωu } ❊H ∏ Ku (Xi , ξ |η) . i=1  {X1 ,...,Xm }⊂ (n) Xi ∩X j =0/ if i= j  The probability to encounter the contour Γ is (n)  νn (Γ) =  (n)  Zr (Γ) Zu (ωu , Γ|η) + ∑ . ∑ (n) Z Z (n) ωu ∈Ωu (Γ) η∈Σ˜ (ω ) u  u  The expression for νn (Γ) is not yet in the desired form to use the results of Section 4.5. Define some more auxiliary partition functions. (n) (n) Z˜ u (ωu |η) = NH  m  ∑  m≥0  ∑  (n) ❊(n) H ∏ Ku (Xi , ξ |η) .  ❚  i=1  {X1 ,...,Xm }⊂ (n)  Xi ∩X j =0/ if i= j (n)  (n)  In comparison with Zu (ωu , Γ|η) the partition function Z˜ u (ωu ) has dropped the constraint that the (n) contours in ωu must always be present. However, Z˜ u (ωu |η) only has the unrestricted contours in ωu and restricted contours are forced to be located in the complement of sp(ωu ).  ∑  (n) Z˜ u (ωu |η) ≤ Z (n)  and  (n)  Zr ≤ Z (n)  (4.64)  η∈Σ˜ u (ωu )  The bound on νn (Γ) becomes (n)  νn (Γ) ≤  Zr (Γ) (n)  Zr  (n)  +  ∑  ∑  ωu ∈Ωu (Γ) η∈Σ˜ u (ωu  Zu (ωu , Γ|η) . ˜ (n) ) Zu (ωu |η)  (4.65)  Each of the terms on the right-hand side can be bounded using the cluster expansion. In fact it is possible to show (n)  3 Zr (Γ) ≤ e− 2 (n) Z (Γ)  −δ ˆ 1 N(Γ)  (n)  and  ∑  η∈Σ˜ u (ωu  3 Zu (ωu , Γ|η) ≤ e− 2 (n) ˜ ) Zu (ωu |η)  103  −δ ˆ 1 N(sp(ωu )∪Γ)  .  (4.66)  We first show how (4.66) finishes the proof of the Peierls estimate then we turn to the proof of (4.66). Pick a contour Γ and substitute (4.66) into (4.65). 3 −δ ˆ 1 N(Γ)  νn (Γ) ≤ e− 2  1+  ∑  3 −δ ˆ 1 N(sp(ωu )−Γ)  e− 2  ωu ∈Ωu (Γ)  There is the special case when Γ is an unrestricted contour and ωu = {Γ}. For the other cases ωu − Γ (n)  is contained in Ωu . 3 −δ ˆ 1 N(Γ)  νn (Γ) ≤ e− 2  2+  3 −δ ˆ 1 N(sp(ωu ))  e− 2  ∑  ≤ e−  −δ ˆ 1 N(Γ)  ωu ∈Ωnu  For the last inequality Claim 4.31 was used. (n)  The proofs for the inequalities in (4.66) are almost identical. We will discuss the term  Zr (Γ) (n)  Zr  in  full detail and then sketch the argument for the unrestricted partition functions. Suppose Γ is a restricted contour. Recall the definitions (4.25) and (4.28) of the cube-weight and contour-weight, respectively. For t ∈ ❈ construct the t-weights Ft (Cx+ˆ , ξ ) = F(Cx+ˆ , ξ ) et ·W+ (Γ, ξ ) if Γ = Γ  Wt (Γ , ξ ) =  W+ (Γ , ξ ) else  The variable t marks the contour Γ. Create the auxiliary polymer model with polymer activities (n)  Kt (X, ξ ). The activities are are defined as in Definition 4.14 with the weights W+ (ξ ) replaced by the t-weights Wt (ξ ). (n)  Zt  (n)  = NH ·  m  ∑  ∑  m≥0  ❚  (n) ❊(n) H ∏ Kt (Xi , ξ ) ,  {X1 ,...,Xm }⊂ (n)  i=1  Xi ∩X j =0/ if i= j (n)  (n)  If t = 0, then the polymer model Zr is recovered. Moreover, the derivative of Zt to t gets rid of all terms in  (n) Zt  with respect  which do not contain the contour Γ. (n)  Zr (Γ) (n) Zr  =  d (n) log Zt dt  104  t=0  (4.67)  The cluster expansion provides an equivalent expression for the derivative of the logarithm. d (n) log Zt dt  = t=0  =  d dt  1 ∑ m! m≥1 (A  d dt  1 m≥1 m! (A  m  Jc (A1 , . . . , Am )  j=1  1 ,...,Am )  t=0  m  ∑  ˆ = N(Γ)  (n) (n) ∏ ❊H [❑t (A j )]  ∑  (n) (n) ∏ ❊H [❑t (A j )]  ∑ 1 ,...,Am )∈Am (Γ)  d (n) P (Γ) dt t  Jc (A1 , . . . , Am )  j=1  t=0  t=0  The derivative got rid of all sequences (A1 , . . . , Am ) ∈ / Am (Γ). (n)  The function Pt (Γ) is analytic in t. Set α = e3  −δ ˆ 1 N(Γ)  and compute the derivative via Cauchy’s  differentiation formula. d (n) P (Γ) dt t  1 2πi  = t=0  (n)  {|t|=α  Pt (Γ) dt t2  Take absolute values to obtain an upper-bound. d (n) P (Γ) dt t  t=0  (n) ≤ α −1 sup P˜t (Γ) = e−3  −δ ˆ 1 N(Γ)  t:|t|=α  (n) sup P˜t (Γ) t:|t|=α  (n) We next show that the polymer model with weights K˜t (X, ξ ) fits into the framework of the  generalized polymer model of Section 4.5. Theorem 4.19 ensures that the contour-weights satisfy |Wt (Γ, ξ )| ≤ e  ˆ 3 δ1 N(Γ)  |Wη (Γ|ξ )| ≤ e · 3  ∑  ˇ N(Γ) (3 δ1 −  e  3δ 2 ˆ 1 )N(Γ)  η∈Σ+ c (Γ)  ≤e  ˆ − δ1 N(Γ)  exp λ  1 ε 1  exp λ 4  ∑ ξx2 x∈Γ  1 ε 4 1  ∑  ξx2  ,  x∈Γ  while the cube-weights satisfy |Ft (Cx+ˆ , ξ )| ≤ Take  1  −d/5 1  |C+ |  ∑  exp  x∈Cx+ ˆ  1−λ 4  ∑  ψy2 + ϕx (y)2 .  y∈Cx  large enough, then Theorem 4.18 shows that there exists some constant a0 such that d (n) P (Γ) dt t  t=0  ˆ ≤ a0 N(Γ)  The final step is to substitute the inequality into (4.67). (n)  Zr (Γ) ˆ ≤ a0 N(Γ) · e−3 Z (n) (Γ)  105  −δ ˆ 1 N(Γ)  3 −δ ˆ 1 N(Γ)  ≤ e− 2  (4.68)  Consider the second inequality in (4.66). Let ωu ∈ Σu (Γ) and fix η ∈ Σ˜ u (ωu ). Define t-weights which not only mark the contour Γ but every single contour in ωu as well. If Γ is restricted let (n)  ωu = ωu ∪ {Γ} otherwise ωu = ωu . Using the exact same method as for the ration  Zr (Γ) (n)  Zr  one finds  (n)  Zu (ω, Γ) ≤ e−2 (n) Z˜ u (ωu )  δ N(sp(ω ˆ u )∪Γ) 1  ˆ The final step is to sum over η ∈ Σ˜ u (ωu ). The set Σ˜ u (ωu ) contains at most 3N(sp(ωu )∪Γ) spin config-  urations. Substitute this into the bound.  4.7  Phase coexistence  In this section we present the proof of Theorem 4.3 on the coexistence of phases. The first step is to make the important reduction that unrestricted contour ensembles can be ignored in the analysis of νn as n → ∞. The convergence theorem for cluster expansion Theorem 4.18 is used to prove both the local convergence of measures and the decay of correlations.  4.7.1  Reduction and restatement of theorem  Both restricted and unrestricted contours can occur. However, it turns out that for local convergence the unrestricted contours do not play a role. Define the restricted measure νnr by dνnr (ξ ) =  1  d ❚ (ξ )e−S(❚ (n)  ∑  ∑  (n) Zr ω∈Ω(n) η∈Σ(n) (ω) r  (n) ;ξ )  χ η (ξ )❚(n) .  (4.69)  It is shown that to prove local convergence it suffices to prove the following seemingly weaker result. Theorem 4.3 . If the cube partition function Z (C) is well-adapted, then (i) the measures νnr converge locally to ν∞ = 12 ν + + 21 ν − . (ii) The measures ν ± are ergodic, translation invariant with an exponential decay of correlations (iii) Let δ =  d 12 ,  then there exists ξ0 > 0 such that ν ± ({ξ : | sup ψx ∓ ξ0 | < γ d/2−δ }) > 1 − e− 1 , δ  x∈C  for any cube C.  106  It suffices to consider the measure νnr because we claim that |νnr (G) − νn (G)| → 0, as n → ∞ for any local function G. Define the auxiliary partition functions d ❚ (ξ )G(ξ )e−S(❚ (n)  (n)  Zr (G) =  ∑  ∑  (n) ;ξ )  χ η (ξ )❚(n) ,  (4.70)  (n) (n) ω∈Ωr η∈Σ (ω)  (n)  and for ωu ∈ Ωu  (n)  Zu (ωu , G) =  d ❚ (ξ )G(ξ )e−S(❚ (n)  ∑  (n) ;ξ )  χ η (ξ )❚(n) ,  (n)  η∈Σu (ωu )  The unrestricted decomposition (4.14) shows that (n)  Zr (G) + νn (G) = Z (n)  (n)  =  νnr (G)  (n)  (n)  (n)  (n)  Zu (ωu , G) Zr (G) Zr Zu (ωu , G) = + ∑ ∑ (n) (n) (n) Z Z Z (n) (n) (n) Zr ωu ∈Ωu ωu ∈Ωu  Zr −1+1 + Z (n)  (n)  Zu (ωu , G) ∑ Z (n) (n)  ωu ∈Ωu  Use the fact that |G| ≤ 1 to bound the difference. (n)  |νnr (G) − νn (G)|  Zr ≤ (n) − 1 + Z  (n)  Zu (ωu ) ∑ Z (n) = 2 (n)  ωu ∈Ωu  (n)  Zu (ωu ) ∑ Z (n) (n)  ωu ∈Ωu  The final equality is due to the unrestricted decomposition (4.14). The difference can be interpreted in terms of unrestricted contours. |νnr (G) − νn (G)| ≤ 2νn ∃ contour Γ : diam(Γ) >  L(n) ≤2 10  ∑  νn (Γ)  (u)  Γ∈Ψn  The right-hand side is twice the probability that there exists an unrestricted contour. Claim 4.7 shows that the right-hand side tends to zero as n → ∞.  4.7.2  Finite volume measures νn± and local convergence  In this part we define finite volume measures νn± and proof part (i) of Theorem 4.3 . The measures ν ± mentioned in Theorem 4.3 arise as the local limits of finite volume measures νn± . (n)  Recall from Definition 4.11 that I0 (ω) is the exterior component and that η ∈ Σr (ω) is constant  107  either ±1 in I0 (ω). The measures νn± distinguish the cases where η is ±1 in I0 (ω). (n)  (n)  Σ± (ω) = {η ∈ Σr (ω) : η ≡ ±1 in I0 (ω)} Define the finite volume measures ν ± dνn± (ξ ) =  1  d ❚ (ξ )e−S(❚ (n)  ∑  ∑  (n) (n) Z± ω∈Ω(n) r η∈Σ± (ω)  (n) ;ξ )  χ η (ξ )❚(n) .  (4.71)  Comparing the definitions (4.71) with (4.69) it clearly follows that νnr = 12 νn+ + 21 νn− . Hence, to complete the proof of (i) it suffices to show that νn± converge locally. By symmetry it suffices to consider the measure νn+ .  Take any local function G and suppose sp(G) contains the origin. Fix t ∈ ❈ with |t| ≤ 2 and let  a0 > 0 be a small constant independent of γ. Define the partition function (n)  Zt  =  d ❚ (ξ )a0  ˇ 4N(sp(G)) tG(ξ ) −S(❚(n) ;ξ ) χ  (n)  ∑  ∑  e  e  ❚(n) .  η (ξ )  (4.72)  (n) (n) ω∈Ωr η∈Σ+ (ω)  The measure of G can be obtained via a derivative of the logarithm. νn+ (G) =  d (n) log Zt dt  t=0 (n)  The steps to prove convergence are clear. First re-express the partition function Zt  as a Gaussian (n)  polymer model. Then use the cluster expansion to show that the derivative of log Zt  converges  uniformly in n. (n)  Polymer model. The proof that Zt  can be written as a Gaussian polymer model is very similar (n)  to the argument for the partition function Z± in Lemma 4.9. We state the result and highlight the places where the proofs differ. (n)  Lemma 4.32. There exist polymer activities Kt (X, ξ ) such that (n)  Zt  (n)  = NH · ∑  n≥0  Symmetry reduction.  ∑  ❚  ✶{  n  Xi ⊃sp(G)}  (n) ❊(n) H ∏ Kt (Xi , ξ ) .  (4.73)  i=1  {X1 ,...,Xn }⊂ (n) Xi ∩X j =0/ if i= j  The first step for restricted contours was to use symmetry. By flipping the  field ξ the compatibility condition between contours could be released. The argument relied on the fact that S(ξ ; x) ˇ is a symmetric function of ξ . However, the function etG is not symmetric. Therefore, the contours who have the factor etG in their interior have to be treated separately. The 108  (a) The enlarged contour Γ(e)  (b) An example of a G-contour  Figure 4.7: Contours for a local function G coming technical definition of enlarged contours is easily illustrated in Figure 4.7(a). Definition 4.33 (Enlarged contour). (n)  (i) A contour Γ ∈ Ωr surrounds G if sp(G) is strictly contained in one of the interior components of Γ. Let IG (Γ) denote this unique interior component. (ii) Let Q be the collection of cubes which are within distance 2  +  of the origin. Define Qmax as  the maximum max{x1 : (x1 , . . . , xd ) ∈ Q}. (iii) Define Γmax = max{x1 : (x1 , . . . , xd ) ∈ Γ} and let J(Γ) be the set of all x = (x1 , . . . , xd ) ∈ IG (Γ) which satisfy  Qmax ≤ x ≤ Γmax , 1 |x | ≤ 2 for all j = 1. j  +  (n)  (iv) Given Γ ∈ Ωr which surrounds G define the enlarged contour Γ(e) = Γ ∪ Q ∪ J(Γ) (n)  For contours Γ ∈ Ωr who do not surround G define Γ(e) = Γ. It can be assumed that the support of G is a polymer in P (n) . The definition of enlarged contours clears the way to define G-contours by also incorporating the support of G into the contours. The technical definition is illustrated in Figure 4.7(b). Definition 4.34 (G-contours). (n) (i) Given η ∈ Σr a cube C+ ∈ C + is said to be +-correct if ηxˇ = +1 for all xˇ ∈ Vˇ (C+ ∪ ∂2ext+ (C+ )).  Similarly, a cube can be −-correct. An incorrect cube is a cube which is neither ±-correct. 109  (n)  (ii) Given η ∈ Σr the pre-contour ensemble ω(η) consists of the union of all incorrect cubes with sp(G). The G-contour ensemble ωG (η) is the set of the connected components of the enlarged contours {Γ(e) : Γ ∈ ω}. A G-contour Γ is a connected component of ωG . Let sp(ωG ) be the union of the contours Γ contained in ωG . (n)  (n)  (iii) The set of all contour ensembles is ΩG = {ωG (η) : η ∈ Σr }. The set of all contours is (n)  (n)  ΨG = {Γ : Γ ∈ ωG for some ωG ∈ ΩG }. (n)  (n)  (iv) For a contour ensemble ωG define ΣG (ωG ) = {η ∈ Σr : ωG (η) = ωG } 1 (n) 10 L . Only restricted 3 (n) bounded by 10 L . Therefore,  Pick n large enough such that the diameter of sp(G) is smaller than contours are allowed and it follows that diam(ωG ) for ωG ∈  (n) ΩG  is  the definition of exterior and interior components in Definition 4.11 still applies. Just as restricted contours, the G-contours have a sign. Define the set of configurations (n)  Γ Σ+ G (Γ) := η : η ∈ Σr with Γ ∈ ωG (η) and Γ is a +-contour .  Proposition 4.35. Let  +1 if η = 0, +1 xˇ σx = −1 if η = −1, xˇ (n)  where xˇ is the center of Cx . The partition function Zt (n)  Zt  =  d ❚ (ξ )e−S(ω (n)  ∑  c ;ξ +ξ  0)  satisfies  χ+(ξ + ξ0 )ω c  (n)  ω∈ΩG  ∑  ∏  ˇ 4N(sp(G)) tG(σ ξ )  a0  e  ✶{sp(G)⊂Γ} e−S(Γ;σ (ξ +ξ0 )) χ η (σ (ξ + ξ0 ))Γ  Γ∈ω η∈Σ+ G (Γ) (n)  Proof. For any contour Γ ∈ ΨG which neither intersects with sp(G) nor surrounds G the symmetry argument of Proposition 4.12 applies directly. Consider the unique contour Γ which contains sp(G). The claim is that this contour is an exterior contour. Suppose there would be another contour Γ which surrounds Γ. In that case Γ surrounds G. However, this leads to a contradiction because Γ must contain the set Q ⊂ sp(G) ⊂ Γ. The measure νn+ ensures that all exterior contours are +-contours. Let F(Cx+ˆ , ξ ) be the cube-weights defined by (4.25). The contour-weights depend on whether Γ contains sp(G) or not. If sp(G) ⊂ Γ and η ∈ Σ+ G (Γ) define the contour weight Wη(G) (Γ,t|ξ ) =  1 (n) NH (Γ|ξ )  ˇ 4N(sp(G)) tG(σ ξ ) −S(Γ;σ (ξ ⊕ξ +ξ0 )) χ  d Γ (ξ )a0  e  110  e  η (σ (ξ  ⊕ ξ + ξ0 ))Γ , (4.74)  where the normalization constant is defined by (4.26). In all other cases define Wη(G) (Γ,t|ξ ) =  1  d Γ (ξ )e−S(Γ;σ (ξ ⊕ξ +ξ0 )) χ η (σ (ξ ⊕ ξ + ξ0 ))Γ ,  (n) NH (Γ|ξ )  (4.75)  Define the cumulative weight WG (Γ|ξ ) by WG (Γ,t|ξ ) =  ∑  Wη(G) (Γ,t|ξ ).  η∈Σ+ G (Γ)  The proof of Lemma 4.32 is completed by defining local polymer activities as sums of conditional expectations. (n)  Definition 4.36 (Local polymer activities). Let FG (X) = Fm (X, ΩG ), the restricted polymer activity is (n)  Kt (X, ξ ) =  Y ω ❊(n) H F(ξ ) WG (t, ξ ) ξ ,  ∑ (ω,Y )∈FG (X)  where ξ = ξ  ∂ int/2 (X)  . The sum over the empty set is zero.  1  Cluster expansion. Apply the cluster expansion techniques to the Gaussian polymer model of Lemma 4.32. The cluster expansion provides an equivalent expression for the derivative of the logarithm. d (n) log Zt dt  = t=0  =  d dt  1 ∑ m! m≥1 (A  d dt  1 ∑ m! m≥1 (A  ˆ = N(sp(G))  m  (n) (n) ∏ ❊H [❑t (A j )]  ∑ 1 ,...,Am )  Jc (A1 , . . . , Am )  j=1  t=0  m  (n) (n) ∏ ❊H [❑t (A j )]  ∑ 1 ,...,Am )∈Am (sp(G))  d (n) P (sp(G)) dt t  Jc (A1 , . . . , Am )  j=1  t=0  t=0  The derivative got rid of all sequences (A1 , . . . , Am ) ∈ / Am (sp(G)). The core of the proof of convergence is the claim that the cube-weights F(Cx+ˆ , ξ ) and the contour-weights W˜ G (Γ,t|ξ ) satisfy the constraints of Theorem 4.18. For the cube-weights this was achieved in Theorem 4.19. It remains to show that W˜ G (Γ,t|ξ ) satisfies a proper bound. Claim 4.37. For any t with |t| = 1, the contour-weight WG (Γ,t|ξ ) satisfies ˆ  1 ε 1  WG (Γ,t|ξ ) ≤ (6ea0 )N(Γ) exp λ 4  ∑ ξx2 x∈Γ  If Claim 4.37 holds, then by choosing a0 small enough the cluster expansion Theorem 4.18 can 111  be used. We first finish the proof of local convergence under the assumption that Claim 4.37 holds. Then we come back to prove the claim. (n)  Claim 4.38. For |t| ≤ 2 the function Pt (Γ) is analytic in t. (n)  Proof. By Morera’s theorem Pt (sp(G)) is analytic in |t| ≤ 2 if and only if (n)  Pt (sp(G))dt = 0, ∆  for any triangle ∆ which lies in the disk of radius 2 centered at the origin. (n)  The function Pt (sp(G)) has a very complicated description with factors etF(σ ξ ) deeply buried away. However, by Theorem 4.18 and Fubini’s theorem the integrals can be interchanged. (n)  Pt (sp(G))dt = ∆  1 m! m≥1 (A  m  ∑  ∏ ❊H  ∑ 1 ,...,Am )∈Am (sp(G))  (n)  [ ∆  j=1  ❑t(n) (A j )dt] Jc (A1 , . . . , Am )  Trivially, for a fixed field ξ the function etG(σ ξ ) is analytic in t.  ∆  ❑t(n) (A j )dt = 0  (n)  The function Pt (sp(G)) is analytic. Compute the derivative via Cauchy’s differentiation formula. d (n) P (sp(G)) dt t  = t=0  1 2πi  (n)  Pt (sp(G)) dt t2 {|t|=1}  (4.76)  Take the limit as n → ∞. First apply Theorem 4.18 and the Dominated Convergence Theorem to (n)  interchange the limit and integral. Apply Theorem 4.18 to conclude that Pt (sp(G)) converges uniformly in n for any fixed t with |t| = 1. This completes the proof of local convergence except for the proof of Claim 4.37. (n)  Proof of Claim 4.37. Let Γ be a G-contour in ΩG . There are two options. The easy case is when Γ does not contain sp(G). Apply the pre-Peierls estimate of Theorem 4.19 directly. W+ (Γ|ξ ) =  ˇ  Wη (Γ|ξ ) ≤ 3N(Γ) e−  ∑  3δ 2 ˆ 1 )N(Γ)  1 ε 1  exp λ 4  (n)  ∑ ξx2 x∈Γ  η∈Σ+ (Γ)  ≤ e−  δ N(Γ) ˆ 1  1 ε 1  exp λ 4  ∑ ξx2 x∈Γ  For a contour Γ containing sp(G) we would have to derive the pre-Peierls estimate again. For a fixed η we give a heuristic proof by picture using Figure 4.8. The argument can easily be made rigorous. 1 ε  The factor exp λ 4 1 ∑x∈Γ ξx2 came from the fact that in ∂ int (Γ) there were no large fluctuations + from ±ξ0 . The path J was chosen thick enough such that this is still the case. Next we concentrate on the decay factors. 112  The exact same argument which was used in the pre-Peierls estimate shows that one obtains factors e−  −3 2 δ (N(Γ ˆ 1 )+N(Γ ˆ 2 )) 1  .  In the region X = sp(G) ∪ J − Γ1 − Γ2 the spin variable η is either ±1. By Proposition 4.22 the contribution from the region X is at most ˇ ˆ ˆ N(sp(G)) N(sp(G)) N(J)  2  ea0 ˇ N(sp(G))  The factor a0  2  ˇ ˆ ˆ N(sp(G)) N(sp(G)) N(J)  ≤ ea0  2  2  .  (n)  occurred in the definition of Zt . The factor e comes from etG(σ ξ ) and the fact  that both |t| ≤ 1 and |G| ≤ 1. ˆ  The factor 2N(J) is harmless because J is only there to connect a contour to sp(G). In this case J connects Γ2 to sp(G). The volume of Γ2 is at least the volume of J because Γ2 surrounds sp(G). Borrow a bit from the exponential decay of Γ2 . ˆ  2N(J) e−  − 32 δ ˆ 2) N(Γ 1  3 1 −2δ ˆ N(J) 1  ≤ e− 4  3 1 −2δ ˆ N(Γ2 ) 1  e− 4  Finally we have to count all possible spin configurations η which could produce the contour Γ. ˇ  There are at most 3N(Γ) of those. Combining the bounds yields ˆ  1 ε 1  WG (Γ|ξ ) ≤ (6a0 )N(Γ) exp λ 4  ∑ ξx2  .  x∈Γ  Figure 4.8: Example of G-contour which contains sp(G)  4.7.3  Decay of correlations and concentration of Gaussian averages  In this section we prove parts (ii) and (iii) of Theorem 4.3. The proof of part (ii) goes exactly along the lines of the proof of local convergence in Section 4.7.2. The Peierls estimate is used to prove  113  that the Gaussian field averages concentrate near ±ξˆ in the measures νn± . The measures ν ± are defined as the local limit of νn± and thus will satisfy the same bound. Correlations. Translation invariance follows by the definition of the finite volume measures. Ergodicity of the measures ν ± follows immediately if there is an exponential decay of correlations. Let G1 and G2 be two local functions. Analogous to (4.72), define a grand partition function with two complex parameters s,t. (n)  Zt1 ,t2 =  d ❚ (ξ )a0  ˇ ˇ 4N(sp(G ❚(n) . 1 )) t1 G1 (ξ ) 4N(sp(G 2 )) t2 G2 (ξ ) −S(❚(n) ;ξ ) χ e a0 e e η (ξ )  (n)  ∑  ∑  (4.77)  (n) (n) ω∈Ωr η∈Σ+ (ω)  The correlation between G1 and G2 is obtained via a double derivative. νn+ (G1 G2 ) − νn+ (G1 )νn+ (G2 ) =  d d (n) log Zt1 ,t2 dt1 dt2  t1 =t2 =0  (n)  The usual strategy is employed. Rewrite Zt1 ,t2 as a Gaussian polymer model. Show that the weights of the polymer model satisfy the constraints of Theorem 4.18, apply Theorem 4.18. (n)  We give a short description how to find the polymer activities Kt1 ,t2 (X, ξ ) for the Gaussian (n)  polymer model of Zt1 ,t2 . The support of both G1 and G2 can be assumed to be non-empty polymers in P (n) . Pick cube-centers xˆ1 and xˆ2 with xˆi ∈ sp(Gi ). Define the enlarged contours as was done in Definition 4.33. If sp(Gi ) is contained in the interior of Γ, then connect Γ via the unique path Ji to Cx+ˆi . Some contours Γ might get connected to sp(G1 ), some so sp(G2 ) and some to both sp(G1 ) and sp(G2 ). Repeat the definition of Definition 4.34 to define G12 -contours. The symmetry argument (n)  can be applied to obtain a final definition of Kt1 ,t2 (X, ξ ) alike to Definition 4.36. Having the polymer activities one can use the cluster expansion and then take the derivatives with respect to s and t. d d (n) logZt1 ,t2 dt1 dt2 =  t1 =t2 =0  d d dt1 dt2  =  d d dt1 dt2  1 ∑ m! m≥1 (A  1 m! m≥1 (A  ∑  m  ∑ 1 ,...,Am )  (n) (n) ∏ ❊H [❑s,t (A j )]  Jc (A1 , . . . , Am )  j=1  t1 =t2 =0  m  (n) (n) ∏ ❊H [❑s,t (A j )]  ∑ 1 ,...,Am )∈Am (sp(G1 ),sp(G2 ))  d d (n) ˆ = N(sp(G Pt1 ,t2 (sp(G1 ), sp(G2 )) 1 )) dt1 dt2  j=1  Jc (A1 , . . . , Am )  t1 =t2 =0  t1 =t2 =0  If either a factor et1 G1 (ξ ) or et2 G(ξ ) is missing, the double derivative will be zero. The only terms which survive both derivatives must be in Am (sp(G1 ), sp(G2 )). (n)  Morera’s theorem can be used to show that the function Pt1 ,t2 (sp(G1 ), sp(G2 )) is analytic in t1 ,t2 .  114  Use the multi-dimensional Cauchy’s differentiation theorem. d d (n) Ps,t (sp(G1 ), sp(G2 )) dt1 dt2  (n)  Pt1 ,t2 (sp(G1 ), sp(G2 )) dt1 dt2 s2t 2 |t1 +t2 |=1  1 = t1 =t2 =0 (2πi)2  A simple bound yields d d (n) P (sp(G1 ), sp(G2 )) dt1 dt2 s,t  (n)  t1 =t2 =0  ≤ max |Pt1 ,t2 (sp(G1 ), sp(G2 ))|. t1 .t2  By the argument provided for the proof of local convergence the polymer activities satisfy the assumptions of Theorem 4.18 which in turn states 1 −1 + dist(sp(G1 ),sp(G2 )  (n)  |Pt1 ,t2 (sp(G1 ), sp(G2 ))| ≤ a1 λ − 4  .  Concentration. The concentration of Gaussian averages is a standard counting exercise using the Peierls estimate. To prove part (iii) it suffices to show that ηx = +1 with high probability. Consider the measure νn+ . If x is neither contained in a contour nor surrounded by a contour, then ηx = +1. Use the Peierls estimate and the combinatorial bound Lemma B.1. νn+ ({x contained in contour}) ≤  ∑  νn+ (Γ) ≤  (n) Γ∈Ψr  δ N(Γ) ˆ 1  (n) Γ∈Ψr  Γ x  ≤  e−  ∑ Γ x  ∑ (e3d e−  δ N(Γ) ˆ 1  k≥m  1 1 )k ≤ e− 2 2  δ N(Γ) ˆ 1  .  Let L = {xˆ ∈ Vˆ (❚(n) ) : xˆ = (xˆ1 , . . . , xˆd ) with xˆ1 = x1 } be the set of cube-centers on a line from x. If Γ surrounds Cx+ , then it must cross through one the cube-centers xˆ in L. It may do so multiple times, let yΓ be the point which has smallest distance to x. If Γ has crossing point yΓ , then it must contain at least  −1 dist(x, y ) Γ +  translating x by a distance k  +  cubes. Given x and k ∈ ◆, let xk denote the cube-center obtained by along the line L. L(n)  νn+ ({x  surrounded by contour}) ≤  ∑ ∑  L(n)  νn+ (Γ)  k=1 Γ∈Ψ(n) r Γ xk  ≤  ∑ ∑ (e3d e−  k≥1 k ≥k  Combine the two bounds to finish the proof.  115  ≤  ∑ ∑  e−  δ N(Γ) ˆ 1  k=1 Γ∈Ψ(n) r Γ xk δ N(Γ) ˆ 1  1 1 )k ≤ e− 2 2  δ N(Γ) ˆ 1  .  4.7.4  Equivalent definition of ν ±  The measures ν ± are the local limits of the finite volume measures νn± . In this section we define measures {ν˜ n± } and show that they converge locally to ν ± as well. The measures ν˜ n± are simpler than νn± . In the upcoming Chapter 5 the measures ν˜ n± are used to prove a phase transition for the particle models introduced in Chapter 2. The condition used to define the measures νn± was that all contours are  Finite volume measures.  restricted and that η ≡ ±1 in I0 (ω). The new measures will have an easier condition namely that η is fixed at ±1 in Bcn where Bn  ❚(n) is a sufficiently large region.  Given X let xˆmin (X) be the minimal element of Vˆ (X) with respect to the lexicographical order. Given a local function F let Bn (F) be the set of cubes C+ which are within distance of xˆmin (sp(F)). Recall that µn (dξ ) is the Gaussian measure of {ξx , x ∈ ❚ standard Gaussian random variables. Define the measures ν˜ n± by  (n) }  ν˜ n± (F) =  1 ˜ Zn  µn (dξ )F(ξ )  x∈ ˇ Vˇ (❚(n) )  Z (C) (ψx1 +xˇ , . . . , ψxM +xˇ )χ ±n  ·  +  where ξx are i.i.d.  B (F)c  ∏  n 20  ,  (4.78)  and Z˜ n± =  µn (dξ )  B (F)c  ∏  x∈ ˇ Vˇ (❚(n) )  Z (C) (ψx1 +xˇ , . . . , ψxM +xˇ )χ ±n  .  (4.79)  Observe that by translation invariance the grand partition function Z˜ n does not depend on F. Given a local function F define the set of spin configurations (n) Σ˜ ± (F) = {η ∈ Σ(n) : ηx = ±1 in Bn (F)c }.  ˜ (n) Contours are still defined by Definition 4.4. Let Ω ± be the set of all contour ensembles obtained (n) (n) (n) (n) from Σ˜ ± . Similarly, for ω ∈ Σ˜ ± let Σ˜ ± (ω) denote all η ∈ Σ˜ ± whose contour ensemble is given by ω. Repeating the argument used in Section 4.1 the measures ν˜ n± are equivalently defined by d ν˜ n± (ξ ) =  1  d ❚ (ξ )e−S(❚ (n)  ∑  ∑  (n) Z˜ ± ω∈Ω˜ (n) ˜ (n) r η∈Σ ± (ω)  (n) ;ξ )  χ η (ξ )❚(n) .  (4.80)  Recall the identity (4.76) in Section 4.7.2 which is νn± (F) =  1 2πi  {|t|=1}  1 1 ∑ m! t 2 m≥1 (A  m  (n) (n) ∏ ❊H [❑t (A j )]  ∑ 1 ,...,Am )∈Am (sp(F))  Jc (A1 , . . . , Am ) dt. (4.81)  j=1  (n) n Any contour in Σ˜ ± must be restricted because the box Bn (F) has radius 20 · + . Hence, the argument ± ˜ t . The only in Section 4.7.2 shows that the measure ν˜ n (F) satisfies (4.81) with ❑t replaced by ❑  116  ˜ t no contour is allowed to intersection with Bn (F)c . difference in the weight is that in ❑ ˜ t(n) (A j ) = ❑t(n) (A j ) provided sp(A j ) ⊂ Bn (F). This causes a The latter observation shows that ❑ sp(A j ) ⊂ Bn (F). A trivial bound is  large cancellation of all terms in the sum where 1 ∑ m! t:|t|=1 m≥1 (A  m  |ν˜ n± (F) − νn± (F)| ≤ sup  + sup  |Jc (A1 , . . . , Am )|  c 1 ,...,Am )∈Am (sp(F),Bn (F) ) j=1 m  1 m≥1 m! (A  ∑ t:|t|=1  (n) (n) ∏ ❊H [|❑˜ t (A j )|]  ∑  ∑  c 1 ,...,Am )∈Am (sp(F),Bn (F) )  (n) (n) ∏ ❊H [|❑t (A j )|]  |Jc (A1 , . . . , Am )| .  j=1  Recycle the argument in Section 4.7.2 that Kt satisfies the assumptions of Theorem 4.18. Hence, the polymer activities K˜t satisfy these assumptions as well. Apply Theorem 4.18 to bound the difference 1  |ν˜ n± (F) − νn± (F)| ≤ a1 λ − 4 which tends to zero as n → ∞.  117  +  −1 dist(sp(F),B  c n (F) )  ,  Chapter 5  Proof of phase transitions In this chapter we provide the proofs for Theorem 2.9 and Theorem 2.11 for a phase coexistence in the Ising and Widom-Rowlinson model. The main ingredient for both proofs is Theorem 4.3 which lists conditions under which a Gaussian measure exhibits a phase transition. The main task then is to show that these conditions are satisfied for both the Ising and WR model. Once the conditions are satisfied a proof for the coexistence of phases in the Ising and WR model is readily obtained. We recall some geometric definitions needed to state the results. The Ising and WR finite volume  ❚(n) . The torus ❚(n) for n ∈ ◆ is the graph with vertices  particle models are placed on the torus  [−L(n) , L(n) ]d ∩ ❩d where opposite faces are identified. The length-scale  −  ≈  1−ε 1  with ε positive  but small. Let Cn be the partition of ❚ into cubes C ⊂ ❚ of side-length −. Given a set X let Vˇ (X) be the set the cube-centers of all cubes C contained in X. For x ∈ ❚(n) let Cx be the unique (n)  −  (n)  cube in Cn− such that x ∈ Cx .  5.1  Ising model  Given ξ ∈ ❘❚  (n)  the field average at x was defined by ψ˚ x =  1 ξy w(x, ˚ y), w˚ y∈∑ (n) ❚  with w, ˚ w(x, ˚ y) given by (2.20). Let µn denote the standard Gaussian measure on the torus  ❚(n) .  The finite volume measures are defined by  Pn,γ,β (σ ) =  2−|❚  (n) |  (n) Zγ,β  √  ❊(n) ∏ e−  β σx ψ˚ x  ,  (5.1)  x∈❚(n)  and Zγ,β = ❊(n) (n)  ∏  x∈❚(n)  118  eV (ψ˚ x ) ,  (5.2)  where V : ❘ → ❘ is V (ζ ) = log cosh( β ζ ) and ❊(n) denotes the expectation with respect to µn . Recall that the lattice sites {x1 , . . . , xM } are the lattice sites contained in Co the cube at the origin. Define the closely related measure νn,γ,β (dξ ) =  1 (n)  Zγ,β  Z (C) (ψ˚ x1 +xˇ , . . . , ψ˚ xM +xˇ )µn (dξ ),  ∏  x∈ ˇ Vˇ (❚  (n) )  where  M  Z (C) (ζ1 , . . . , ζM ) = ∏ eV (ζi ) .  (5.3)  i=1  ± If Z (C) is well-adapted, then Theorem 4.3 can be applied to νn,γ,β . In that case let νγ,β denote the  two Gibbs states mentioned in Theorem 4.3. In order to define infinite volume measures we study local functions G(σ ). A local function G(σ ) is a bounded function which only depends on the spins in a finite set sp(G) ⊂ ❩d . We show that any local function G(σ ) can be expressed as a local function of the field ξ . This will be the reason why a phase transition in the field ξ directly translates to a phase translation in the Ising model. Claim 5.1. Given a local function G(σ ) let G f (ξ ) be defined by √ G f (ξ ) =  G(σ )  ∑ σ ∈{−1,+1}sp(G)  ∏ x∈sp(G)  e √  e−  β σx ψ˚ x  β ψ˚ x  √ +e  β ψ˚ x  .  If n is large enough such that sp(G) ⊂ ❚(n) , then  ❊n,γ,β [G(σ )] = νn,γ,β [G f (ξ )],  (5.4)  where ❊n,γ,β is the expectation with respect to Pn,γ,β . Proof. Start from the definition of the finite volume measure.  ❊n,γ,β [G(σ )] =  2−|sp(G)| (n)  Zγ,β  ❊(n)  √  ∑ σ ∈{−1,+1}sp(G)  G(σ )  ∏ x∈sp(G)  e  β σx ψ˚ x  ∏  eV (ψ˚ x )  x∈sp(G) /  The equality was obtained by using the equality √ √ √ − β ψ˚ x + β ψ˚ x e + e 2−|X| ∑ e β σx ψ˚ x = ∏ , ∏ 2 x∈X σ ∈{−1,1}X x∈X with X = ❚(n) − sp(G). Divide and multiply by the factor ∏x∈sp(G) eV (ψ˚ x ) to finish the proof. Claim 5.1 provides a clear pathway for the proof of a phase transition in the Ising model. If Z  (C)  is well-adapted, then the measures νn,γ,β have a local limit which defines a local limit for the 119  measures Pn,γ,β . Moreover, Theorem 4.3 implies the existence of a phase transition for the field ξ which then immediately transfers to a phase transition in the Ising model. We first show that if Z (C) is well-adapted, then the statement of Theorem 2.9 immediately follows from Theorem 4.3. The proof of Theorem 2.9 is then completed by showing that indeed Z (C) is well-adapted. Lemma 5.2. Suppose Z (C) is well-adapted, then Theorem 2.9 holds with the Gibbs states  P±∞,γ,β  defined by ± ❊±∞,γ,β [G(σ )] = νγ,β (G f (ξ )), ± where G is an arbitrary local function and ❊± ∞,γ,β denotes expectation with respect to P∞,γ,β .  Proof. Apply Claim 5.1 to change the local function G(σ ) into the local function G f (ξ )  ❊n,γ,β [G(σ )] = νn,γ,β (G f (ξ )). If Z (C) is well-adapted, then Theorem 4.3 applies for the field ξ and the limit as n → ∞ exists 1 2  1 2  − + ❊∞,γ,β [G(σ )] = νγ,β (G f (ξ )) + νγ,β (G f (ξ )).  Hence, parts (i) and (ii) of Theorem 2.9 follow immediately from Theorem 4.3. Compute the expected spin at location x by taking the local function G(σ ) = σx .  ❊±∞,γ,β [σx ] = ❊±∞,γ,β [σx ✶{|ψx ∓ξˆ |≤2γ d/2−δ } ] + ❊±∞,γ,β [σx ✶{|ψx ∓ξˆ |>2γ d/2−δ } ] 1 δ  By part (iii) of Theorem 4.3 the second contribution is of O(e− 2 1 ). Use Claim 5.1 to compute the expectation of σx . √ − β ψx 1 δ − e ± √ √ ✶{|ψ ∓ξˆ |≤2γ d/2−δ } + O(e− 2 1 ) ❊±∞,γ,β [σx ] = νγ,β x e β ψx + e− β ψx √ ˆ √ ˆ e β ξ − e− β ξ √ =± √ ˆ + O(γ d/3 ) βξ − β ξˆ e +e √  e  β ψx  d The final part of the proof is to show that P± ∞,γ,β define Gibbs states. Fix a finite set Λ ⊂ ❩ and  let Λ ⊂ Λc be the set of sites which have at most distance  1 2 1  to Λ. Given a spin configuration σ Λ  on Λ and σ define σ on Λ ∪ Λ by  σ Λ x σx = σ x  if x ∈ Λ if x ∈ Λ  Recall that Gibbs states are defined via (2.19). Hence, to verify that P± ∞,γ,β defines a Gibbs state one 120  must show that  1  P±∞,γ,β (σ Λ |σ ) =  2  β  ± Zγ,β (Λ|σ )  · e− 2 ∑x∈Λ mx (σ ) ,  (5.5)  where 2  β  ± Zγ,β (Λ|σ ) =  e− 2 ∑x∈Λ mx (σ ) .  ∑ σ Λ ∈{−1,+1}Λ  The identity (5.5) is easily verified using the limit definition of ν ± given in Section 4.7.4. Given X let xˆmin (X) be the minimal element of Vˆ (X) with respect to the lexicographical order. Given a local function G let Bn (G) be the set of cubes C+ which are within distance configurations on  ❚  ❚  ·  +  of xˆmin (sp(G)).  (n) .  Let Σn (σ ) be the set of all spin whose restriction to Λ is exactly σ . For the Ising model the measures ν˜ n±  Recall that µn is the standard Gaussian field on the torus (n)  n 20  defined by (4.80) are ν˜ n± =  1 µn Zn  Bc  G(σ n )e∑x∈❚(n) σx ψx χ+G , n  ∑  ˚  (5.6)  σ n ∈Σn (σ )  where Zn = µn  Bc  e∑x∈❚(n) σx ψx χ+G n  ∑  ˚  σ n ∈Σn (σ )  In (5.6) use the identity  ∑  x∈❚(n)  σxn ψ˚ x =  ∑  x∈❚(n)  m˚ x (σ n )ξx .  Integrate out the Gaussian random variables {ξx : x ∈ Λ} 2  β  Zn · ν˜ n± = ∑ G(σ Λ )e− 2 ∑x∈Λ m˚ x (σ ) µ˜ n σΛ  ∑  e∑x∈❚(n) −Λ  m˚ x ξx χ BcG +  ,  (5.7)  σ˜ n ∈Σ˜ n (σ )  where µ˜ n is the measure of the Gaussian variables {ξx : x ∈ ❚(n) − Λ} and Σ˜ n (σ ) is the set of spin configuration on ❚(n) − Λ which are fixed at σ in Λ. Similarly, one obtains that β  2  Zn = ∑ e− 2 ∑x∈Λ m˚ x (σ ) µ˜ n σΛ  ∑  e∑x∈❚(n) −Λ  m˚ x ξx χ BcG +  ,  (5.8)  σ˜ n ∈Σ˜ n (σ )  Observe that dividing out (5.7) by (5.8) yields the desired measure (5.5).  5.1.1  Well-adapted  By Lemma 5.2 it suffices to show that the cube partition function Z (C) : ❘M → ❘ defined by (5.3) is well-adapted to some symmetric double-well function Φ. The conditions to be well-adapted are  121  listed in Definition 4.2. Claim 2.14 provides a candidate for the double-well function, choose ζ2 −V (ζ ). 2  Φ(ζ ) =  The fact that Φ(ζ ) is indeed a double-well function for β > 1 is proved in Section C.1. One by one we go by the conditions for the cube partition function to be well-adapted. • Condition 0 (Symmetry) The function V (ζ ) is symmetric thus so is Z (C) . • Condition 1 (Taylor) For the Ising model choose ξ0 = ξˆ and apply the Taylor expansion for V (ζ ). M  log Z (C) (ζ1 , . . . , ζM ) = ∑ V (ξˆ ) +V (ξˆ )(ζi − ξˆ ) +V (ξˆ )(ζi − ξˆ )2 + R(ζi ) i=1  The minimizer ξˆ of Φ(ζ ) satisfies Φ (ξˆ ) = ξˆ −V (ξˆ ) = 0 or V (ξˆ ) = ξˆ . By Taylor’s Theorem the remainder satisfies M  ∑ R(ζi ) ≤ max |V  i=1  M  M  i=1  i=1  (ζ )| ∑ |ζi − ξˆ |3 ≤ β 3/2 ∑ |ζi − ξˆ |3  The third derivative of V is bounded by β 3/2 . Hence, all requirements for Condition 1 are satisfied. • Condition 2 (Finite range) Proposition C.2 shows that V (ξˆ ) = λ < 1 and ˚ · Av)(x, ˚ y) = λ α˚ x (Co ), ∑ |Axy | = λ ∑ (Av y∈Co  y∈Co  for all x ∈ Co . • Condition 3 (Double well) The condition is easily satisfied because Z (C) factors over x ∈ C M  exp  ∑  i=1  −  ζi2 ξˆ 2 +V (ζi ) + −V (ξˆ ) 2 2  122  M  = exp − ∑ (Φ(ζi ) − Φ(ξˆ )) . i=1  5.2  WR model  Given ξ ∈ ❘❚  (n)  the field average at x was defined by •  ψx = •  1 ξy w(x, y), w y∈∑ ❚(n) •  •  •  with w, w(x, y) given by (2.24). The set of lattice spin configurations is ˇ (n) ) = {σxˇi : xˇ ∈ ❚ ˇ (n) , σxˇi = ±1, 1 ≤ i ≤ kxˇ with kxˇ ∈ ◆0 }. Σσ (❚ ˇ (n) define the particle density by At each lattice point there are kxˇ ∈ ◆0 spin particles. For xˇ ∈ ❚ ρxˇ (k) =  kxˇ |C| .  The Hamiltonian in an auxiliary field ξ was defined by Hn,γ,υ,r,β (σ , ξ ) = −υβ  ∑  •  kxˇ −  β  ∑  σxˇ ψxˇ +  ˇ (n) x∈ ˇ ❚  ˇ (n) x∈ ˇ ❚  kβ |C| 2  ∑  ρxˇ (k)2 ,  ˇ (n) x∈ ˇ ❚  xˇ σxˇi . Let µn denote the standard Gaussian measure on the torus where σxˇ = ∑ki=1  ❚(n) . The finite  volume measure is defined by  Pn,γ,υ,r,β (σ ) =  1 (n)  Zγ,υ,r,β  |C|kxˇ  ❊(n) ∏ kxˇ e−Hn,γ,υ,r,β (σ ,ξ ) , ˇ (n) 2 kxˇ ! x∈ ˇ ❚  (5.9)  where Zγ,υ,r,β is the grand partition function and ❊(n) denotes the expectation with respect to µn . (n)  Recall that Zγ,υ,r,β factors over the cubes C ∈ Cn− as was shown in Lemma 2.16. For k ∈ ◆0 and (n)  ζ ∈ ❘ define the Hamiltonian  Hγ,υ,r,β (k, ζ ) = −υβ k − kV (ζ ) +  rβ |C|ρ(k)2 , 2  and the cube partition function |C|k −Hγ,υ,r,β (k,ζ ) e . ∑ k=0 k! ∞  (C) Zγ,υ,r,β (ζ ) =  The grand partition function is the product of the cube partition functions Zγ,υ,r,β = ❊(n) (n)  •  ∏  ˇ (n) ) x∈( ˇ ❚  123  (C) Zγ,υ,r,β (ψxˇ ) .  Define the closely related measure on ξ νn,γ,υ,r,β (dξ ) = To study the measure  1  •  ∏  (n) Zγ,υ,r,β x∈ ˇ Vˇ (❚(n) )  Z (C) (ψxˇ )µn (dξ ).  Pn,γ,υ,r,β on local functions it suffices to study the local functions for the  measures νn,γ,υ,r,β (dξ ). The proof of the following claim uses the computations of Lemma 2.16. Claim 5.3. Let G(σ ) be a local function of the spin configuration σ , define the local function G f (ξ ) of the field ξ by •  G f (ξ ) =  Z (C) (ψxˇ )  ∏  −1  G(σ )  ∑  σ ∈Σσ (sp(G))  x∈sp(G) ˇ  |C|kxˇ −Hn,γ,υ,r,β (σ ,ξ ) e . 2kxˇ kxˇ ! x∈sp(G) ˇ  ∏  ˇ (n) , then If n is large enough such that sp(G) ⊂ ❚  ❊n,γ,υ,r,β [G] = νn,γ,υ,r,β (G f ), where ❊n,γ,υ,r,β denotes the expectation with respect to Pn,γ,υ,r,β . Proof. The proof follows the exact same steps as Lemma 2.16.  ❊n,γ,υ,r,β [G] =  1 (n)  Zn,γ,υ,r,β  ❊(n)  G(σ )  ∑  |C|kxˇ −Hn,γ,υ,r,β (σ ,ξ ) ∏ 2kxˇ kxˇ! e ˇ (n)  x∈ ˇ ❚  ˇ (n) ) σ ∈Σσ (❚  Follow the exact same step as in the proof of Lemma 2.16 and sum over all the possible spins in the ˇ (n) − sp(G). Integrating out the spins in ❚ ˇ (n) − sp(G) we are left with a spin configuration region ❚ σ ∈ Σσ (sp(G)). The Hamiltonian for such a spin configuration in an auxiliary field ξ is Hsp(G),γ,υ,r,β (σ , ξ ) = −υβ  kxˇ −  ∑  •  x∈sp(G) ˇ  σxˇ ψxˇ +  ∑  β  x∈sp(G) ˇ  kβ |C| ∑ ρxˇ (k)2 . 2 x∈sp(G) ˇ  The expectation of G can be written as  ❊n,γ,υ,r,β [G] =  1 (n) Zγ,υ,r,β  ❊(n)  ∑  σ ∈Σσ (sp(G))  G(σ )  |C|kxˇ −Hsp(G),γ,υ,r,β (σ ,ξ ) ∏ 2kxˇ kxˇ! e ∏ c Z (C) (ψxˇ) . x∈sp(G) ˇ x∈sp(G) ˇ •  By the definition of the function G f (ξ ) it follows that  ❊n,γ,υ,r,β [G] = ❊(n) G f (σ ) ∏ Z (C) (ψxˇ) = νn,γ,υ,r,β (G f ). •  ˇ (n) x∈ ˇ ❚  The function G f only depends on ξ in the finite set of lattice sites which are within distant γ −1 of sp(G). Moreover, G f is bounded by |G|. Adopt the same strategy as for the Ising model. First show that if Z (C) is well-adapted, then the 124  statement of Theorem 2.11 immediately follows from Theorem 4.3. The proof of Theorem 2.11 is then completed by showing that indeed Z (C) is well-adapted. Lemma 5.4. Suppose Z (C) is well-adapted, then Theorem 2.11 holds with the Gibbs states P± ∞,γ,υ,r,β defined by ± ❊±∞,γ,υ,r,β [G] = νγ,υ,r,β (G f ), ± where G is an arbitrary local function and ❊± ∞,γ,υ,r,β denotes expectation with respect to P∞,γ,υ,r,β .  Proof. Apply Claim 5.3 to change the local function G(σ ) into the local function G f (ξ )  ❊n,γ,υ,r,β [G(σ )] = νn,γ,υ,r,β (G f (ξ )). If Z (C) is well-adapted, then Theorem 4.3 applies for the field ξ 1 2  1 2  − + (G f (ξ )) + νγ,υ,r,β (G f (ξ )). ❊∞,γ,υ,r,β [G(σ )] = νγ,υ,r,β  Hence, parts (i) and (ii) of Theorem 2.9 follow immediately from Theorem 4.3. For part (iii) recall the definition of the spin average  σ xˇ =    σxˇ  if kxˇ > 0,  0  else.  kxˇ  Compute the expected spin average at location x. ˇ  ❊±∞,γ,υ,r,β [σ xˇ] = ❊±∞,γ,υ,r,β [σ xˇ✶{|ψxˇ∓ξˆ |≤2γ d/2−δ } ] + ❊±∞,γ,υ,r,β [σ xˇ✶{|ψxˇ∓ξˆ |>2γ d/2−δ } ] 1 δ  By part (iii) of Theorem 4.3 the second contribution is of O(e− 2 1 ). Let G(σ ) = σ xˇ , then ± ❊±∞,γ,υ,r,β [σ xˇ] = ν∞,γ,υ,r,β [G f (σ )✶{|ψxˇ ∓ξˆ |≤2γ d/2−δ } ] + O(e− 2 1 ), 1 δ  where G f (σ ) is defined in Claim 5.3. The local function G f (σ ) is √ i k β σxˇ ψxˇ k r 2 |C| e G f (σ ) = Z (C) (ψxˇ )−1 ∑ eυβ k− 2 |C|ρ(k) σ xˇ ∏ ∑ 2 i=1 k≥0 k! σ ∈{−1,+1}k •  •  √ i k β σxˇ ψxˇ k r 2 |C| e √ , = Z (C) (ψxˇ )−1 ∑ eυβ k+kV (ψxˇ )− 2 |C|ρ(k) σ xˇ ∏ √ ∑ i=1 e+ β ψxˇ + e− β ψxˇ k≥0 k! σ ∈{−1,+1}k •  •  •  •  •  •  •  where V (ψxˇ ) = log cosh( β ψxˇ ). For a fixed k the sum over σ ∈ {−1, +1}k is an expectation with 125  respect to k i.i.d. Bernoulli random variables which are ±1 with probability √ e± β ψxˇ •  p± (ψ˚ xˇ ) =  √ √ e+ β ψxˇ + e− β ψxˇ •  •  The expectation with respect to the Bernoulli random variables is easily computed. √  k  ∑  σ xˇ ∏  √  i=1 e+  σ ∈{−1,+1}k  Let  √ √ e+ β ψxˇ − e− β ψxˇ √ √ = e+ β ψxˇ + e− β ψxˇ  •  β σxˇi ψxˇ  e •  β ψxˇ  √ + e− β ψxˇ •  •  •  •  •  ✶{k>0}  √ − e− β ζ √ , m(ζ ) = √ e+ β ζ + e− β ζ √  βζ  e  under the assumption that |ψxˇ ∓ ξˆ | ≤ 2γ d/2−δ it follows that •  |C|k υβ k+kV (ψxˇ )− r |C|ρ(k)2 2 e . k≥1 k! •  G f (σ ) = ± m(ξˆ ) + O(γ d/3 ) Z (C) (ψxˇ )−1 ∑ •  •  The definition of Z (C) (ψxˇ ) shows that G f (σ ) = ± m(ξˆ ) + O(γ d/3 ) 1 −  1 . Z (C) (ψxˇ ) •  The final step is to apply Lemma 2.18 to show that Z (C) (ψxˇ )−1 is exponentially small. •  ˜ −|C|Fψ• (ρ)  •  Z (C) (ψxˇ ) ≥ a0 e  xˇ  .  ˜ is negative and Z (C) (ψxˇ )−1 is negligiFinally, apply part (ii) of Lemma C.3 which states that Fψxˇ (ρ) ble. Combining the constants, this shows that ν ± (G f (σ )) = ±m(ξˆ ) + O(γ d/3 ). •  •  Finally, the proof that P  γ,υ,r,β  ± ∞,γ,υ,r,β  indeed define Gibbs states follows the exact same argument as  provided earlier in the proof of Lemma 5.4 for the Ising model.  5.3  WR model well-adapted  In the next few sections we show that the cube partition function Z (C) for the Widom-Rowlinson model is well-adapted to a double-well function Φ. Before stating the results we remind the reader of the exact definition of Z (C) and the candidate for Φ. The cube partition function is Z (C) (ζ ) =  |C|k |C| e k≥0 k!  ∑  2 β υρ(k)+V (ζ )ρ(k)− kβ 2 ρ(k)  126  (5.10)  where ρ(k) =  k |C|  is the particle density. Claim 2.17 on p. 27 provides a candidate for Φ which is ˜ ) be the minimizer of defined in terms of a minimization problem. Given ζ ∈ ❘ let ρ˜ = ρ(ζ Fζ (ρ) = −υβ ρ −V (ζ )ρ +  rβ 2 log ρ ρ + ρ(log ρ − 1) + 2 2|C|  (5.11)  The function F is defined by F(ζ , ρ) =  ξ2 rβ 2 log ρ − υβ ρ −V (ξ )ρ + ρ + ρ(log ρ − 1) + . 2 2 2|C|  (5.12)  ˆ are the two global minimizers of F. Define the function Φ(ζ ) by Recall that (±ξˆ , ρ) ˜ )). Φ(ζ ) = F(ζ , ρ(ζ It will be shown that Z (C) is well-adapted to Φ(ζ ). In Lemma C.3 it is shown that Φ is a symmetric double-well function for ν large enough, r > 1 and γ small enough.  5.3.1  Main results  Symmetry of Z (C) immediately follows because the function V (ζ ) is symmetric. The real work is to verify that Z (C) satisfies Condition 1 to Condition 3. Theorem 5.5. There exist positive constants a0 , ξ0 ∈ ❘ with |ξ0 − ξˆ | ≤ a0 (log |C|)2 |C|−1 such that log Z (C) (ζ ) = log Z (C) (ξ0 ) + ξ0 |C|(ζ − ξ0 ) + λ |C|(ζ − ξ0 )2 + R(ζ ), with λ < 1. Furthermore, there exists a constant a1 such that remainder satisfies |R(ζ )| ≤ a1 (log |C|)2 |C||ζ − ξ0 |3 , provided |ζi − ξ0 | ≤  ξ0 4.  Theorem 5.5 verifies Condition 1 and Condition 2. The next theorem finishes the proof that Z (C) is well-adapted to Φ by verifying Condition 3. Theorem 5.6. If |ξ0 − ξˆ | ≤ a0 (log |C|)2 |C|−1 for some constant a0 , then there exists a constant a such that  1  2  e− 2 |C|ζ Z (C) (ζ ) 1  2  e− 2 |C|ξ0 · Z (C) (ξ0 )  ≤ a0 exp − |C| Φ(ζ ) − Φ(ξˆ ) .  127  Incongruence. The statements of Theorem 5.5 and Theorem 5.6 are equivalent to Condition 1 •  and Condition 2 because the block averages ψ are constant on cubes C. Recall that for the block •  averages ψ we consider the cube partition function Z (C) (ζ1 , . . . , ζM ) with ζ1 = . . . = ζM . Hence, the statements of Theorem 5.5 and Theorem 5.6 can be equivalently stated as M  M  log Z (C) (ζ ) = log Z (C) (ξ0 ) + ξ0 ∑ (ζi − ξ0 ) + λ ∑ (ζi − ξ0 )2 + R(ζ ), i=1  and  1  M  2  e− 2 ∑i=1 ζi Z (C) (ζ ) 1  i=1  2  e− 2 |C|ξ0 · Z (C) (ξ0 )  M  ≤ a0 exp − ∑ Φ(ζi ) − Φ(ξˆ ) . i=1  Finally, Condition 2 is satisfied because λ < 1 and •  •  •  ∑ |Axy | = λ ∑ (Av · Av)(x, y) = λ αx (Co ),  y∈Co  y∈Co  for all x ∈ Co .  5.3.2  Derivatives as moments  In this section we show that the derivatives of log Z (C) (ζ ) are linear combinations of the moments of a Gibbs measure  P(C) . In subsequent sections we first compute the moments and then use the ζ  moments to compute the derivatives. Let K be a random variable with probability measure (K = k) = P(C) ζ Let  1 |C|k |C| e Z (C) (ζ ) k!  2 β υρ(k)+V (ζ )ρ(k)− rβ 2 ρ(k)  .  (5.13)  is a Gibbs measure in a ❊ζ denote the expectation w.r.t. the measure Pζ(C) . The measure P(C) ζ  fixed auxiliary field ζ . For short-hand notation the subscript ζ is suppressed. The partial derivatives of log Z (C) (ζ ) are closely related to the moments of K. The appearance of V (ζ ) in the exponent eV (ζ )k provides some extra derivatives of V . V (ζ ) = V (ζ ) =  β tanh( β ζ ), β cosh( β ζ )2  V (ζ ) = −β 3/2  ,  tanh( β ζ ) cosh( β ζ )2  128  (5.14) .  The derivatives of log Z (C) (ζ ) are d log Z (C) (ζ ) = V (ζ )❊(C) [K] dζ d2 log Z (C) (ζ ) = V (ζ )2 Var(K) +V (ζ )❊(C) [K], dζ 2  (5.15)  where Var(K) is the variance of K. The third order derivative requires taking the derivative of Var(K). d Var(K) = V (ζ ) dζ  ❊(C) [K 3 ] − ❊(C) [K 2 ]❊(C) [K] − 2❊(C) [K]Var(K)  = V (ζ )❊(C) (K − ❊(C) [K])3 Use the relation V (ζ ) = −V (ζ )V (ζ ) to find the third order derivative of log Z(ζ ). d3 log Z (C) (ζ ) = V (ζ )❊(C) [K] −V (ζ )Var(K) +V (ζ )3 ❊(C) (K − ❊(C) [K])3 dζ 3  5.3.3  (5.16)  Computing moments  ˜ )m concentrate around ρ(ζ In this section we compute the moments of ρ(K). The moments of P(C) ζ ˜ ) is the minimizer of Fζ (ρ). where ρ(ζ Lemma 5.7. For υ large enough, r > 1 and γ small enough, there exist positive constants a1 and a3 , which depend on β , r and υ, such that ˜ ))m ]| ≤ am log(|C|)3/2 |C|−(m+1)/2 . |❊(C) [(ρ(K) − ρ(ζ for m = 1, 3. The quadratic moment satisfies ˜ ))2 ] ≤ |C|−1 ❊ζ [(ρ(K) − ρ(ζ  (rβ )−3/2 − |C|−1/3 3 rβ + ρ( ˜ ζ¯ )  −1/2  − |C|−1/2  .  Moreover, the choice of υ does not depend on γ. The proof of Lemma 5.7 requires estimates on Z (C) (ζ ). In particular, we validate Lemma 2.18 1 ˜ )). which states that |C| log Z (C) (ζ ) is concentrated around Fζ (ρ(ζ Lemma 5.8. The cube partition function satisfies ˜ )) Z (C) (ζ ) ≤ 2(rβ )−1/2 e−|C|Fζ (ρ(ζ .  129  and Z (C) (ζ ) ≥  rβ +  3 ˜ ) ρ(ζ  −1/2  ˜ )) − |C|−1/2 e−|C|Fζ (ρ(ζ .  R EMARKS 5.9. • The proofs of Lemma 5.7 and Lemma 5.8 extensively use that the random variable ρ(K) is ˜ ) and with variance proportional to well approximated by a Gaussian measure centered at ρ(ζ |C|−1 . The variable ρ(K) is discrete. The vast majority of the proofs is to show that discrete Gaussian sums are well-approximated by Gaussian integrals. Once this has been achieved the proof is readily completed. The random variable ρ(K) has a particularly simple distribution (5.13). The proofs thus show the difficulty of working with discrete random variables which are nearly Gaussian. • The bounds for odd m are better than those for even m, roughly it is |C|−(m+1)/2 versus |C|−m/2 . This is due to the close relation with cumulants κi . Given a random variable X, the cumulant κi with i an integer is κi =  di log(etX ) dt i  t=0  .  Except for the term V (ζ ), we are computing cumulants. The cumulants of a Gaussian variables with mean µ and variance σ 2 are all zero except for the first two which are µ and σ 2 . Hence, for m = 1, 3 only the error term of the Gaussian approximation contributes.  ♦  Proof of Lemma 5.8. A proof for the lower-bound is given in full detail. The upper-bound is obtained by using almost the exact same methods as those used for the lower-bound. Hence, some details for the upper-bound are omitted. Lower-bound. The partition function Z (C) (ζ ) splits Z (C) (ζ ) =  ∑Z  (C)  (ζ , k),  k≥0  where the contribution for a fixed k is Z (C) (ζ , k) =  |C|k rβ exp − |C|(−υβ ρ(k) −V (ζ )ρ(k) + ρ(k)2 ) . k! 2  (5.17)  ˜ where ρ˜ = ρ(ζ ˜ ). Apply Stirling’s formula [22, Section II.9] Pick any k such that ρ(k) ≥ ρ/2 k! =  √ 2π k1/2 exp(log k − k + λk )  130  with 0 ≤ λk ≤  1 , 12k  to the factorial. log  |C|k 1 1 1 ≥ k log |C| − log(2π) − k(log k − 1) − log k − k! 2 2 12k 1 1 1 log ρ(k) − log |C| − − log(2π). = −|C| ρ(k)(log ρ(k) − 1) − ˜ 2|C| 2 6ρ|C| 2  (5.18)  ˜ In the final step we used that ρ(k) ≥ ρ/2. Substitute the bound (5.18) into (5.17). −1 |C|−1/2 6ρ|C| Z (C) (ζ , k) ≥ √ e ˜ e−|C|Fζ (ρ(k)) . 2π  The minimizer ρ˜ solves Fζ (ρ) = 0 which justifies ρ(k)  ˜ + Fζ (ρ(k)) = Fζ (ρ) where  Fζ (t)(ρ(k) − t)dt,  ρ˜  1 1 . Fζ (t) = rβ + − t 2|C|t 2  ˜ the second derivative satisfies Fζ (t) < rβ + ρ3˜ while Fζ (t) > rβ By the assumption that ρ(k) ≥ ρ/2 since t > |C|−1 . ˜ + Fζ (ρ(k)) ≤ Fζ (ρ)  1 3 ˜ 2 rβ + (ρ(k) − ρ) 2 ρ˜  (5.19)  The bound on Z (C) (ζ , k) becomes −1 |C|−1/2 6ρ|C| ˜ 2 ˜ − 12 |C|(rβ + ρ3˜ )(ρ(k)−ρ) e . Z (C) (ζ , k) ≥ √ e ˜ e−|C|Fζ (ρ) 2π  Let N − =  ρ˜ 2 |C|  Z (C) (ζ ) ≥  , then −1 1 3 |C|−1/2 6ρ|C| ˜ 2 ˜ (C) √ Z (ζ , k) ≥ e ˜ e−|C|Fζ (ρ) e− 2 |C|(rβ + ρ˜ )(ρ(k)−ρ) . ∑− ∑ 2π k≥N k≥N −  ˜ Define N 0 = ρ|C| and split the sum accordingly.  ∑  k≥N −  e  ˜ 2 − 12 |C|(rβ + ρ3˜ )(ρ(k)−ρ)  N 0 −1  =  ∑  3  1  k=N −  ˜  1  2  e− 2 |C|(rβ + ρ˜ )(ρ(k)−ρ) +  ∑  3  2  e− 2 |C|(rβ + ρ˜ )(ρ(k)−ρ) ˜  k≥N 0  The sums on the right-hand side can be bounded by integrals. For example, if a function f is positive  131  k k−1  and increasing on the interval (k − 1, k), then f (k) ≥  ∑  e  N 0 −1  ˜ 2 − 21 |C|(rβ + ρ3˜ )(ρ(k)−ρ)  ≥  k≥N −  N − −1 ∞  ≥  3  1  ∞  2  e− 2 |C|(rβ + ρ˜ )(ρ(x)−ρ) dx + ˜  3  1  2  e− 2 |C|(rβ + ρ˜ )(ρ(x)−ρ) dx ˜  N0 3  1  N − −1  f (x)dx.  2  e− 2 |C|(rβ + ρ˜ )(ρ(x)−ρ) dx − 1 ˜  The bound on the partition function becomes −1 |C|−1/2 6ρ|C| ˜ Z (C) (ζ ) ≥ √ e ˜ e−|C|Fζ (ρ) 2π  Make the change of variables x →  x |C|  ∞ N − −1  e  x ˜ 2 −ρ) − 12 |C|(rβ + ρ3˜ )( |C|  dx − 1  and then translate to ρ˜ while using  −1 |C|−1/2 6ρ|C| ˜ e ˜ e−|C|Fζ (ρ) |C| Z (C) (ζ ) ≥ √ 2π  ∞  1  3  N − −1 |C|  ˜  ≤ ρ2 .  2  e− 2 |C|(rβ + ρ˜ )x dx − 1  ˜ −ρ/2  Let W be a zero-mean Gaussian with variance σ 2 = |C|−1 (rβ + ρ3˜ )−1 , then by symmetry |C| √ 2π  ∞  1  3  2  e− 2 |C|(rβ + ρ˜ )x dx =  ˜ −ρ/2  |C|1/2 ˜ 1 − P(W ≥ ρ/2) . (rβ + ρ3˜ )1/2  Use a classical tail bound for the Gaussian distribution [22, Section VII.1]. ˜ P(W ≥ ρ/2) ≤ e− 2 1  ρ˜ 2σ  2  .  −1  1 ˜ ≥ 1 − 6ρ|C| and collecting the error terms Combining the bound for the tail probability with e 6ρ|C| ˜  completes the proof. Upper-bound. All the bounds used in the derivation of the lower-bound can be reversed at the cost of some negligible error terms. Recall the definition (5.17) of Z (C) (ζ , k). Apply Stirling’s formula with the bound λk ≥ 0. |C|−1/2 −|C|Fζ (ρ(k)) Z (C) (ζ , k) ≥ √ e . 2π Recall that Fζ (t) > rβ provided t > |C|−1 . Hence, for k ≥ 1 the equivalent lower-bound for (5.19) becomes ˜ + Fζ (ρ(k)) ≤ Fζ (ρ) ˜ ). where ρ˜ = ρ(ζ  132  rβ ˜ 2, (ρ(k) − ρ) 2  The term with k = 0 is Z (C) (ζ , 0) = 1. −1/2 rβ ˜ |C| ˜ 2 √ Z (C) (ζ ) ≤ 1 + e−|C|Fζ (ρ) e− 2 |C|(ρ(k)−ρ) . ∑ 2π k≥1  The sum can be dealt with by a comparison with an integral. For example if a function f is positive and increasing on the interval (k, k + 1), then f (k) ≤  k+1 k  |C|−1/2 ˜ ˜ Z (C) (ζ ) ≤ e−|C|Fζ (ρ) e|C|Fζ (ρ) + √ 2π  f (x)dx. ∞  rβ  2  ˜ e− 2 |C|(ρ(x)−ρ) dx  −∞  The Gaussian integral can be explicitly calculated and is equal to (rβ )−1/2 . By Lemma C.3 part (ii) ˜ ˜ is negative which makes the term e|C|Fζ (ρ) the factor Fζ (ρ) exponentially small. Proof of Lemma 5.7. The expectation can be written as a sum ˜ m] = ❊(C) [(ρ(K) − ρ)  1 ∑ Z (C) (ζ , k), Z (C) (ζ ) k≥0  where  |C|k rβ 2 exp − |C|(−υβ ρ −V (ζ )ρ + ρ ) . k! 2 ˜ ). We used the short-hand notation ρ = ρ(k) and ρ˜ = ρ(ζ ˜ m Z (C) (ζ , k) = (ρ − ρ)  (5.20)  First pick m = 1, 3, then we come back to the case m = 2. Substitute Stirling’s formula [22, Section II.9] k! =  √ 2π k1/2 exp(log k − k + λk )  with 0 ≤ λk ≤  1 , 12k  (5.21)  into the logarithm. log  |C|k 1 1 = k log |C| − log(2π) − k(log k − 1) − log k − λk k! 2 2 1 1 log ρ = −|C| ρ(log ρ − 1) − − log |C| − log(2π) − λk . 2|C| 2 2  Substitute (5.22) into (5.20). |C|−1/2 ˜ m e−|C|Sζ¯ (ρ)−λk Z (C) (ζ , k) = √ (ρ − ρ) 2π  133  (5.22)  The minimizer ρ˜ solves Fζ (ρ) = 0 which justifies ρ  ˜ + Fζ (ρ) = Fζ (ρ) ˜ + = Fζ (ρ)  ρ˜  Fζ (t)(ρ − t)dt  ˜ Fζ (ρ) 2  ˜ 2 + R(ρ) (ρ − ρ)  where 1 1 , Fζ (t) = rβ + − t 2|C|t 2  (5.23)  ρ  R(ρ) = ρ˜  Fζ (t) 2  (ρ − t)2 dt.  (5.24)  (5.25)  We outline the remainder of the proof. Partition the set {k : k ≥ 1} by fixing a large constant Aγ ˜ ≤ Aγ |C|1/2 } while S2 is the which does depend γ. Define the set of natural numbers S1 = {k : |ρ − ρ| complement ◆ − S1 . A symmetry argument is used to show that most mass in the sum over k ∈ S1 cancels and the sum is of order |C|−(m+1)/2 . The sum over k ∈ S2 is bounded by the tail probability of a Gaussian measure and is also of the order |C|−(m+1)/2 . ˜ −1/2 and set σC = σ |C|−1/2 . Set Aγ = (rβ /2)−1/2 log(|C|)1/2 Mass cancellation. Let σ = Fζ (ρ) ˜ ≤ Aγ |C|1/2 . and consider those k such that |ρ − ρ| −1/2 ˜ 2 (ρ−ρ) ˜ |C| ˜ m e−|C| 2σ 2 e−λk −|C|R(ρ) √ Z (C) (ζ , k) = e−|C|Fζ (ρ) (ρ − ρ) 2π  The term e−λk −|C|R(ρ) is an error term which can be bounded by using (5.21) and combining (5.24) with (5.25). ˜ = O(|C|−1/2 log |C|) and ρ˜ > υ/(2r) by part (i) For υ large enough Fζ (t) ≤ 1 because |t − ρ| of Lemma C.3. (1 − 2A3γ |C|−1/2 ) ≤ e  1 − 6ρ|C| −|C|·|R(ρ)| ˜  Define the functions f (x) = xm e  −|C|  x2 2σ 2  ≤ e−λk −|C|R(ρ) ≤ e|C|·|R(ρ)| ≤ (1 + 2A3γ |C|−1/2 )  and g(x) = |x|m e  −|C|  x2 2σ 2  (5.26)  . The bound (5.26) shows that  −1/2 ˆ ¯ |C| ˜ + 2A3γ |C|−1/2 g(ρ − ρ) ˜ . Z (C) (ζ , k) ≤ e−|C|S(ζ ) √ f (ρ − ρ) 2π  (5.27)  ˜ and ∑ g(ρ − ρ). ˜ We concentrate on finding bounds on ∑ f (ρ − ρ) √ √ The function f is increasing on the interval (−σC m, σC m) and decreasing elsewhere. Therefore, define the natural numbers NA− = (ρ˜ − Aγ |C|−1/2 )|C| , NA+ = (ρ˜ + Aγ |C|−1/2 )|C| ,  134  √ Nσ− = (ρ˜ − σC m)|C| , √ Nσ+ = (ρ˜ + σC m)|C| ,  (5.28)  ˜ ˜ and consider the sum and let N0 = ρ|C| . For short-hand notation let f˜(x) = f (x − ρ) NA+  ∑  Nσ−  f˜(ρ) =  ∑  f˜(ρ) +  k=NA−  k=NA−  N0  Nσ+  f˜(ρ) +  ∑  k=Nσ− +1  NA+  f˜(ρ) +  ∑  k=N0 +1  ∑  f˜(ρ)  k=Nσ+ +1  The sum has been split over the intervals of increase and decrease of f and where f is negative or positive. Each of the sums on the right-hand side can be compared with an integral e.g., if f is increasing and negative on [k − 1, k], then f (k) ≤  k k−1  f (x)dx. The sum can thus be bounded by an  integral. NA+  ∑  f˜(ρ) ≤  k=NA−  Nσ− −1 NA−  +  N0  f˜(ρ(x))dx + Nσ+ −1  NA+  f˜(ρ(x))dx +  Nσ+ +1  N0 +1  ≤  NA+ NA−  f˜(ρ(x))dx +  f˜(ρ(x))dx + f˜(ρ(Nσ− )) + f˜(ρ(Nσ− + 1))  Nσ− +1  Nσ− +1 Nσ− −1  f˜(ρ(x))dx + f˜(ρ(Nσ+ )) + f˜(ρ(Nσ+ + 1))  f˜(ρ(x))dx  + f˜(ρ(Nσ− )) + f˜(ρ(Nσ− + 1)) + f˜(ρ(Nσ+ )) + f˜(ρ(Nσ+ + 1)) The fact that f is an odd function simplifies the bound. NA+  ∑  f˜(ρ) ≤  k=NA−  Nσ− +1 Nσ− −1  f˜(ρ(x))dx + f˜(ρ(Nσ− )) + f˜(ρ(Nσ− + 1)) + f˜(ρ(Nσ+ )) + f˜(ρ(Nσ+ + 1))  Finally, note that if |x| ≤ 2mσC then f˜(x) ≤ g(x) ≤ (2m)m σCm . NA+  ∑  f˜(ρ) ≤ 6(2m)m σCm .  (5.29)  k=NA−  Decomposing the similar sum of g(x) yields NA+  ∑  k=NA−  ∞  ˜ ≤ g(ρ − ρ) −∞  ˜ g(x − ρ)dx + 2(2m)m σCm .  Substitute the bounds (5.29) and (5.30) into (5.27). NA+  −1/2 ˆ ¯ |C| Z (C) (ζ , k) ≤ e−|C|S(ζ ) √ 2A3γ |C|−1/2 2π − k=N  ∑  A  135  ∞ −∞  ˜ g(x − ρ)dx + 8(2m)m σCm  (5.30)  Make the change of variables x → |C|−1/2 √ 2π  x |C| ,  translate to ρ˜ and compute the Gaussian integral  ∞  σ ˜ g(x − ρ)dx =√ −∞ 2πσC  ∞  |x|m e  −|C|  x2 2σ 2  −∞  dx = σ σCm  2m/2 Γ m+1 √ 2 , π  where Γ(·) is the Gamma function. Hence, by choosing a large enough constant A one obtains NA+  ∑  Z (C) (ζ , k) ≤ A log(|C|)3/2 |C|−(m+1)/2 .  (5.31)  k=NA−  ˜ > ρ˜ − A|C|−1/2 but k = 0. Combine Gaussian tails. Consider the S2 of those k such that |ρ − ρ| (5.23) with the fact that Fζ (t) > rβ for t > |C|−1 . ˜ + Fζ (ρ) ≥ Fζ (ρ)  rβ ˜ 2 (ρ − ρ) 2  (5.32)  The bound on Z (C) (ζ , k) for k ≥ 1 becomes rβ |C|−1/2 −|C|Fζ (ρ) ˜ ˜ 2 ˜ m e− 2 |C|(ρ−ρ) e |ρ − ρ| Z (C) (ζ , k) ≤ √ . 2π  Define h(x) = |x|m e  |C|  x2 2σ˜ 2  (5.33)  with σ˜ = (rβ )−1/2 and set σ˜C = σ |C|1/2 . The function h is increasing  on the interval (−∞, −A|C|−1/2 ] while it is decreasing on [A|C|−1/2 , ∞). Recall the definition (5.28) and use integral bounds. |C|−1/2 −|C|Fζ (ρ) ˜ (C) √ (ζ , k) ≤ Z e ∑ 2π k∈S2 Make the change of variables x →  ∑Z  (C)  k∈S2  k∈S2  (C)  ∞  ˜ h(ρ(x) − ρ)dx +  ˜ h(ρ(x) − ρ)dx  NA+ −1  −∞  translate to ρ˜ and use symmetry.  2|C|1/2 −|C|Fζ (ρ) ˜ e (ζ , k) ≤ √ 2π 2σ˜ ˜ =√ e−|C|Fζ (ρ) 2π σ˜C  Make the change of variables x →  ∑Z  x |C| ,  NA− +1  ∞ Aγ |C|−1/2  xm e  ∞  −|C|  x2 2σ˜ 2  −|C|  Aγ |C|−1/2  xm e  dx  x2 2σ˜ 2  dx  √x . 2σ˜C  (ζ , k) ≤  2σ˜ σ˜Cm 2m/2 −|C|Fζ (ρ) ˜ √ e 2π  Use the trivial bound xm ≤ 2m m! ex  2 /2  ∞  x √ Aγ /( 2σ˜ )  m −x2  e  dx  and a classical tail bound for the Gaussian normal distri-  136  bution [22, Section VII.1]. 1 √ 2π  ∞ √ Aγ /( 2σ˜ )  e−x  2 /2  1  dx ≤ e− 2  √ (Aγ / 2σ˜ )2  1  = e− 2 log |C|  Choose A large enough such that  ∑Z  (C)  (ζ , k) ≤ A |C|−(m+1)/2 .  (5.34)  k∈S2 ˜ m The term k = 0 was ignored but Z (C) (ζ , 0) = ρ˜ m . The term e|C|Fζ (ρ) ρ˜ is exponentially small ˜ is negative. It has been shown that there exists because by Lemma C.3 part (ii) the constant Fζ (ρ)  a constant Am such that  ∑Z  (C)  ˜ ˜ m (ζ , k) ≤ e−|C|Fζ (ρ) e|C|Fζ (ρ) ρ˜ + A log(|C|)3/2 |C|−(m+1)/2 + A |C|−(m+1)/2  k≥0 ˜ ≤ Am e−|C|Fζ (ρ) log(|C|)3/2 |C|−(m+1)/2  The proof for m = 1, 3 is completed by ˜ ζ¯ )|m ] = ❊(C) [|ρ(K) − ρ(  1 Z (C) (ζ )  ∑Z  (C)  (ζ , k),  k≥0  and Lemma 5.8. The case m = 2 is easier since here there are no cancellations as in the case of odd m. Using the upper-bound (5.32) and by comparing sums with integrals one finds that there exists a constant A such that |C|1/2 ˆ ζ¯ ) ˜ (C) −|C|Fζ (ρ) 2 |C|S( ˜ √ Z (ζ , k) ≤ e ρ e + ∑ 2π k≥0  ∞  x2  x2 e−(rβ )|C| 2 dx + A|C|−(m+1)/2  −∞  ˜ ≤ e−|C|Fζ (ρ) |C|−1 (rβ )−3/2 + (A + 1)|C|−3/2  The fact that r > 1 and Lemma 5.8 finishes the proof for m = 2.  5.3.4  Proof of Theorem 5.5  All the necessary tools are in place to prove Theorem 5.5. The linear, quadratic and remainder terms are handled by separate propositions. Proposition 5.10 (Linear term). There exists υ0 such that for r > 1 and υ > υ0 and γ = γ(υ, r, β ) small enough 1 d log Z (C) (ζ ) |C| dζ 137  = ξ0 , ζ =ξ0  where |ξ0 − ξˆ | ≤ a0 log(|C|)3/2 |C|−1 for some constant a0 = a0 (υ, r, β ). Proof. Recall that the first derivative was given by 1 d log Z (C) (ζ ) |C| dζ  = V (ξ0 )❊(C) [ρ(K)]. ζ =ξ0  For ζ ∈ ❘ define the function GC (ζ ) = ζ −V (ζ )❊(C) [ρ(K)]. and the closely related function ˜ ). G(ζ ) = ζ −V (ζ )ρ(ζ Invoke Lemma 5.7 to bound the difference between GC (ξ ) and G(ξ ). By (5.14) the derivative |V (ζ )| is bounded by  β.  ˜ )]| ≤ [ρ(N) − ρ(ζ |GC (ζ ) − G(ζ )| ≤ V (ζ )| ❊(C) ζ  β a1 log(|C|)3/2 |C|−1  Observe that the bound is independent of ζ . ˜ ) makes sure that Evaluation at the minimizer ρ(ζ G(ζ ) =  ∂ F(ζ , ρ) ∂ζ  = ρ=ρ˜  d Φ(ζ ). dζ  The continuous function G(ζ ) satisfies G(ξˆ ) = 0 and the derivative G (ξˆ ) = Φ (ξˆ ) is strictly positive because ξˆ is a minimizer. There exist ε > 0 and a positive constant a such that G (ζ ) > a > 0 provided |ζ − ξˆ | < ε. Set the constant A = 2a−1 β a1 and then define the constants ζ1 = ξˆ − A log(|C|)3/2 |C|−1 and ζ2 = ξˆ + A log(|C|)3/2 |C|−1 . Let γ be small enough such that A log(|C|)3/2 |C|−1 ≤ ε. It will be shown that GC (ξ1 ) < 0 while GC (ξ2 ) > 0 such that there exists some ξ0 ∈ (ζ1 , ζ2 ) where GC (ξ0 ) = 0. GC (ζ1 ) = GC (ζ1 ) − G(ζ1 ) + G(ζ1 ) − G(ξˆ ) ≤ ( β a1 − aA) log(|C|)3/2 |C|−1 GC (ζ2 ) = GC (ζ2 ) − G(ζ2 ) + G(ζ2 ) − G(ξˆ ) ≥ (aA − β a1 ) log(|C|)3/2 |C|−1 The choice of A guarantees that GC (ζ1 ) < 0 while GC (ζ2 ) > 0. Proposition 5.11 (Quadratic term). There exists υ0 such that for r > 1 and υ > υ0 and γ ≤ γ0 (υ, r, β ) d log Z (C) (ζ ) dζ 2  ≤ λ |C|, ζ =ξ0  138  where λ < 1. Proof. Recall the second derivative is 1 d log Z (C) (ζ ) |C| dζ 2  = |C|V (ξ0 )2 Var(ρ(K)) +V (ξ0 )❊(C) [ρ(K)].  (5.35)  ζ =ξ0  We deal separately with the terms on the right-hand side. Let λi , i = 1, 2 denote the first and second term on the right-hand side of (5.35). Consider the first term on the right-hand side of (5.35). The derivative V (ξ0 )2 is bounded by β . ˜ 0 )) ≤ |C|β ❊(C) [(ρ(K) − ρ(ξ ˜ 0 ))2 ]. λ1 ≤ |C|β Var(ρ(K)) = |C|β Var(ρ(K) − ρ(ξ Apply Lemma 5.7 and note that, by Lemma C.3 part (i) ˜ 0 ))2 ] → β ❊(C) [(ρ(N) − ρ(ξ  1 1 , r + O(|C|−1/3 ) |C|  (5.36)  as υ → ∞. It follows that λ1 is bounded by a constant which converges to 1r + O(|C|−1/3 ) as υ → ∞. The constant λ2 satisfies ˜ 0 ) +V (ξ0 )|❊(C) [ρ(N) − ρ(ξ ˜ 0 )]|. λ2 ≤ V (ξ0 )ρ(ξ ˜ 0 ) ≤ √2 V (ξˆ /2)ξˆ . Moreover, Proposition C.4 By Lemma C.3 part (iii) it follows that V (ξ0 )ρ(ξ β  υ shows that ξˆ ≈ β ρˆ and part (i) of Lemma C.3 shows that ρˆ ≥ 2r . All these facts combined show ˜ 0 ) → 0 as υ → ∞. Moreover, by Lemma 5.7 the second term on the right-hand side that V (ξ0 )ρ(ξ  vanishes as γ → 0. Hence, λ1 can be chosen as small as possible by choosing υ large enough and γ small enough. Setting λ = λ1 + λ2 finishes the proof. Proposition 5.12. There exists a constant a1 such that the remainder term satisfies |R(ζ )| ≤ a1 (log |C|)3/2 |C||ζ − ξ0 |3 .  Proof. Recall that the third derivative is given by d3 log Z (C) (ζ ) = V (ζ )|C|❊(C) [ρ(K)] −V (ζ )|C|2 Var(ρ(K)) dζ 3 3  3  +V (ζ ) |C|  139  ❊  (C)  (5.37) (ρ(K) − ❊ [ρ(K)]) (C)  3  ˜ ). The moments can all be related to ρ˜ = ρ(ζ ˜ ❊(C) [ρ] = ρ˜ + ❊(C) [ρ − ρ]  (5.38)  By Lemma 5.7 the first term on the right-hand side of (5.37) is O(|C|). For the variance the bound becomes ˜ ≤ ❊(C) [(ρ − ρ) ˜ 2 ]. Var(ρ) = Var(ρ − ρ)  (5.39)  Lemma 5.7 shows that the second term on the right-hand side of (5.37) is O(|C|). Finally, consider the third central moment. ˜ − (❊(C) [ρ − ρ])) ˜ 3] ❊(C) [((ρ − ρ) (5.40) 3 ˜ 3 ] − 3❊(C) [(ρ − ρ) ˜ 2 ]❊(C) [(ρ − ρ)] ˜ + 2❊(C) ˜ = ❊(C) [(ρ − ρ) [(ρ − ρ)] † ζ Apply Lemma 5.7 for the last time to show that the third term on the right-hand side of (5.37) is of order (log |C|)3/2 |C|−2 .  5.3.5  Proof of Theorem 5.6  Lemma 5.8 shows that there exist constants a ≤ a such that for any ζ 1  2  1  2  1  2  ˜ )) e− 2 |C|ζ Z (C) (ζ ) ≤ a e− 2 |C|ζ e−|C|Fζ (ρ(ζ = a e−|C|Φ(ζ ) .  and 1  2  ˜ )) e− 2 |C|ζ Z (C) (ζ ) ≥ ae− 2 |C|ζ e−|C|Fζ (ρ(ζ = ae−|C|Φ(ζ ) .  The inequalities form the backbone of the proof for Theorem 5.6. 1  2  e− 2 |C|ζ Z (C) (ζ ) 1 2 e− 2 |C|ξ0  · Z (ξ0 )  1  2  ≤ a−1 e− 2 |C|ζ Z (C) (ζ )e|C|Φ(ξ0 )  (C)  Next apply the upper-bound. 1  2  e− 2 |C|ζ Z (C) (ζ ) 1 2 e− 2 |C|ξ0  · Z (ξ0 )  1  ≤ a · a−1 e|C|( 2 ζ  2 +Φ(ξ )) 0  ≤ a · a−1 e−|C|(Φ(ζ )−Φ(ξ0 ))  (C)  The last step is to replace Φ(ξ0 ) by Φ(ξˆ ). Using Taylor’s theorem and the fact that |ξ0 − ξˆ | is bounded by a0 (log |C|)2 |C|−1 . There exists some constant a such that for γ small enough |C| · |Φ(ξ0 ) − Φ(ξˆ )| ≤ a |C| a0 (log |C|)2 |C|−1  140  2  ≤ 1.  The proof of Theorem 5.6 has been completed.  141  Chapter 6  Conclusions We study systems consisting of interacting spin particles which can have a positive or negative spin. The main achievement in this thesis is the development of new method to prove phase transitions. The method is applied to an Ising model and a type of Widom-Rowlinson (WR) model where the interactions are regulated by Kac potentials. It is shown that the infinite volume Gibbs state of these models is a mixture of two ergodic states. One Gibbs state predominantly has positive spin particles, the other Gibbs state predominantly has negative spin particles. In the physics literature the coexistence of two Gibbs states is called a phase transition. The proof uses a new expansion around mean field theory which is related to the Glimm Jaffe Spencer expansion around mean field theory [27, 28]. We first discuss the results for the Ising and WR model and then comment on the method we developed in this thesis to prove phase transitions. Ising and WR model. For the Ising model the mean field inverse critical temperature βc is equal to 1. It is shown that for β > 1 and γ > 0 small enough, the Ising model exhibits a phase transition. The method of proof can be readily extended to show that the infinite volume Gibbs state is unique for β ≤ 1. This is achieved by replacing the double-well function in Chapter 4 by a single-well function. A single-well function has a unique minimizer and allows for a parabolic lower-bound. The phase diagram of the Ising model has been constructed before by Bodineau and Presutti [4]. The WR model is defined on continuum spin particle configurations and is a variation on existing work. However, our method is not yet able to handle a purely continuum system. The WR model should be seen as an intermediate step towards a true continuum model as is argued in Remark 2.12. The WR model has parameters υ and r which are the chemical potential and the repulsion parameter, respectively. It is shown that for υ large enough, r > 1 and for γ > 0 small enough, the WR model exhibits a phase transition. Method of proof. Chapter 4 contains the main result in this thesis: it lists four conditions under which a Gaussian field exhibits a phase transition. In particular, the result can be applied to the Ising 142  and Widom-Rowlinson model. The proof of the main result is based on a recent cluster expansion technique for Gaussian fields [31]. The cluster expansion technique is presented in Chapter 3. In Chapter 5 it is shown how a phase transition in the Gaussian field implies a phase transition in the original particle models. The first novelty of our research is that we expand the field averages around their mean field theory. In the existing literature the particle densities are expanded around their mean field theory. Both mean field theories for the field averages and the particles densities are nearly Gaussian. However, particle densities are discrete variables while field averages are continuous variables. As mentioned in Remark 5.9 it is much easier to deal with continuous variables than discrete variables. The second difference is that we choose periodic boundary conditions. To our knowledge the existing work on phase transitions exclusively deals with the issue of a coexistence of phases by use of fixed boundary conditions. In this setting each bounded set Λ comes with a fixed boundary condition in Λc . For examples see Section 2.1.1 for continuum models and Remark 2.10 for Ising models. In this thesis we express the finite volume measures in two equivalent ways. We start with the spin particle model without an auxiliary field. The Kac-Siegert transformation inserts an auxiliary field in the spin particle model. Inverting the Kac-Siegert transformation when the auxiliary field is fixed in Λc yields strange unphysical conditions for the model without the auxiliary field. The periodic boundary condition allows for a clean definition for both descriptions of the model: with and without the auxiliary field. The Gaussian field model in Chapter 4 is analyzed by studying contours. We deal with cluster expansions for a mixed state that have not, to our knowledge, been discussed in a similar context before. We derive a Peierls estimate which shows that contours are exponentially unlikely. This uncovers two ergodic states which correspond to the two global minimizers of the mean field theory. We obtain the ergodic decomposition of the Gibbs state which by Theorem 2.4 is unique. The Peierls estimate is then used to show that the field averages in these ergodic states concentrate around the corresponding global minimizer. In this thesis we only focused on the question whether there is a range of parameters where two phases coexist. However, our method readily extends to complete the phase diagram where a unique phase is expected. The region where a unique phase is expected corresponds to the region where the mean field theory has a unique global minimizer. Hence, the only adjustment would be to replace the double-well function with a single-well function. The estimates in this thesis still apply and uniqueness follows. We believe that periodic boundary conditions have an advantage over the currently employed method of constructing the phase diagram by choosing fixed boundary conditions. With fixed boundary conditions the proofs of uniqueness and phase coexistence require different techniques. We illustrate this with Theorem 2.8 for the continuum LMP model. To show a phase coexistence Lebowitz, Mazel and Presutti define finite volume measures with two boundary conditions. The 143  boundary conditions are constructed from the two global minimizers ρ ± of the mean field theory for particle densities. The boundary condition q+ has particle densities sufficiently close to ρ + while the boundary condition q− has particle densities close to ρ − . They derive a Peierls estimate which shows that contours are exponentially unlikely. This shows the boundary conditions propagate inwards and the particle density concentrates near ρ ± if the chosen boundary condition is q± . This proves prove that there exist at least two distinct Gibbs states. This method cannot be extended to prove the existence of a unique Gibbs state. To prove uniqueness one has to show that, regardless of the chosen boundary condition q, there exists a unique Gibbs state e.g., see Bodineau and Presutti [4]. In summary, with the method we developed for periodic boundary conditions it is possible to use the Peierls estimate to uncover the ergodic Gibbs state(s). In case of phase coexistence the Peierls estimate also shows field averages concentrate near the global minimizer(s) of the mean field theory. The complete phase diagram can be uncovered by a single technique. With fixed boundary conditions the Peierls estimate is only used to show that two Gibbs states are different. A different technique is needed to show uniqueness. Future directions. The main focus of future research would be to extend the proof of phase transitions for the current WR model to prove phase transitions for continuum particle models such as the LMP model. We envision to achieve this in three steps. In the current setup space is partitioned into cubes such that one cube is centered at the origin. The repulsive potential in the WR model factors over the cubes e.g., there are no repulsive interactions between particles in neighbouring cubes. The first adaptation would be to remove this unphysical behaviour by including repulsive interactions between cubes which are within a certain distance of each other. Call this the extended lattice WR model. Following Lebowitz, Mazel and Presutti [41] the extended lattice WR model could then be viewed as the marginal distribution of a symmetric continuum model, see Remark 2.12. Hence, the next step would be to consider a continuum model whose marginal distribution is approximated by the extended lattice WR model. The Kac-Siegert transformation can also be applied to continuum models [10, Chapter 2]. Hence, it is expected that the new expansion around mean field theory we developed in this thesis would extend to the continuum model. The final step is to remove the symmetry and study a continuum particle model, such as the LMP model, instead of a Widom-Rowlinson model. The method we propose in this thesis is related to the mean field expansion of Glimm, Jaffe, and Spencer [27, 28]. The model studied by these authors is symmetric. Imbrie [33, 34] and Borgs and Imbrie [5] successfully applied Pirogov-Sina˘ı theory to extend the results of Glimm, Jaffe and Spencer to models without symmetry. We expect that, following the work of Borgs and Imbrie, it would be possible to prove phase transitions for continuum particle models without symmetries.  144  Bibliography [1] A BDESSELAM , A., P ROCACCI , A., AND S COPPOLA , B. Clustering bounds on n-point correlations for unbounded spin systems. J. Statist. Phys. 136 (2009), 405–452. ¨ [2] A DAMS , S., KOTECK Y´ , R., AND M ULLER , S. Finite range decomposition for families of gradient Gaussian measures. Preprint, 2012. [3] BAUERSCHMIDT, R. 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The results in this appendix are self-contained except for Section A.3.1. Here the notation for the flat and block averages is needed. These averages were introduced in Section 4.1.  A.1  Finite range decomposition  Consider the matrix W = (D − A)−1 where D is a diagonal matrix and A a finite range positive definite matrix. We show that under some conditions on D and A the matrix W is a positive definite matrix which allows for a finite range decomposition. Definition A.1. Take X ⊆ ❚n and let D be a diagonal matrix satisfying Dxx ≥ Dmin > 0 for all x ∈ X. The matrix D and a symmetric positive definite matrix A are called (Dmin , λ , L)-compatible if 1 ∑ |Axy | ≤ λ < 1, Dxx y∈X  (A.1)  for all x ∈ X and A has finite range L. The matrices A and D are X × X matrices. The next proposition shows that if a diagonal matrix D and a positive definite matrix A are (Dmin , λ , L)-compatible, then (D − A)−1 exists and has a finite range decomposition. Proposition A.2. Suppose D and A are (Dmin , λ , L)-compatible, then W = (D − A)−1 is symmetric s positive definite and has a finite range decomposition with λs = D−1 min λ and Ls = sL.  149  Proof. The matrix W has the formal expansion ∞  W = D−1 ∑ (AD−1 )s s=0  When multiplying the series by (D − A) from either left or right one obtains the identity matrix. The formal expansion shows that W is symmetric and positive definite. Define Cs = λ −s D−1 (AD−1 )s , then the matrix Cs has range sL. Use a walk representation to compute the entries of the matrix Cs . Let Wxys denote the set of all sequences in X of length s of the form ω = (ω0 , . . . , ωs ), such that ω0 = x and ωs = y and dist(ωi−1 , ωi ) ≤ L for all i = 1, . . . , s. The walk representation is Cs (x, y) =  ∑  ω∈Wxys  1 s Aωi−1 ωi . Dxx ∏ i=1 Dωi−1 ωi−1  Take absolute values to obtain an upper-bound. |Cs (x, y)| ≤  ∑  ω∈Wxys  1 s Aωi−1 ωi Dxx ∏ i=1 Dωi−1 ωi−1  The assumption (A.1) allows to interpret the bound in terms of a random walk. The random walk has state space X ∪ {ν} where ν is a dummy state called the graveyard. The transition probabilities are given by pxy =   |Axy |     Dxx  1 − ∑y∈X    1  if x, y ∈ X, |Axy | Dxx  if x ∈ X and y = ν  (A.2)  if x = y = ν.  Assumption (A.1) guarantees that the probabilities are well-defined and that the probability to jump from x ∈ X to the graveyard is at least 1 − λ . Let Wxs =  s y∈X Wxy ,  then the upper-bound becomes  ∑ |Cs (x, y)| = ∑ Px (ω),  y∈X  where  ω∈W s (x)  Px is the probability measure of the random walk started at x with transition probabilities  (A.2). The set W s (x) only contains walks which avoid the graveyard. Let the size of a walk |ω| be 150  measured in the number of steps. At each step the random walk has probability at least 1 − λ to jump to the graveyard.  ∑ |Cs (x, y)| ≤ Px (ω : |ω| = s and ω avoids the graveyard) ≤ λ s  y∈X  A.2  Positive definite minimizers with boundary conditions  Fix a region X ⊆ ❚n and suppose that D and A are (Dmin , λ , L)-compatible. Let W = D − A and consider the regions ∂ (X) = ∂Lext (X)  and  X = X ∪ ∂ (X).  (A.3)  Fix a boundary condition ξ¯ in ∂ (X). Study the minimizer g(ξ ˆ ) := argmin g,W g g  X  where the minimum is taken over all g ∈ ❘X such that g ≡ ξ in ∂ (X). Uniqueness of gˆ follows because the matrix W is positive definite. The main result obtained in this section is that the dependence on the boundary condition ξ decays exponentially with the distance to ∂ (X). Lemma A.3. Take ξ ∈ ❘X with ξx = 0 for x ∈ X c , then ξ ,W gˆ = 0. If Xk is a subset of X and dist(Xk , ∂ (X)) ≥ kL, then  ∑  g(ξ ˆ )2x ≤  x∈Xk  λ k−1 ∑ ξ 2. D2min (1 − λ )2 x∈∂ (X) x  The first step in the proof of Lemma A.3 is to find an expression for g. ˆ We next show that the expression also proves the statement that ξ ,W gˆ = 0. Proposition A.4. Let w ∈ ❘X be given by  ξ x wx = 0  if x ∈ ∂ (X), else.  The vector g(ξ ˆ ) is given by g(ξ ˆ ) = W −1 w, and ξ ,W gˆ = 0 if ξ ∈ ❘X with ξx = 0 for all x ∈ ∂ (X). Proof. For g ∈ ❘X with g ≡ ξ in ∂ (X) define the function G(g) = g,W g X . Use directional deriva-  151  tives to find g(ξ ˆ ). The minimizer satisfies 0 = ∇G(g), ˆ h  X  d G(gˆ + εh) dε  =  ε=0  ,  for all h ∈ ❘X with hx = 0 for x ∈ ∂ (X). Compute the right-hand side. d G(gˆ + εh) dε  ε=0  = h,W gˆ  X  + g,W ˆ h x.  By symmetry of W it follows that d G(gˆ + εh) dε  ε=0  = 2 h,W gˆ  X  =0  for all h ∈ ❘X with hx = 0 for x ∈ ∂ (X). This shows that h,W gˆ  X  = 0 for all h ∈ ❘X with hx = 0  for x ∈ ∂ (X). Moreover, the vector gˆ satisfies gˆ = W −1 w. Existence of gˆ is guaranteed by Proposition A.2. Proposition A.4 provides an explicit expression for g(ξ ˆ ). Combine this with the finite range decomposition Proposition A.2 to prove exponential decay of g(ξ ˆ ). Proof of Lemma A.3. Proposition A.4 explicitly defines gˆ as gˆ = W −1 w. By Proposition A.2 the matrix W −1 exists and satisfies W −1 =  1 Dmin  ∑ λ sCs ,  s≥0  where Cs has range sL and  ∑ |Cs (x, y)| ≤ 1,  y∈X  for all x ∈ X. Compute the sum of squares for gˆ in the region Xk . 1  ∑ gˆ2x ≤ D2  x∈Xk  ∑ ∑ λ s ∑ |Cs (x, y)| · |wy |  min x∈Xk  s≥0  2  y∈X  The matrices Cs satisfy  ∑ λ s ∑ |Cs (x, y)| ≤ (1 − λ )−1 . y∈X  s≥0  Apply Jensen’s inequality and use the symmetry of Cs . 1 ∑ w2y ∑ λ s ∑ |Cs (y, x)| (1 − λ ) y∈X x∈Xk s≥0 min  ∑ gˆ2x ≤ D2  x∈Xk  152  If s < k, then Cs (y, x) = 0 for all x ∈ Xk because Cs has finite range sL.  ∑  x∈Xk  A.3  gˆ2x ≤  1 λ k−1 2 s w |C (y, x)| = λ ∑ ∑ ∑ s D2min (1 − λ ) y∈∂ (X) y s≥(k−1) x∈X D2min (1 − λ )2  ∑  w2y  y∈∂ (X)  Gaussian integrals  The lemma presented in this section is an adaptation of Lemma 6.29 in [14]. It provides a sufficient condition for the integrability of exponential moments of a Gaussian measure and provides a bound. In Section A.3.1 we give a specific example where the latter condition is satisfied. Here the notation of flat and block averages introduced in Section 4.1 is needed. Lemma A.5. Let {Zx }x∈X be a finite set of Gaussian random variables with covariance matrix W . Suppose  ∑ |W (x, y)| ≤ ν,  (A.4)  y∈X  for all x ∈ X, then for any κ > ν  ❊ exp  1 2κ  ∑ Zx2  ≤ exp  x∈X  1 ∑ W (x, x) . 4(κ − ν) x∈X  Proof. Define the matrix Wt = (W −1 − κ −1tI)−1 . The claim is that Wt is a symmetric positive definite matrix. Symmetry of Wt is obvious. Let (νi )1≤i≤n with n = |X| be the eigenvalues of W in decreasing order. The eigenvalues of Wt are νit = (νi−1 − tκ −1 )−1 = νi (1 − tκ −1 νi )−1 ≤  κ νi = νi . 1 − ν1 κ −1 κ − ν1  (A.5)  The Schur test [61, Corollary 3.7] and (A.4) shows that W x 22 ≤ ν x 22 . Hence, the largest eigenvalue ν1 is bounded by ν. Hence, the eigenvalues of Wt are all positive and Wt is a positive definite matrix. Define Z(t) = ❊[et(2κ)  −1  ∑ Zx2 ],  1 Z(1) log = Z(0) 2κ  the aim is to bound Z(1)/Z(0). 1  ∑  x∈X 0  ❊[Zx2 et(2κ) ∑ Zx ] 1 t· dt = ∑ −1 ∑ Z 2 t(2κ) x] 2κ x∈X ❊[e −1  2  1 0  t ❊t [Zx2 ]dt  where ❊t is the expectation with respect to the Gaussian measure on X with covariance measure Wt . The sum of diagonal elements of Wt , or equivalently the sum of eigenvalues of Wt , are bounded by  153  using (A.5) log  Z(1) 1 = Z(0) 2κ  1  ∑  x∈X 0  n  1 2κ  tWt (x, x)dt ≤  1  κ κ − ν1  ∑ νi  i=1  0  n  νi . i=1 4(κ − ν1 )  tdt = ∑  The sum ∑ νi is the trace of W and (κ − ν1 )−1 ≤ (κ − ν)−1 .  A.3.1  Application to Gaussian averages  Let X be any fixed subset of ❚n and take σx ∈ {−1, +1}X to be any spin configuration on X. Let  ❘X which is zero everywhere except at x ∈ X where it is equal to one. Given y, let ϕyσ denote the matrix corresponding to the linear transformation which maps (ξx , x ∈ ❘X ) to  ex be the vector in  (Avx [σ ξ ] − Avy [σ ξ ], x ∈ X). Recall that Av and ϕ both refer simultaneously to the site and block average. Proposition A.6. Take any sequence {yxˇ }x∈ ˇ Vˇ (X) with yxˇ ∈ Cxˇ , then −ε/2  ∑ ∑ |ϕyσˆ (x, z)| ≤ αx (X) 1  x∈ ˇ Vˇ (X) z∈Cxˇ  for all x ∈ X. The inner products satisfy Avσ (ex ), Avσ (ex )  X  = |C(1) |−1  and  ∑  ϕyσˇ (ex ), ϕyσˇ (ex ) xˇ ≤  −ε/2 (1) −1 |C | 1  x∈ ˇ Vˇ (X)  Proof. The statements for ϕ are trivially true when using the block averages. If x and y are in the •  •  same cube, then Avx [ξ ] − Avy [ξ ] = 0 . Consider the case when using the site averages. 1  ∑ ∑ |ϕyσˆ (x, z)| ≤ |C  (1)  x∈ ˇ Vˇ (X) z∈Cxˇ  =  ∑ | zˇ∈∑ Vˇ (X) z∈Czˇ  |✶{x∈C(1) } − ✶{x∈C(1) } | z  yzˇ  1 ∑ |✶{z∈Cx(1) } − ✶{yzˇ∈Cx(1) } | |C(1) | zˇ∈∑ Vˇ (X) z∈Czˇ  Most terms will cancel out because dist(z, zˇ) ≤  −  and z can only contribute when z ∈ Cx(1) and zˇ ∈ / Cx(1)  or vice versa. Hence, z must be contained in ∂2int−(Cx(1) ) or ∂2ext−(Cx(1) ). When normalized by |C(1) |−1 these sets contain at most 2(2d)  −  d−1 1  d 1  = 4d  lattice points. This proves the first part of the proposition.  154  −ε 1 ,  (A.6)  Compute the diagonal elements of Avσ . Avσ (ex ), Avσ (ex )  X  ≤  1 α (X) ✶{z∈Cx(1) } = x (1) ≤ |C(1) |−1 (1) 2 ∑ |C | z∈X |C |  Finish by considering the linear transformation ϕ˚ yσ (ex ), ϕ˚ yσ (ex )  X  ≤  4d −ε 1 1 ✶ ✶ + ✶ ✶ ≤ , (1) (1) (1) ∑ {z∈Cx(1) } {z∈C / yxˇ } {z∈C / x } {z∈Cyxˇ } |C(1) |2 z∈X |C(1) |  where in the last line we re-used the bound (A.6).  155  Appendix B  Combinatorial result Partition ❩d into cubes C of arbitrary fixed integral side-length such that one cube is centered at the origin. All spatial sets X are assumed to be unions of cubes and connected. Let N(X) denote the number of cubes contained in the set X. Set Co to be the cube centered at the origin. The constant nd , depending on dimension d, denotes the number of cubes which are contiguous to Co . Lemma B.1. Let α be positive, then ∞  ∑  α N(X) ≤  ∑ (end α)n .  n≥m  X:X⊃Co  N(X)≥m  The sum runs over connected sets X which contain the origin. R EMARK . For an integer n let An denote the number of connected sets which contain the origin. In the context of this thesis it suffices to show that |An | ≤ a1 an2 ,  (B.1)  for some positive constants a1 and a2 . Connected sets containing the origin are also known as lattice animals. It has been long known that the limit of |An |1/n as n → ∞ exists [38]. Hence, the desired ♦  inequality (B.1) follows immediately.  Proof. Number the cubes in any order such that the cube at the origin has label 0. Each set X can be uniquely associated with a tree T on the labels of cubes in X by a recursive procedure. Let V (T ), E(T ) denote the vertex set, respectively edge set of T . Set V (T ) = {0} and E(T ) = 0. / Suppose V (T ) = {0, i1 , . . . , in } and consider all cubes contiguous to V (T ). From these cubes select the cube with the lowest label j which did not occur yet. Similarly, select the cube in V (T ) which is contiguous to j and has the lowest label, say this cube has label ik . Let e be the edge { j, ik }. Update vertex set V (ξ ) → V (ξ ) ∪ { j} and edge set E(ξ ) → E(ξ ) ∪ {e}. Repeat this procedure until V (ξ ) contains all the labels of the cubes in Γ. 156  The construction allows to associate a unique tree to each connected set X. Hence, α N(X) =  ∑  α N(T ) ,  ∑  (B.2)  T : 0∈T  X:X⊃Co  |V (T )|≥m  N(X)≥m  where N(T ) is the number of vertices of the tree. Rewrite the sum over trees. α N(T ) =  ∑ T : 0∈T  |V (T )|≥m  αn ∑ ∑ ∏ ✶{xi ∼x j } , n=m n! T on {1,...,n} x2 ,...,xn (i, j)∈T ∞  ∑  where T runs over all rooted trees on n vertices rooted at 1 and x1 = 0. The set {x1 , . . . , xn } is a subset of the labels. Induction on n is used to show that for a fixed tree T with n vertices  ∑ ∏ ✶{x ∼x } ≤ (nd )n−1 . i  j  x2 ,...,xn (i, j)∈T  Trivially, the above holds for n = 1 since the label x1 is fixed. Fix n > 1 and suppose w.l.o.g. the vertex with label n is one of the leaves in T . Suppose it is joined by the edge e = (k, n) and let T = T − e.  ∑ ∏ ✶{x ∼x } ≤ ∑ i  x2 ,...,xn (i, j)∈T  j  ∏ ✶{x ∼x } ∑ ✶{x ∼x } i  j  k  x2 ,...,xn−1 (i, j)∈T  n  ≤ nd  xn  ∑  ∏ ✶{x ∼x } i  j  x2 ,...,xn−1 (i, j)∈T  By Cayley’s theorem there are nn−2 trees of size n which in combination with (B.2) yields α N(T ) ≤  ∑ T : 0∈T  (nd α)n nn−2 ∑ n! n=m ∞  |V (T )|≥m  Using Stirling’s formula it follows that  nn−2 n!  ∑  ≤ en and α N(T ) ≤  ∞  ∑ (end α)n  n=m  T : 0∈T  |V (T )|≥m  157  Appendix C  Variational problems In this chapter we study the variational problems we encountered in Chapter 2. The main aim is to show that the functions Φ defined in Chapter 2 are symmetric double-well functions. The chapter is self-contained, the necessary definitions are repeated. Definition C.1. A function Φ : ❘ → ❘ is called a symmetric double-well function if (i) Φ(ζ ) = Φ(−ζ ), (ii) Φ admits two global minimizers ±ξˆ , (iii) Φ admits parabolic lower-bounds underneath each of its wells, i.e., there exists κ > 0 such that for ζ > −ξˆ /2 κ Φ(ζ ) − Φ(ξˆ ) ≥ (ζ − ξˆ )2 , 2 and for ζ < ξˆ /2 κ Φ(ζ ) − Φ(ξˆ ) ≥ (ζ + ξˆ )2 . 2 The function V (ζ ) = log cosh( β ζ ) and its derivatives play an important role. V (ζ ) = V (ζ ) =  β tanh( β ζ ), β cosh( β ζ )2  V (ζ ) = −β 3/2  ,  tanh( β ζ ) cosh( β ζ )2  158  (C.1) .  C.1  Ising model  Define the function Φ : ❘ → ❘ by Φ(ζ ) =  ζ2 −V (ζ ). 2  We will show that for β > 1 the function Φ is a symmetric double-well function. Proposition C.2. For β ≤ 1 the function Φ has a unique minimum at ζ = 0. For β > 1, Φ is a symmetric double-well function. Moreover, at the wells ±ξˆ the derivative satisfies V (ξˆ ) < 1. Proof. By symmetry it suffices to take ζ > 0. Observe that Φ (ζ ) = 1 − V (ζ ) is non-negative when β ≤ 1. It shows that Φ is a convex function which reaches a unique minimum at ζ = 0. If β > 1, then Φ (0) = 1 − β is negative. The derivative Φ (ζ ) = −V (ζ ) is positive and Φ (ζ ) is a strictly increasing function which tends to 1 as ζ → ∞. Observe that because Φ (0) = 0 and Φ(0) = 0, there exists ξˆ > 0 such that Φ reaches a strictly negative minimum at ζ = ξˆ . Consider V (ξˆ ) and note that Φ (ξˆ ) has to be positive because otherwise Φ (ξˆ ) < 0. The derivative satisfies Φ (ξˆ ) = 1 −V (ξˆ ) > 0 or V (ξˆ ) < 1. Continue with the proof of the lower-bound on Φ(ζ ) − Φ(ξˆ ). The first step is to show there exists κ such that Φ(ζ ) ≥  κ (ζ − ξˆ )2 + Φ(ξˆ ), 2  for all ζ ≥ 0. Define g(ζ ) = Φ(ζ ) − Φ(ξˆ ) − κ2 (ζ − ξˆ )2 , the aim is to show that g(ζ ) ≥ 0. The derivative g (ζ ) = −V (ζ ) is positive and, for κ small enough, g (0) < 0 while g (ξˆ ) > 0. Hence, the function g(ζ ) has a unique inflection point ζ0 which lies somewhere in the interval (0, ξˆ ). The function g satisfies g(ξˆ ) = g (ξˆ ) = 0 which justifies ξˆ  g (t)(t − ζ )dt.  g(ζ ) = ζ  If ζ ≥ ζ0 then the integral representation immediately gives g(ζ ) ≥ 0. If ζ ∈ [0, ζ0 ), then the integral representation yields ζ0  g(ζ ) = ζ  ξˆ  g (t)(t − ζ )dt +  ξˆ  g (t)(t − ζ )dt ≥  g (t)tdt = g(0). 0  ζ0  The quantity g(0) = −Φ(ξˆ ) − κ2 ξˆ 2 is strictly positive for κ small enough. The lower-bound derived so far is only valid for ζ > 0. However, the bound can be extended to  159  hold for ζ > −ξˆ /2 by picking 3 κ = min κ , 2 · Φ(−ξˆ /2) ξˆ 2  −2  .  By symmetry the lower-bound also holds for the well −ξˆ .  C.2  WR model  ˜ ) be In the WR model the minimizers were found in a two-tier approach. Given ζ ∈ ❘ let ρ˜ = ρ(ζ the minimizer of Fζ (ρ) = −υβ ρ −V (ζ )ρ +  rβ 2 log ρ ρ + ρ(log ρ − 1) + , 2 2|C|  (C.2)  where F : [|C|−1 , ∞) → ❘. F(ζ , ρ) =  log ρ rβ 2 ζ2 − υβ ρ −V (ζ )ρ + ρ + ρ(log ρ − 1) + 2 2 2|C|  (C.3)  Define the function Φ(ζ ) by ˜ )). Φ(ζ ) = F(ζ , ρ(ζ  Lemma C.3. For r > 1 and β > 0, there exists a υ0 such that for υ > υ0 ˜ ) is an increasing function of ζ and ρ(0) ˜ (i) ρ(ζ >  υ 2r ,  ˜ )) is strictly negative, (ii) the function Fζ (ρ(ζ ˜ ) < √2 V (ξˆ /2)ξˆ for all ζ ≥ 12 ξˆ . (iii) Φ(ξˆ ) is negative and V (ζ )ρ(ζ β  (iv) The function Φ(ζ ) is a symmetric double-well function with minimizers ±ξˆ , The proof of the lemma is executed in two steps. First we derive an auxiliary result that F(ξ , ρ) ˆ This information allows to prove the has two symmetrically located global minimizers (±ξˆ , ρ). statements about Φ listed in Lemma C.3. Proposition C.4. For r > 1 and β > 0, there exists a υ > υ0 such that F(ξ , ρ) has two global ˆ Moreover, these minimizers satisfy minimizers (±ξˆ , ρ). √ β ρˆ − e− β ρˆ < ξˆ <  160  ˆ β ρ.  Proof. The minima of F(ζ , ρ) are recovered by first minimizing over ζ and consecutively minimizing over ρ. By symmetry it suffices to consider the case when ζ > 0. Fix ρ and consider Fρ (ζ ) =  ζ2 −V (ζ )ρ. 2  Repeating the argument of Proposition C.2 it follows that Fρ has a unique positive minimizer ζ˜ (ρ) when ρβ > 1. However, when ρβ ≤ 1 the function Fρ has a unique minimizer ζ˜ (ρ) = 0. The minimizer of F must lie somewhere on the curve (ρ, ζ˜ ) where ζ˜ = ζ˜ (ρ) is the unique positive minimizer of Fρ (ζ ). Therefore, define the function G(ρ) = F(ρ, ζ˜ ) with ρ > |C|−1 . It will ˆ First, we show that G (ρ) is negative for ρ < be shown that G(ρ) has a unique minimizer ρ. Then we proceed by showing that G(ρ) is a convex function on for ρ > Because ζ˜ is a minimizer one finds that the derivative satisfies ∂F dG = dρ ∂ρ For υ sufficiently large and ρ <  = −V (ζ˜ ) − υβ + rβ ρ + log ρ + ζ =ζ˜ υ 2r  υ 2r .  1 . 2|C|ρ  the derivative is negative 1 υ υ 1 < − β + log + < 0, 2|C|ζ 2 2r 2  G (ρ) < −υβ + rβ ζ + log ζ + where we used that |C|−1 < ρ <  υ 2r .  υ 2r .  Next we show that G is convex for ρ >  υ 2r .  ˜ d2G ˜ ) d ζ + rβ + 1 − 1 = −V ( ζ dρ 2 dρ ρ 2|C|ρ 2 The derivative F (ζ˜ ) = 0 or  d ζ˜ dρ  can be found using implicit differentiation. The minimizer ζ˜ of Fρ (ζ ) satisfies  ρ  ζ˜ − ρV (ζ˜ ) = 0.  (C.4)  Use equality (C.4) to find the derivative of ζ˜ (ρ). d ζ˜ V (ζ˜ ) = dρ 1 − ρV (ζ˜ ) Substitution yields d2G V (ζ˜ )2 1 1 1 = − + rβ + − > r− β, 2 2 ˜ dρ ρ 2|C|ρ 1 − ρV (ζ ) 1 − ρV (ζ˜ ) where the inequality was obtained because ρ >  υ 2r  and |V (ζ )| ≤  161  (C.5)  β . The proof of the uniqueness  of the global minimizer for ζ ≥ 0 is finalized by showing that ρV (ζ˜ ) → 0 as υ → ∞. If the latter υ 2r  is true, then there exists an υ0 such that G is convex provided ρ >  and υ > υ0 .  First we show that, provided υ is sufficiently large, β ρ − exp[− β ρ] < ζ˜ <  (C.6)  β ρ.  By (C.1) it follows that V (ζ ) = Substitute ζ =  β tanh( β ζ ) =  √ 2e− β ζ √ β 1− √ ≥ e β ζ + e− β ζ  √ β (1 − e−2 β ζ ).  β ρ − exp[− β ρ] into Fρ (ζ ) and note that for υ, and thus ρ, large enough √ ζ − ρV (ζ ) ≤ −e− β ρ +  Next substitute ζ =  √  β ρe−2β ρ+2e  −  βρ  < 0.  β ρ into Sρ (ζ ). ζ − ρV (ζ ) =  β ρ(1 − tanh(β ρ)) > 0  ˆ Furthermore, it shows that This shows (C.6) indeed holds and proves the statement about ξˆ and ρ. ˜ ˜ ζ ≈ β ρ and it follows that ρV (ζ ) → 0 as υ → ∞. ˜ ) is well-defined. Proof of Lemma C.3. The first aim is to show that ρ(ξ Fζ (ρ) = −V (ζ ) − υβ + rβ ρ + log ρ +  1 2|C|ρ  1 1 Fζ (ρ) = rβ + − . ρ 2|C|ρ 2  (C.7)  The second derivative is strictly positive because ρ > |C|−1 . The function Fζ is convex and has a ˜ ). unique minimizer ρ(ξ Use (C.7) to prove part (i). d ρ˜ V (ζ ) = dζ rβ + ρ1˜ − 2|C|1 ρ˜ 2  (C.8)  ˜ ) is increasing. Moreover, at ζ = 0 The derivative V (ζ ) is non-negative for ζ ≥ 0. The function ρ(ζ the derivative is F0  υ υ υ ≤ −β + 2 log < 0, 2r 2 2r  ˜ for υ large enough. Hence, ρ(0) must be larger than  162  υ 2r .  ˜ ) solves Part (ii) follows easily. The minimizer ρ(ζ V (ζ ) + νβ = rβ ρ˜ + log ρ˜ +  1 2|C|ρ  Substitute this identity into Fζ (ρ) and use that ρ˜ is large for υ large. ˜ = −rβ ρ˜ 2 + Fζ (ρ)  rβ 2 rβ 2 ˜ =− ˜ < 0. ρ˜ + O(ρ˜ log ρ) ρ˜ + O(ρ˜ log ρ) 2 2  ˜ ξˆ ) satisfies F ˆ (ρ) ˆ =0 Use Proposition C.4 to prove that Φ(ξˆ ) is negative. The minimizer ρˆ = ρ( ξ  or rβ ρˆ + log ρˆ +  1 = V (ξˆ ) + υβ . 2|C|ρˆ  Substitute this equality into Φ(ξˆ ) and use Proposition C.4. Φ(ξˆ ) ≤  rβ 1−r 2 β 2 ˆ = ˆ < 0. ρˆ − rβ ρˆ 2 + ρˆ 2 + O(ρˆ log ρ) β ρˆ + O(ρˆ log ρ) 2 2 2  ˜ ) and take the derivative by using (C.8). Consider the term V (ζ )ρ(ζ ˜ ) +V (ζ ) V (ζ )ρ(ζ  V (ζ ) 1 ˜ )− = V (ζ ) ρ(ζ 1 1 1 rβ + ρ˜ − 2|C|ρ˜ 2 rβ + ρ˜ − 2|C|1 ρ˜ 2  ˜ ) is a decreasing function because V By part (i) it follows that V (ζ )ρ(ζ  is negative. Combine this  knowledge with Proposition C.4. 2 ˜ ξˆ /2) ≤ V (ξˆ /2)ρ( ˜ ξˆ ) ≤ V (ξˆ /2)ρ( V (ξˆ /2)ξˆ β ˜ )) = 0 and (C.8) to conclude Use the fact that Fζ (ρ(ζ ˜ ) −V (ζ ) Φ (ζ ) = 1 −V (ζ )ρ(ζ  d ρ˜ V (ζ )2 ˜ )− = 1 −V (ζ )ρ(ζ dζ rβ + ρ1˜ − 2|C|1 ρ˜ 2  A trivial bound shows ˜ )− Φ (ζ ) ≥ 1 −V (ζ )ρ(ζ  1 r  Taking all of this information into account it follows that the right-hand side converges to 1− 1r > 0 as υ → ∞. Choose υ large enough such that Φ (ζ ) ≥  163  1 2  1 − 1r .  ˜ )) = 0 and (C.8) to conclude Use the fact that Fζ (ρ(ζ ˜ ) −V (ζ ) Φ (ζ ) = 1 −V (ζ )ρ(ζ  d ρ˜ V (ζ )2 ˜ )− = 1 −V (ζ )ρ(ζ dζ rβ + ρ1˜ − 2|C|1 ρ˜ 2  A trivial bound shows ˜ )− Φ (ζ ) ≥ 1 −V (ζ )ρ(ζ  1 r  The final part of the proof is to show that Φ is a double-well function. Define the function κ g(ζ ) = Φ(ζ ) − Φ(ξˆ ) − (ζ − ξˆ )2 , 2 ˆ The first aim is to show that g(ζ ) ≥ 0 for ζ ≥ 0. Later this can be extended where Φ(ξˆ ) = Φ(ξˆ , ρ). ˆ to hold for ζ ≥ −ξ /2. The fact that g(ξˆ ) = g (ξˆ ) = 0 justifies ξˆ  g (t)(t − ζ )dt.  g(ζ ) =  (C.9)  ζ  In Proposition C.2 the proof continued by showing that g (t) ≥ 0 for t ≥ 0. However, in this case we are unable to show such a result and a detour is required. The second derivative can be computed using (C.7) and (C.8). g (ζ ) = 1 −V (ζ )ρ˜ −  V (ζ )2 −κ rβ + ρ1˜ − 2|C|1 ρ˜ 2  (C.10)  A trivial lower-bound shows that g (ζ ) ≥ f (ζ ) where f (ζ ) = 1 −  V (ζ )2 −V (ζ )ρ˜ − κ . rβ  Use (C.1) and (C.8) to compute the derivative of f . f (ζ ) = −V (ζ )  2 1 3 + − ρ˜ . − ρ˜ > −V (ζ ) rβ rβ + ρ1˜ − 2|C|1 ρ˜ 2 rβ  The derivative f (ζ ) is non-negative because V (ζ ) ≤ 0 for ζ ≥ 0 and ρ˜ >  υ 2r  >  3 rβ  for υ large  enough. Before proceeding some more information about f is needed. At ζ = 0 one can immediately conclude that f (0) = 1 − ρ˜ − κ is strictly negative. Moreover, part (iii) can be used to show that  164  f (ξˆ /2) is strictly positive for υ large enough. 1 2 V (ξˆ /2)ξˆ . f (ξˆ /2) ≥ 1 − − κ − r β By (C.4) it follows that ξˆ ≈  β ρˆ and part (iii) shows that ρˆ >  υ 2r .  Therefore, √2 V (ξˆ /2)ξˆ tends β  to zero as υ → ∞. Take υ large enough and κ small enough such that 1 1 f (ξˆ /2) ≥ (1 − ) 2 r  (C.11)  For κ small enough and υ large enough there exists a unique ζ0 ∈ [0, ξˆ /2) such that f (ζ0 ) = 0. The integral representation (C.9) then shows that for ζ ≥ ζ0 ξˆ  g(ζ ) ≥  f (t)(t − ζ )dt ≥ 0. ζ  Consider the case when ζ ∈ [0, ζ0 ) and observe that f (t) ≤ 0 for t ∈ [0, ζ0 ). ξˆ  g(ζ ) ≥  ξˆ  f (t)(t − ζ )dt ≥  f (t)tdt 0  ζ  Use the triangle inequality to recover g (t). ξˆ  g(ζ ) ≥  ξˆ  g (t)tdt − 0  ξˆ  | f (t) − g (t)|tdt = g(0) − 0  | f (t) − g (t)|tdt  (C.12)  0  Next we show that the right-hand side is strictly positive. Consider g(0) = Φ(0) − Φ(ξˆ ) − κ2 ξˆ 2 and observe that on the interval [0, ξˆ ] the function Φ decreases to Φ(ξˆ ). Hence, one can conclude that κ g(0) ≥ Φ(ξˆ /2) − Φ(ξˆ ) − ξˆ 2 = 2  ξˆ  ξˆ g (t)(t − )dt 2 ξˆ /2  Apply the bound g (t) ≥ f (t) and use (C.11). ξˆ  g(0) ≥  ξˆ /2  1 1 ξˆ f (t)(t − )dt ≥ (1 − )ξˆ 2 2 16 r  (C.13)  ˆ It already has been shown that for g (t) ≥ f (t) ≥ f (ξˆ /2) ≥ 12 (1 − 1r ) provided t > ξ2 .  Consider the second term in the far right-hand side of (C.12). Recall (C.10), the definition of f  165  and the fact that V ≤  β by (C.1). | f (t) − g (t)| ≤ β  1 1 2 1 ≤ − 1 rβ rβ − 2ρ˜ 2ρ˜ r2 β  (C.14)  Substituting (C.13) and (C.14) into (C.12) yields g(0) ≥  1 1 1 ˆ2 1 ξ . (1 − ) − 16 r 2ρ˜ r2 β  Again use part (iii) to conclude that the right-hand side is strictly positive for υ large enough.  166  

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