Renormalization group analysis ofself-interacting walks and spin systemsbyBenjamin WallaceB.Sc. Honours, Queen’s University, 2012M.Sc., Queen’s University, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)The University Of British Columbia(Vancouver)June 2017c© Benjamin Wallace 2017AbstractThe central concern of this thesis is the study of critical behaviour in models of statisticalphysics in the upper-critical dimension. We study a generalized n-component lattice |ϕ|4 modeland a model of weakly self-avoiding walk with nearest-neighbour contact self-attraction on theEuclidean lattice Zd. By utilizing a supersymmetric integral representation involving bosonand fermion fields, the two models are studied in a unified manner.Our main result, which is contingent on a small coupling hypothesis, identifies the preciseleading-order asymptotics of the two-point function, susceptibility, and finite-order correlationlength of both models in d = 4. In particular, we show that the critical two-point functionsatisfies mean-field scaling whereas the near-critical susceptibility and finite-order correlationlength exhibit logarithmic corrections to mean-field behaviour. The proof employs a renormal-ization group method of Bauerschmidt, Brydges, and Slade based on a finite-range covariancedecomposition and requires two extensions to this method.The first extension, which is required for the computation of the finite-order correlationlength (even for the ordinary weakly self-avoiding walk and |ϕ|4 model), is an improvement ofthe norms used to control the evolution of the renormalization group. This allows us to obtainimproved error estimates in the massive regime of the renormalization group flow.The second extension involves the identification of critical parameters for models initializedwith a non-zero error coordinate coupled to a marginal/relevant coordinate. This allows us, forexample, to realize the two-point function and susceptibility for the walk with self-attraction asa small perturbation of the corresponding quantities without self-attraction, whose asymptoticbehaviour was determined by Bauerschmidt, Brydges, and Slade. This establishes a form ofuniversality.iiLay SummaryIn this thesis, we study two models from statistical physics: the weakly self-avoiding walk withself-attraction, which is a model of a linear polymer in a poor solution; and the lattice |ϕ|4model, which can be understood as a model of a ferromagnet. Both models are expected toundergo a phase transition at which certain precise quantitative properties, known as criticalexponents, should become independent of the fine model specifications. Our main result pro-vides a mathematically rigorous confirmation of several predicted values of critical exponentsfor both models.iiiPrefaceSections 1–1.6 are a general introduction to the subject matter of this thesis and we do notclaim any originality here. Section 1.7 states our main result, which combines and slightlyextends results from the following:• the article [12], written jointly with Roland Bauerschmidt, Gordon Slade, and AlexandreTomberg and published in Annales Henri Poincare´; and• the article [13], written jointly with Roland Bauerschmidt and Gordon Slade and publishedin Journal of Statistical Physics.Section 1.8 includes part of [13].Chapters 2–5 are based on [12,13]:• Chapter 2 includes part of [12,13], and discusses the general theory developed in [10,28–31]and applied and extended in [8, 9, 108];• Chapter 3 includes part of [12, 13] and Section 3.1 includes an additional discussion re-garding an argument of [9];• Chapter 4 includes part of [12]; in addition, Sections 4.1 and 4.3.4 include discussionsregarding some of the ideas in [31] and [29], respectively; and• Chapter 5 includes part of [13].ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Asymptotic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Generating function and Laplace transform . . . . . . . . . . . . . . . . . 41.2 Equilibrium statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Relation to quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Functions on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 The graph Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.3 The Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Examples of spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Phase transition in the Ising model . . . . . . . . . . . . . . . . . . . . . 121.4.3 Infinite-volume spin systems . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Critical behaviour and universality . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5.1 The critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5.2 Critical behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5.3 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5.4 The renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . 181.6 Self-interacting walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6.1 Weakly self-avoiding walk with self-attraction . . . . . . . . . . . . . . . 211.6.2 Predicted behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24vTable of Contents1.7 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.8 Relations between models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.8.1 The n→ 0 limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.8.2 Self-avoiding walk representation . . . . . . . . . . . . . . . . . . . . . . . 311.9 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Renormalization group method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 Reformulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3 Progressive integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 The space of field functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4.2 The Tφ seminorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4.3 Norm weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 Perturbative coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5.2 Local field polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.5.3 Perturbative flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6 Non-perturbative coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.6.1 Initial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 Renormalization group step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.8 Renormalization group flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.9 Bibliographic remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Analysis of critical behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1 Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1.1 Change of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1.2 Conclusion of the argument . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 Two-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3 Finite-order correlation length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 The renormalization group step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.1 Simplified renormalization group step . . . . . . . . . . . . . . . . . . . . . . . . 664.1.1 Main contributions to K+ . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Improved norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2.1 Covariance bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2.2 New choice of norm beyond the mass scale . . . . . . . . . . . . . . . . . 704.3 Proof of Theorem 2.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.1 Norm parameter ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73viTable of Contents4.3.2 Stability domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.3 Extension of stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 754.3.4 Extension of the crucial contraction . . . . . . . . . . . . . . . . . . . . . 765 Critical initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Initial coordinates for the renormalization group . . . . . . . . . . . . . . . . . . 815.1.1 Properties of the Tϕ seminorm . . . . . . . . . . . . . . . . . . . . . . . . 815.1.2 Bounds on K0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.1.3 Unified bound on K0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1.4 Smoothness of K0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Renormalization group flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3 Critical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.1 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.1.1 Long-range models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.1.2 The O(n) model and self-avoiding walk . . . . . . . . . . . . . . . . . . . 936.2 Other observable quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.1 The correlation length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.2.2 Inversion of the Laplace transform . . . . . . . . . . . . . . . . . . . . . . 956.2.3 The broken symmetry phase . . . . . . . . . . . . . . . . . . . . . . . . . 95Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96AppendicesA Finite-volume approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A.1 A monotonicity lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A.2 Convergence of the finite-volume approximation . . . . . . . . . . . . . . . . . . 105B Moments of the free Green function . . . . . . . . . . . . . . . . . . . . . . . . . 107B.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107B.2 Riemann sum approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107B.3 Covariance decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108B.4 Proof of main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109C An implicit function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111C.1 Implicit function theorem with a parameter . . . . . . . . . . . . . . . . . . . . . 111C.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111viiList of FiguresFigure 1.1 Phase diagram of H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Trapped SAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 1.3 Discrete WSAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 1.4 Phase diagram of WSAW-SA . . . . . . . . . . . . . . . . . . . . . . . . . 27viiiAcknowledgementsI would like to thank my advisor Gordon Slade for his invaluable guidance and support duringmy time at UBC. I cannot overstate how much I have benefited from his patience and generosity,without which this thesis would certainly not have been possible. I would also like to thankJoel Feldman and Ed Perkins for serving on my thesis committee. The work discussed in thisthesis resulted from collaborations with Gord, Roland Bauerschmidt, and Alex Tomberg and Iam grateful towards all of them for fruitful discussions as well as for a lot of helpful advice. Iam also grateful to David Brydges for a number of enlightening and inspiring conversations aswell as for suggesting a wealth of promising research directions. I thank Maxime Bergeron andTom Hutchcroft for helpful comments on Chapter 1 of this thesis and I am thankful towardsthe UBC probability group as a whole for providing such a stimulating and friendly researchenvironment.Lastly, I am grateful for the unwavering support I have received from my family and friendsthroughout the course of my studies.ixChapter 1IntroductionThermodynamics originated as the study of heat and its transformations into other forms ofenergy, but can more generally be described as the study of bulk matter. That is, it seeksto answer questions regarding those properties of matter that are defined in terms of largecollections of particles. Two familiar examples are density and temperature. The density of asubstance is its mass per unit volume. Such a quantity is sometimes said to be intensive: itdepends only on the nature of the substance under consideration, not on the amount that ispresent.There is a small caveat that should be added to this definition: the mass per unit volumeof a sample of some substance will only become independent of the volume once this volume issufficiently large. This makes density a macroscopic quantity (as opposed to, e.g. the atomicnumber). To say that thermodynamics is the study of bulk matter amounts to saying that itdeals in macroscopic quantities.This raises a number of simple questions. For instance, if these quantities are truly intensive,then it should be possible to derive them from the microscopic behaviour of the substance’sconstituent particles; yet above, we have resorted to discussing “large” quantities of matterin order to make sense of density. In fact, this leads us to another natural question: whatconstitutes a sample sufficiently large to be considered bulk matter and why should certainmeasurements of a sample stabilize when the sample is large?A natural response is that some sort of law of large numbers must be at work. Althoughthe behaviour of a system of particles is not random, it may be sufficiently complex that it isreasonable to view it as such. This is the basic idea of equilibrium statistical mechanics: toleverage the complexity of large systems of particles in order to explain the apparent simplicityof bulk matter as arising from the kind of self-averaging that pervades probability theory. Fromthis point of view, macroscopic quantities are types of averages that arise from some probabilitydistribution. These averages typically depend on temperature or other parameters, so in thissense the main objects of study in statistical mechanics are certain parameterized families ofprobability measures.It is useful to organize these measures according to their qualitative properties. Qualitativelysimilar measures correspond to a phase of a substance. For instance, water at 20◦C is notsignificantly different from water at 50◦C. On the other hand, steam and water have verydifferent behaviours despite both being H2O. In fact, the boiling of water is signalled by anabrupt drop in density by a factor of around 1/1000 at normal atmospheric pressure.1Chapter 1. IntroductionGasLiquidSolidTemperature (T )Pressure (P )Tc(P )PcTc(Pc)Figure 1.1: The phase diagram of H2OA similar effect occurs under different atmospheric conditions as well. The temperature atwhich water boils varies as a function of pressure Tc = Tc(P ) and a phase transition is said tooccur when the pressure and/or temperature are varied in such a way as to cross the graph ofthis function; see Figure 1.1. However, as P is increased, the density difference along this curvedecreases and there is a critical point (Tc(Pc), Pc) at which this difference vanishes.Since statistical mechanics deals with extremely large, complicated systems of interactingparticles, only simplified models of real materials can usually be studied in detail. Such modelsare useful for building a qualitative understanding of the phases of matter and phase transitions.However, the simplifications inherent in their definitions mean that they are usually not suitablefor making quantitative predictions.A remarkable phenomenon, known as universality, is that this is no longer entirely true atthe critical point. At criticality, many quantities behave in a way that is independent of the finedetails of the model being used. Thus, some of the quantitative properties of real materials canin principle be predicted exactly by studying models only roughly resembling these materialsIn the 1970’s Ken Wilson, inspired by ideas in quantum field theory, gave an explanation ofuniversality in terms of the renormalization group: an abstract dynamical system that acts onmodels by averaging out their fine details. This idea was enormously successful and led to his1982 Nobel Prize.There have been several rigorous implementations of Wilson’s ideas, some of which willbe mentioned in Section 1.5.4. The main purpose of this thesis is to discuss extensions of arigorous renormalization group method of Bauerschmidt, Brydges, and Slade that have beenused to study the critical behaviour of a generalized |ϕ|4 model and a model of weakly self-avoiding walk with contact self-attraction (WSAW-SA). We will introduce these models andour main results in the present chapter and discuss the proofs in the remainder of the thesis.21.1. AsymptoticsWe begin in Section 1.1 with some general background on asymptotic notation and themethod of generating functions and Laplace transforms. In Section 1.2, we briefly introducesome of the basic ideas of equilibrium statistical physics. The models we study are defined ongraphs, which we discuss in Section 1.3. We define spin systems, in particular the |ϕ|4 model, inSection 1.4. Critical behaviour and the renormalization group is probably most easily explainedin the context of such systems, and we give an informal description of these ideas in Section 1.5.In Section 1.6, we introduce the WSAW-SA and discuss its critical behaviour. This gives usall the necessary definitions to state our main results in Section 1.7. Before proceeding to themethod of proof, we discuss the close relationship between models of walks and spin systems inSection 1.8; in particular, we recall a representation of the WSAW-SA in terms of a spin systemrelated to the |ϕ|4 model that allows us to unify our treatment of both models. The remainderof the thesis is outlined in Section 1.9.1.1 AsymptoticsWe begin with a short discussion of some useful notation and mathematical background.1.1.1 Asymptotic notationLet F and G be a functions on a subset of the real line. For a ∈ [−∞,∞], we writeF (x) ∼ G(x), x→ a (1.1.1)F (x) = o(G(x)), x→ a (1.1.2)if, respectively,limx→aF (x)G(x)= 1 (1.1.3)limx→aF (x)G(x)= 0. (1.1.4)We also writeF (x) = O(G(x)) or F (x) ≤ O(G(x)), x→ a (1.1.5)if there is a constant C ≥ 0 such that |F (x)| ≤ CG(x) for all x in some neighbourhood of a (inthe extended real line). Lastly, we writeF (x) G(x), x→ a (1.1.6)if F (x) = O(G(x)) and G(x) = O(F (x)) as x→ a, possibly with different constants for the twoinequalities. We will sometimes write F (x) ≈ G(x) in heuristic arguments where the hope isthat a rigorous argument might replace ≈ by or ∼.31.2. Equilibrium statistical mechanics1.1.2 Generating function and Laplace transformThe generating function of a sequence an is the function g(z) defined by the power series withcoefficients an:g(z) =∞∑n=0anzn. (1.1.7)If the function g is sufficiently well understood, then the coefficients can be recovered by differ-entiation:an =1n!g(n)(0). (1.1.8)This is known as the method of generating functions [114].In many cases, g cannot be computed exactly. Nevertheless, there is a close relationshipbetween the asymptotics of the sequence an as n→∞ and the function g(z) near its dominantsingularities, i.e. its singularities closest to the origin [52]. For instance, if an ∼ r−nnα, thenthe root test implies that the generating function f has radius of convergence r. If, moreover,an ≥ 0, then g(z) ∼ C(r − z)−(α+1) as z ↑ r. This is an example of an Abelian theorem. Theconverse does not always hold; a theorem providing conditions under which the converse is trueis known as a Tauberian theorem, and is generally harder to prove than its Abelian counterpart.Likewise, the asymptotics of a function f(T ) as T → ∞ can sometimes be recovered fromthe behaviour of its Laplace transformG(ν) =∫f(T )e−νT dT (1.1.9)near ν0 := inf{ν : G(ν) < ∞}. There are Abelian and Tauberian theorems for the Laplacetransform analogous to those for generating functions. For instance, f(T ) ∼ ATα implies thatG(ν) ∼ A′(ν − ν0)−(α+1) as ν ↓ ν0 and the converse holds when f is monotone (see [113]).1.2 Equilibrium statistical mechanicsLet (Ω, λ) be a measure space. We view λ as some “natural” measure on Ω. The dynamics ofa physical system with state space Ω are often determined by a function H on Ω, known as theHamiltonian. Typically, H(ω) represents the total energy of the system in state ω; it is thusreasonable to assume that H is bounded below1.Example 1.2.1. The canonical example is a system of n point particles in a domain U ⊂ R3,for which Ω = (U × R3)n and λ is Lebesgue measure on Ω. A state ω ∈ Ω consists of thepositions qi ∈ U and momenta pi ∈ R3 of the n particles. The Hamiltonian is usually smooth1It is convenient to allow H to take on negative values.41.2. Equilibrium statistical mechanicsand the dynamics are determined by Hamilton’s equationsdqdt= ∇pH (1.2.1)−dpdt= ∇qH. (1.2.2)Typically, the Hamiltonian has the formH(q, p) =12m|p|2 + U(q). (1.2.3)The first term is the usual definition of kinetic energy and the second term, which depends onlyon q, is a potential energy function.In statistical mechanics, the systems of concern consist of a very large number of particles(n large in the previous example). Typically, the state space Ω is very high-dimensional andstudying the exact dynamics is infeasible.A common simplifying assumption is that after a long time has passed, the system willsettle into a state of thermal equilibrium, meaning that there is no net flow of heat between thesystem and its surroundings. Thus, the temperature is constant; we denote by β the inversetemperature. The Gibbs measure for a system with Hamiltonian H at inverse temperature β isthe probability measure on Ω given byµβ(dω) =1Ze−βH(ω)dλ(ω) (1.2.4)when this is well-defined. The normalizing constantZ =∫e−βH dλ (1.2.5)is known as the partition function.1.2.1 Relation to quantum theoryLet us briefly describe the relationship between equilibrium statistical mechanics and quantumfield theory. Our presentation is heuristic and involves manipulations of a priori ill-definedobjects. A rigorous development of these ideas goes back to [95, 97, 98, 110]; useful referencesare [105] (for quantum mechanics) and [60] (for quantum field theory).Let H(x, p) be the Hamiltonian (1.2.3) (we have written x instead of q). In quantummechanics, the state space is replaced by a Hilbert space, usually L2 = L2(R3n); the position xjis replaced by the position operator xˆj given by multiplication by xj ; and the momentum pj isreplaced by the momentum operator pˆj = −i~∂/∂xj (we ignore details regarding the domainsof these operators). The resulting operator Hˆ = H(xˆ, pˆ) on L2 determines the evolution of the51.2. Equilibrium statistical mechanicswave function ψ ∈ L2 via the Schro¨dinger equationi~dψdt= Hˆψ (1.2.6)whose solution with initial condition ψ(0) is given by ψ(t) = e−itHˆ/~ψ(0).Suppose that the solution operator is an integral operator with kernel Kt:e−itHˆ/~f =∫Kt(·, y)f(y) dy. (1.2.7)The Feynman path integral formulation of quantum mechanics [50] involves writing the kernelasKt(a, b) =∫Wt(a,b)e(i/~)∫ t0 L(x(s),x˙(s)) ds Dx, (1.2.8)whereWt(a, b) is a space of paths [0, t]→ R3n from a to b equipped with a “Lebesgue measure”Dx andL(x, x˙) =12m|x˙|2 − U(x) (1.2.9)is the Lagrangian of the classical system we started with. As such, the integral representationof Kt is ill-defined; for instance, the measure Dx on paths does not exist. However, let us notconcern ourselves with this.Suppose that ψ can be analytically continued to a region of the complex plane containingthe positive imaginary axis. Then we might hope the function t 7→ ψ(−it) has solution operatorwith kernel of the formK−it(a, b) =∫W−it(a,b)e(i/~)∫−it0 L(x(s),x˙(s)) ds Dx. (1.2.10)By the change of variables s = −iu,(i/~)∫ −it0L(x(s), x˙(s)) ds = (1/~)∫ t0L(x˜(u), i ˙˜x(u)) du = −(1/~)∫ t0H(x˜(u),m ˙˜x(u)) du(1.2.11)with x˜(t) = x(−it). Thus, by the fictive change of variables in which Wit(a, b) is replaced byWt(a, b), we getK−it(a, b) =∫Wt(a,b)e−(1/~)∫ t0 H(x(u),mx˙(u)) du Dx (1.2.12)This procedure is known as a Wick rotation.The analogue of this idea in quantum field theory involves replacing integration over pathswith integration over fields ϕ, which are functions on Rd. The corresponding “measures” of theform e−(1/~)∫HDϕ should be compared with the Gibbs measures (1.2.4) with Planck’s constant~ playing the role of temperature. Although H has been replaced by the integral∫H, we willdiscuss how natural choices for the Hamiltonian in a Gibbs measure are given by sums over61.3. Graphsspaces of fields on discrete approximations to Rd (graphs).Remark 1.2.2. In fact, it turns out that (1.2.12) is not too unreasonable. For simplicity, letus take ~ = 1 and m = 1. Then we would expect ψ˜ to solvedψ˜dt= −Hˆψ˜. (1.2.13)When U = 0, we have H(x, p) = |p|2, so Hˆ = −12∆R3n is the Laplacian on R3n and (1.2.13)is the heat equation. On the other hand,∫ t0 H is formally a positive-definite quadratic form,so (1.2.12) is a formal Gaussian integral. This is consistent with the well-known fact that asolution ψ˜ of the heat equation on a domain can (under appropriate conditions) be writtenas ψ˜(t, x) = E(ψ˜(0, Bt) | B0 = x), where Bt is a Brownian motion. More generally, underappropriate hypotheses, the Feynman-Kac formula expresses solutions to (1.2.13) with U 6= 0as averages with respect to a Brownian motion.1.3 GraphsAn undirected graph or simply a graph is a pair G = (V, E), where V is a set of vertices and Eis a set of edges {x, y} with x, y ∈ V; we will write x ∼ y if {x, y} ∈ E . For simplicity, we willassume that V is countable and that there are no self-loops {x} ∈ E .A graph automorphism is a bijection f : V → V such that x ∼ y if and only if f(x) ∼ f(y).We will assume that G is transitive meaning that for all pairs of distinct vertices a, b ∈ V,there exists an automorphism f with f(a) = b. We fix a vertex 0 ∈ V whose precise choice isimmaterial due to transitivity.1.3.1 Functions on graphsLet us denote the components of an element ϕ ∈ (Rn)V by ϕix ∈ R for x ∈ V and i = 1, . . . , n.The Euclidean inner product and norm on (Rn)V are defined byϕ · ϕ˜ =∑x∈Vϕx · ϕ˜y =n∑i=1∑x∈Vϕixϕ˜ix (1.3.1)|ϕ|2 = ϕ · ϕ. (1.3.2)A V × V matrix M = (Mxy)x,y∈V acts on ϕ component-wise:(Mϕ)x =∑y∈VMxyϕy. (1.3.3)71.3. Graphs1.3.2 The graph LaplacianLet us say that a V ×V matrix M is indexed by E if Mxy 6= 0 if and only if x ∼ y. Throughoutthis chapter, we let J be a matrix indexed by E with nonnegative entries. Thus,Jxy ≥ 0 (1.3.4)with equality if and only if x 6∼ y. The pair (G, J) is an example of a weighted graph. We willusually denote this weighted graph simply as G, with J implicit.Let D be the diagonal V × V matrix with diagonal entriesdx = Dxx =∑y∼xJxy, (1.3.5)where the sum is over all vertices y adjacent to x. The graph Laplacian on G is defined by−∆ = D − J. (1.3.6)An important case is when Jxy = 1x∼y, for which−∆xy = dx1x=y − 1x∼y. (1.3.7)We also define the massive Laplacian with squared mass m2 > 0 by−∆ +m2. (1.3.8)Note thatϕ · (−∆ϕ) = 12∑x,y∈VJxy|ϕx − ϕy|2 ≥ 0, (1.3.9)so −∆ is positive-semidefinite.Example 1.3.1. We will often work on the discrete d-dimensional torus of side LN , definedbyΛ = ΛN = Zd/LNZd (1.3.10)for integers L > 1 and N ≥ 0. We view Λ as a graph with V = Λ and x ∼ y if x and y havedistance 1 on the torus. This graph is transitive and dx = 2d for all x.1.3.3 The Green functionIf m2 > 0, then −∆ +m2 is positive-definite, hence invertible with inverse(−∆ +m2)−1 = (m2 +D)−1∞∑n=0ZnPn, (1.3.11)81.4. Spin systemswhereZ = (m2 +D)−1D, P = D−1J. (1.3.12)Let zx denote the diagonal elements of Z. The Green function of G is the kernel of (−∆+m2)−1,given byCxy = (m2 + dx)−1∞∑n=0znxPnxy. (1.3.13)1.4 Spin systemsWe begin by restricting our attention to spin systems in finite volume:|V| <∞. (1.4.1)Suppose that S ⊂ Rn is equipped with a measure dλ0, let Ω = SV , and define the productmeasuredλ(ϕ) =∏x∈Vdλ0(ϕx), ϕ ∈ Ω. (1.4.2)We refer to the elements of Ω as fields or spin configurations on V with spins in S.Let H : Ω → R be a function and suppose that e−H is integrable with respect to dλ. Thespin system with Hamiltonian H : Ω → R at inverse temperature β is given by the Gibbsmeasuredµβ(ϕ) =1Zβe−βH(ϕ) dλ(ϕ). (1.4.3)We sometimes add an external field h ∈ R by considering the measuredµβ,h(ϕ) =1Zβ,he−β(H(ϕ)−h∑x∈V ϕx) dλ(ϕ). (1.4.4)We will mainly be concerned with ferromagnetic spin systems, for which the Hamiltonian isgiven byH(ϕ) = −ϕ ·Mϕ, Mxy ≥ 0. (1.4.5)The total energy of such a system is a sum of contributions of the form −2ϕx ·Mxyϕy for x ∼ y.Such a contribution attains its minimum when ϕx = ϕy and its maximum when ϕx = −ϕy. Inthis sense, it is energetically favourable for spins to align in ferromagnetic systems.1.4.1 Examples of spin systemsBelow we discuss some common examples of spin systems. In many cases, we discuss Hamilto-nians that depend on one or more parameters for which adjusting the inverse temperature β isequivalent to rescaling these parameters. In these cases, we will fixβ = 1 (1.4.6)91.4. Spin systemswithout loss of generality.Gaussian measuresSuppose S = Rn and λ0 is Lebesgue measure. Then the Hamiltonian H must be boundedbelow for the Gibbs measure (1.4.3) to be well-defined. Essentially the simplest class of non-constant2 Hamiltonians are positive-definite quadratic forms. These are given by a positive-definite symmetric V × V matrix C, called the covariance matrix, and take the formHC(ϕ) =12ϕ · C−1ϕ. (1.4.7)The corresponding Gibbs measures are Gaussian measures.The partition function can be computed explicitly and the Gibbs measure takes the formdϕ√det(2piC)e−12ϕ·Aϕ. (1.4.8)Wick’s theorem gives an expression for the moments: if x1, . . . , x2p ∈ Λ, then∫dµC(ϕ)2p∏k=1ϕixk =∑pi∏kl∈piCxkxl (1.4.9)where the sum is over all pairings pi of {1, . . . , 2p} (i.e. partitions of this set into 2-elementsubsets).The |ϕ|4 modelAs in the previous example, let S = Rn and λ0 be Lebesgue measure. The next step up incomplexity is a quartic Hamiltonian. In particular, we have the Hamiltonian for the lattice |ϕ|4model or Ginzburg-Landau-Wilson model :Hg,ν(ϕ) =∑x∈V(14g|ϕx|4 + 12ν|ϕx|2 + 12ϕx · (−∆ϕ)x), (1.4.10)where g > 0 and ν ∈ R. The expression (1.4.10) should be compared with (1.2.3). With p = mq˙in the latter expression, the kinetic energy takes the form 12m|q˙|2. The lattice analogue of thisquantity in (1.4.10) is12∑x∈Vϕx · (−∆ϕ)x = 14∑x∈V∑y∼x(ϕy − ϕx)2, (1.4.11)where we have applied (1.3.9) to get the right-hand side.2The Gibbs measure with constant Hamiltonian is just the product measure.101.4. Spin systemsExample 1.4.1 (The GFF). With g = 0 and ν > 0,H0,ν(ϕ) = HC(ϕ), C = (−∆ + ν)−1. (1.4.12)In other words, the |ϕ|4 model becomes the Gaussian measure with covariance given by themassive Green function. The corresponding spin system is the discrete massive Gaussian freefield or GFF.The continuum analog of this model with mass 0 (which can be defined on Rd with d > 2)is a simple example of a non-interacting3 quantum field theory. The |ϕ|4 model was introducedin attempts to construct an interacting theory.The O(n) spin modelConsider the |ϕ|4 model and suppose that ∑y∼x Jxy = d0 for all x ∈ V (e.g. this occurs in thesetting of (1.3.7) when every vertex in G has d0 neighbours). In this case, we can writeHg,ν(ϕ) =∑x∈VUg,ν(ϕx)− 12ϕ · Jϕ, (1.4.13)where the single-spin potential Ug,ν is given byUg,ν(t) =14g|t|4 + 12(ν + d0)|t|2, t ∈ Rn. (1.4.14)We can see from (1.4.13) that the |ϕ|4 model is ferromagnetic. When ν + d0 < 0, the potentialhas the shape of a double well with roots at t = 0 and on the sphere |t| = ±√−2(ν + d0)/g.Thus, as g →∞ with ν = −(d0 +g/2), the Gibbs measure for the |ϕ|4 model converges (weakly)to the Gibbs measure with HamiltonianHJ(ϕ) = −12ϕ · Jϕ, ϕ ∈ (Sn−1)V (1.4.15)where Sn−1 ⊂ Rn is the (n−1)-dimensional unit sphere. This is known as the O(n) spin model.The special case n = 1 is the famous Ising model. The cases n = 2 and n = 3 are known as theXY model and the classical Heisenberg model.Remark 1.4.2. The Ising model on Zd (defined by a limiting procedure) was introduced byWillhelm Lenz [87]. Lenz suggested it as a problem for his student Ernst Ising, who determinedin 1924 [75] that there is no (non-trivial) phase transition when d = 1. In [70] Heisenberg,seeking a model that would possess a phase transition, proposed the quantum version of hismodel (see [16] for a discussion on this). Ironically, it turns out that the classical version ofhis model does not possess a phase transition in both d = 1 and d = 2 [92]; and, moreover,in 1936 Rudolf Peierls [100] put forth an argument for the existence of a phase transition in3The corresponding classical model only involves the kinetic term (1.4.11).111.4. Spin systemsthe Ising model if d > 1. Peierls’ argument was made rigorous in [62], but even before thatOnsager [96] provided an exact computation of the free energy in d = 2, which incontrovertiblydemonstrated the existence of a phase transition in two dimensions. The general O(n) modelwas first studied in [109].1.4.2 Phase transition in the Ising modelLet G = ΛN and consider the Hamiltonian for the Ising model with interaction Jxy = 1x∼y inan external field h ∈ R:H(N)h (ϕ) = −12∑x∼yϕxϕy − h∑x∈Vϕx, ϕ ∈ {±1}V . (1.4.16)Let 〈·〉(N)β,h denote the expectation with respect to the corresponding Gibbs measure µ(N)β,h andlet Z(N)β,h be the partition function.The free energy is defined (in finite volume) byF(N)β,h = −1β|V| logZ(N)β,h . (1.4.17)The magnetization is given byM(N)β,h =1|V|∑x∈V〈ϕx〉(N)β,h (1.4.18)and can be written in terms of the free energy asM(N)β,h = −∂∂hF(N)β,h . (1.4.19)When h = 0, the Gibbs measure is invariant under the spin-flip transformation ϕ 7→ −ϕ and itfollows that M(N)β,0 = 0 for all β.In order to study the dependence of the magnetization on h, we define the magnetic sus-ceptibilityχ(N)(β, h) =1β∂∂hM(N)β,h . (1.4.20)By translation-invariance,χ(N)(β, h) =1β2∑x∈VG(N)x (β, h), (1.4.21)whereG(N)x (β, h) = 〈ϕ0ϕx〉β,h − 〈ϕ0〉β,h〈ϕx〉β,h (1.4.22)is the correlation between ϕ0 and ϕx, known as the two-point function. These are all analyticfunctions. In order to detect a phase transition, we must take the infinite-volume limit N →∞.The infinite-volume free energy, magnetization, susceptibility, and two-point function aredefined as the N →∞ limits of their finite-volume counterparts; we denote them by Fβ,h, Mβ,h,121.4. Spin systemsχ(β, h), and Gx(β, h). LetM±β = limh→0±Mβ,h. (1.4.23)Then when β is sufficiently large [62,100], there is a non-zero spontaneous magnetization, mean-ing thatM−β < 0 < M+β . (1.4.24)That is, the magnetization is discontinuous at h = 0. Equivalently, the free energy is notdifferentiable4 at that point. The critical point for the Ising model is the inverse temperatureβc = sup{β > 0 : M+β = 0}. (1.4.25)At the critical point, the magnetization is continuous [4,118], i.e. the free energy is differentiable.However, the susceptibility diverges (see [3]), so the free energy is not twice-differentiable. Thisdivergence corresponds to slow decay of the two-point function, which will be discussed furtherin Section 1.5.1.4.3 Infinite-volume spin systemsA broad distinction can be made between first-order phase transitions in which the free energyhas discontinuous first derivative with respect to an external field h and continuous phasetransitions, in which the free energy is differentiable but non-analytic. As discussed above, theinfinite-volume limit must be taken in order for a phase transition to manifest. For the Isingmodel, we took the infinite-volume limit of the free energy along a sequence of increasing tori.It is worth mentioning that a more general and elegant approach to the study of infinite-volume spin systems was developed by Dobrushin [41] and Lanford and Ruelle [82]. An excellentintroduction to this subject is given in [54] and a comprehensive reference is [59] (see also [86]for spin systems with unbounded 1-component spins).Loosely speaking, this approach take as fundamental not the Hamiltonian but rather a“potential”, which is a collection of functions encoding the microscopic interactions from whichthe Hamiltonian is to be defined; for instance, for the Ising model, the Hamiltonian is a sumof contributions of the form Jxyσxσy for x ∼ y. Given such a potential, a Hamiltonian canbe defined on any finite subgraph of G and a probability measure on Ω is said to be a Gibbsmeasure or Gibbs state whenever its finite-volume conditional measures are of the form (1.4.3).This is somewhat in the spirit of Kolmogorov’s consistency conditions with the importantdifference that the resulting collection Gβ of Gibbs states at inverse temperature β need notconsist of only a single element. This is significant due to the interpretation of distinct elementsof Gβ as corresponding to different phases. For many systems of interest there is a criticalinverse temperature βc such that |Gβ| > 1 if and only if β > βc. The region β > βc is typicallyassociated with first-order phase transitions whereas continuous phase transitions usually occur4There is an interchange of limit and derivative implicit in this statement, but it can be justified.131.4. Spin systemsat the critical point βc.In this thesis, we are mainly interested in the single-phase regime β ≤ βc. Thus, we will avoidthe issue of existence and uniqueness of infinite-volume Gibbs measures by defining observablequantities of interest in infinite volume as limits of their finite-volume counterparts as was donein the previous section. Below we give the definitions we will use throughout the rest of thisthesis for (a generalization of) the |ϕ|4 model.Generalized |ϕ|4 modelWe view ΛN as a subset of Zd approximately centred at the origin (say as [−12LN+1, 12LN ]d∩Zdif LN is even and as [−12(LN − 1), 12(LN − 1)]d ∩ Zd if LN is odd). This allows us to preservetranslation-invariance of the models that concern us when defining them in finite volume.We fix Jxy = 1x∼y so that−∆xy = 2d1x=y − 1x∼y. (1.4.26)We will study a generalization of the |ϕ|4 model whose Hamiltonian on ΛN is given byVg,γ,ν,N (ϕ) =∑x∈ΛN(14(g − γ)|ϕx|4 + 12ν|ϕx|2 + 12ϕx · (−∆ϕ)x + 14dγ(∇|ϕx|2)2), (1.4.27)where(∇|ϕx|2)2 =∑|e|=1(∇e|ϕx|2)2 (1.4.28)and the discrete gradient in the direction of a unit vector e ∈ Zd is defined by∇efx = fx+e − fx. (1.4.29)The expectation with respect to the corresponding Gibbs measure will be denoted 〈·〉g,γ,ν,N .Following (1.4.21) and (1.4.22), we define the two-point function and susceptibility byGx,N (g, γ, ν;n) =1n〈ϕ0 · ϕx〉g,γ,ν,N , Gx(g, γ, ν;n) = limN→∞Gx,N (g, γ, ν;n) (1.4.30)andχ(g, γ, ν;n) = limN→∞∑x∈ΛNGx,N (g, γ, ν;n). (1.4.31)Existence of these limits (which is not known in general) will follow from the proof of our mainresult.Remark 1.4.3. We have omitted the term 〈ϕ0〉g,γ,ν,N · 〈ϕx〉g,γ,ν,N which vanishes in the regimeof interest β ≤ βc.141.5. Critical behaviour and universality1.5 Critical behaviour and universalityMany systems exhibit singular behaviour at or near the critical temperature in the form ofpower law scaling of various observable quantities. This is known as critical behaviour. Forconcreteness, we will discuss the generalized |ϕ|4 model on Zd given by (1.4.27) with Jxy = 1x∼y.1.5.1 The critical pointWe follow the convention (1.4.6) of setting β = 1 and seek a phase transition for |ϕ|4 modelwhen the parameter ν is varied. We define the critical point byνc = νc(g, γ;n) = inf{ν : χ(g, γ, ν;n) <∞}. (1.5.1)By (1.4.31), it is reasonable to expect rapid (i.e. summable) decay of Gx(g, γ, ν;n) in |x| forν > νc and much slower decay at ν = νc.In fact, the two-point function is expected to decay exponentially above νc. The correlationlength ξ is defined to be the reciprocal of the exponential rate of decay of the two-point function;concretely, we letξ(g, γ, ν;n) = lim supk→∞−klogGke(g, γ, ν;n), (1.5.2)where e ∈ Zd is a unit vector whose choice is irrelevant by invariance of this model under latticerotations. Roughly speaking, the correlation length acts as a “macroscopic length scale” of themodel; it is a measure of the largest scale at which spins are very strongly correlated. Basedon the above discussion, we expect ξ to diverge as ν ↓ νc. This divergence is one of the keyfeatures of critical behaviour.A related quantity is the correlation length of order p, defined byξp(g, γ, ν;n) =(∑x∈Zd |x|pGx(g, γ, ν;n)χ(g, γ, ν;n))1/p. (1.5.3)An analysis of ξp requires less control over the behaviour of the two-point function than wouldbe needed to study the correlation length ξ. Our main result includes a statement regardingthe critical behaviour of ξp in d = 4.Remark 1.5.1. There is a simple heuristic relationship between ξ and ξp. Suppose thatthe two-point function decays exponentially at rate 1/ξ, possibly with some sub-exponentialmultiplicative correction; for instance, suppose thatGx(g, γ, ν;n) ≈ C|x|−αe−|x|/ξ(g,γ,ν;n), (1.5.4)in some sense, where α and C are positive constants independent of ν. Then the main con-tributions to the numerator of (1.5.3) should come from |x| ≤ ξ = ξ(g, γ, ν;n). For such |x|,151.5. Critical behaviour and universalityGx(g, γ, ν;n) ≈ C|x|−α and so∑x∈Zd|x|pGx(g, γ, ν;n) ≈ C∑|x|≤ξ|x|−(α−p) ≈ Cξ(g, γ, ν;n)d−α+p. (1.5.5)It follows then from the definition thatξpp(g, γ, ν;n) ≈ ξp(g, γ, ν;n). (1.5.6)1.5.2 Critical behaviourFor simplicity, let us drop g, γ, and n from the notation. It is predicted that there existconstants η, γ¯, and ν¯ (unrelated to γ and ν), known as critical exponents, such thatGx(νc) ∼ C1|x|−(d−2+η), |x| → ∞ (1.5.7)χ(ν) ∼ C2(ν − νc)−γ¯ , ν ↓ νc (1.5.8)ξ(ν) ∼ C3(ν − νc)−ν¯ , ν ↓ νc (1.5.9)ξp(ν) ∼ C4(ν − νc)−ν¯ , ν ↓ νc (1.5.10)where the Ci are constants that may depend on g, γ, and n (and p when i = 4). The criticalexponents, on the other hand, are expected to be universal in the sense that they only dependon “large-scale properties” of the model such as its symmetries and the global geometry of theunderlying graph. In particular, for the n-component |ϕ|4 model on Zd, these exponents shouldonly depend on n and d and be independent of g and γ when g > 0 and γ is sufficiently small(depending on g). In fact, analogous relations are expected to hold for the O(n) spin model,with the same critical exponents.These and other relations are all believed to be manifestations of the existence of a universalscaling limit for the |ϕ|4 model and other models in its universality class. That is, any spinsystem in this class, when appropriately rescaled, is expected to converge in distribution toa unique continuum random field. In this sense, the study of critical behaviour involves far-reaching generalizations of the central limit theorem.Example 1.5.2 (The Gaussian free field). On Zd, (1.3.13) and (1.4.9) imply thatGx(0, 0,m2;n) = (m2 + 2d)−1∞∑k=0zkP k0x, (1.5.11)where z = 2d/(m2 + 2d) and P = (2d)−1J is the transition matrix for the simple random walkX on Zd. Thus, for m2 > 0,χ(0, 0,m2;n) = (m2 + 2d)−1∞∑k=0zk = (m2 + 2d)−1(1− z)−1 = 1m2. (1.5.12)161.5. Critical behaviour and universalityIt follows that νc(0, 0;n) = 0 and γ¯ = 1 for this model. It can also be shown that ν¯ = 1/2 andη = 0 in this case (e.g. see [83] for η). In Appendix B, we give a new proof that ν¯ = 1/2 forξp(0, 0,m2;n).1.5.3 Critical exponentsAs mentioned above, the critical exponents are generally expected to depend on d and n. Belowwe discuss some of the conjectured values and known results.Dimension d = 1In d = 1, nearest-neighbour models typically do not possess a phase transition. However, phasetransitions may occur for sufficiently long-range models [44].Dimensions d = 2In d = 2, the Mermin-Wagner theorem [92] implies that the O(n) model does not possessa first-order phase transition for n ≥ 2. However, the case n = 2 is expected to possessa Kosterlitz-Thouless phase transition [81] (Kosterlitz and Thouless, together with DuncanHaldane, were awarded the 2016 Nobel Prize in Physics for this and related ideas). Rigorousresults relating to this kind of phase transition include [40, 46, 47, 56]. For n = 1, there is aphase transition; in fact, Onsager [96] gave an exact formula for the free energy of the Isingmodel.The 1-component planar models are expected to possess conformally invariant scaling lim-its. As a consequence of these conformal symmetries, the predicted critical exponents arerational numbers: γ¯ = 56/32, ν¯ = 1, and η = 1/4. Recent years have shown rapid progressin this direction, stimulated by the identification by Schramm [104] of a 1-parameter familyof conformally invariant random planar curves now known as the Schramm-Loewner evolutionwith parameter κ, or SLEκ. For instance, it was shown in [34] that the interface curve (between+1 and −1 spins) for the Ising model on a bounded simply connected domain with Dobrushinboundary conditions5 converges to SLE3 in an appropriate scaling limit.Dimension d = 3Very little is known rigorously about three-dimensional models. In fact, it was only recentlyproved that the Ising model’s spontaneous magnetization vanishes continuously at the criticalpoint in three dimensions [3]. The critical exponents are not expected to take on rationalvalues. Approximate values for the exponents have been computed by non-rigorous methodsin [45,64,85].5Positive spins along one side of the boundary and negative spins on the other.171.5. Critical behaviour and universalityDimensions d > 4If d > 4, the critical exponents for the O(n) and |ϕ|4 models are predicted to become indepen-dent of d and n and to take on the values of the corresponding exponents for the Gaussian freefield, i.e. γ¯ = 1, ν¯ = 1/2, and η = 0. This phenomenon is known as mean-field behaviour anddimension 4 is called the upper-critical dimension for this class of models. For n = 1, 2 it isknown that η = 0 for the continuum limit of these models [2, 55] if it exists. On the lattice,it has been shown that η = 0 for a spread-out version of the Ising model [102] and for the 1-component |ϕ|4 model with small coupling strength [103]. Extensions to n = 1, 2 are upcomingin [23].Dimension d = 4Dimension 4 is the case of primary interest in quantum field theory, where one dimensionplays the role of time. It is predicted in dimension 4 that a number of observables possessmultiplicative logarithmic corrections to mean-field scaling. An exception is the two-pointfunction, which is expected to undergo mean-field scaling (η = 0); this was shown for the 1-component |ϕ|4 model in [57] and [48]. Logarithmic corrections to scaling of the susceptibilityand correlation length of the 1-component model were identified in [66,69].Recently, Bauerschmidt, Brydges, and Slade [10, 28–31] have developed a renormalizationgroup method for studying the n-component |ϕ|4 model in 4 dimensions; this method worksfor any n and, in a certain sense, extends to models of self-interacting walks, interpreted asn = 0 (more on this in Section 1.8). In particular, they computed logarithmic corrections toscaling of the susceptibility and specific heat and also identified the continuum massive GFF asthe scaling limit in the near-critical regime [7]. Using an extension of this method, Slade andTomberg computed asymptotics for critical correlation functions in [108]; in particular, theyshowed that η = 0. In this thesis, we discuss extensions of this method that have been used tostudy the finite-order correlation length [12] as well as more general models of walks [13].1.5.4 The renormalization groupIn [79], Leo Kadanoff considered a coarse-graining procedure for studying the Ising model inwhich disjoint blocks in Zd of side L ξ are replaced by single spins. He argued that spinsinside such blocks are so strongly correlated that the model obtained by making this replacementshould behave approximately like an Ising model with a new (“renormalized”) interaction.At the critical point, ξ =∞ and the transformation T can be iterated indefinitely resultingin a dynamical system on a space of models: the one-parameter semigroup (T j)j∈Z+ is known asthe renormalization group6. This was the basis for Ken Wilson’s generalizations of Kadanoff’sidea in [115,116].6The name renormalization group is attributed by Wilson in [115] to the work of Gell-Mann and Low [58].The relationship between these two approaches is discussed in [76].181.5. Critical behaviour and universalityThe coarse-graining procedure of Kadanoff can be viewed as an approximate method forcomputing integrals with respect to a Boltzmann weight e−βH by successively integrating outfluctuations that are “small” in the sense that they are localized in space. In Wilson’s approachfluctuations are instead localized in Fourier space.Wilson’s method results in a dynamical system on a space of models that acts by appropri-ately integrating out small fluctuations, followed by a rescaling step used to make this systemautonomous (i.e. independent of the “scale” j). He argued that the action of this dynamicalsystem would leave the long-range behaviour of the models invariant. Consequently, criticalmodels lying in the same orbit would belong to the same universality class. Thus, such mod-els should possess the same critical exponents and scaling limit. Moreover, this scaling limitshould be invariant under the action of the renormalization group, i.e. it should arise as a fixedpoint of this dynamical system. Therefore, the set of points that flow towards it form its stablemanifold.In addition to these rather broad statements regarding the nature of universality and scalinglimits, Wilson demonstrated that critical exponents could be computed by a careful analysisof the asymptotics of the renormalization group near its fixed points. He claimed that such ananalysis could be performed by approximating this infinite-dimensional dynamical system by afinite-dimensional system spanned by certain marginal and relevant directions. The remainingirrelevant directions would contract under the action of the renormalization group.By analyzing this finite-dimensional approximation, Wilson determined that there is aunique hyperbolic fixed point (in the terminology of dynamical systems theory) in dimensionsd > 4, corresponding to the Gaussian free field and mean-field behaviour. As the dimension islowered below 4, a bifurcation occurs in which the Gaussian fixed point splits into two fixedpoints. One of these corresponds to Gaussian behaviour but is unstable. The other is hyperbolicand corresponds to anomalous scaling behaviour; it is sometimes known as the Wilson-Fisherfixed point [117]. At the bifurcation point d = 4, there is only one (Gaussian) fixed point andlogarithmic corrections to mean-field scaling arise from the fact that this fixed point is nothyperbolic.There are many difficulties in making Wilson’s ideas rigorous and several different ap-proaches exist. For instance, a rigorous implementation of Kadanoff’s block-spin renormaliza-tion group was developed in [57] and used to show that η = 0 if d = 4 and n = 1. This was alsoshown independently in [48], using related ideas. By extensions to the block-spin approach,logarithmic corrections to mean-field scaling in this case were identified in [69]. This thesisconcerns the renormalization group method of Bauerschmidt, Brydges, and Slade [10, 28–31],which we will discuss further in Chapter 2.Example 1.5.3. Let us mention how renormalization group ideas can be used to prove a versionof the classical central limit theorem. This idea appears in [77, 80] and has been extended toprove stable limit laws in [88].LetXn denote a sequence of independent identically distributed continuous random variables191.6. Self-interacting walkswith mean 0 and variance 1. For such Xn, let Yn = 2−n/2(X1 + · · · + X2n). Then the mapYn 7→ Yn+1 induces a renormalization group map R on density functions given by a convolutionand rescaling of the form (Rf)(x) =√2(f ∗ f)(√2x). It is easily verified that the standardGaussian density f∗ is a fixed point of R (in fact, it is the unique fixed point).In this context, a subsequential version of the central limit theorem follows if we can showthat, for all f in an appropriate space of densities, Rnf converges to the fixed point f∗. Alocal version of this statement can be proved by analyzing the linearized map DR(f∗). Acomputation shows that this map has eigenfunctions Hi given by Hermite polynomials witheigenvalues λi = 21−i/2 for integers i ≥ 0. The relevant directions (in the sense of Wilson’srenormalization group) are those in which the linearized map is expanding; thus, the relevantsubspace is spanned in this case by H0 and H1. Similarly, the irrelevant directions are spannedby Hi with i > 2 while H2 spans the marginal subspace.1.6 Self-interacting walksBefore introducing the walks studied in this thesis, we mention the following important exampleof a self-interacting walk.Example 1.6.1 (Self-avoiding walk). Let ω : {0, . . . , n} → V be a discrete-time walk of lengthn on G, meaning that ωi ∼ ωi+1 for all i. Let us denote by Ŵn the collection of such walks andset Ŵ = ⋃n Ŵn. We say that ω is self-avoiding if ωi 6= ωj for all i 6= j. Let Sn denote thecollection of n-step self-avoiding walks with ω0 = 0. We equip Sn with the uniform measureµn for each n. This gives us a simple model of a linear polymer in a good solution. Theself-avoidance constraint models the excluded volume effect of matter.The uniform measures do not form a consistent family due to the possibility of “traps”.That is, the equalityµ|ω|(ω) =∑ω¯⊃ωµ|ω¯|(ω¯) (1.6.1)does not hold for all ω ∈ Ŵ (the sum here is over all self-avoiding walks extending ω). Forinstance, consider the self-avoiding walk ω ∈ Ŵ7 on Zd in Figure 1.2. This walk has positiveprobability under µ7 but, since there are no self-avoiding walks extending ω, the sum on theright-hand side of (1.6.1) is 0.As a result, there is no straightforward way to apply the usual methods of stochastic pro-cesses to study the self-avoiding walk. The existence of traps also contributes to the combina-torial difficulty of this model; for instance, if cn = |Sn|, then it is not clear how to express cn+1as a simple function of cn.We do know, however, that the sequence cn is sub-multiplicative: cm+n ≤ cmcn. This followsfrom the fact that a self-avoiding walk can be split at any point into two self-avoiding walks,but the concatenation of two self-avoiding walks is not necessarily self-avoiding. Thus, log cn is201.6. Self-interacting walksFigure 1.2: A trapped self-avoiding walksubadditive and Fekete’s lemma for subadditive sequences implies the existence of the limitc(G) = limn→∞n−1 log cn. (1.6.2)We call c(G) the connective constant of G. By definition, c(G) is the reciprocal of the radius ofconvergence of the susceptibility, which is defined as the generating functionχ(z) =∞∑n=0cnzn (1.6.3)of the sequence cn. Recalling Section 1.1, it is natural to study the asymptotics of the suscep-tibility as z → c(G)−1.Due to some of the difficulties involved in studying the self-avoiding walk, alternatives tothis model have been proposed. In fact, models of discrete-time walks can be defined in quite abit of generality in terms of measures on Ŵn. It is sometimes useful to work instead with modelsof continuous-time walks parameterized by intervals [0, T ]. In both cases, we can convenientlydefine Gibbs measures directly in infinite-volume with respect to the base measure induced bysimple random walk. However, instead of discussing models of walks in great generality, we willproceed directly to the case of interest in this thesis.1.6.1 Weakly self-avoiding walk with self-attractionLet X denote the continuous-time simple random walk on G conditioned to start at 0. This isthe V-valued Markov process X with generator ∆. In other words,P(Xt = y | X0 = x) = (et∆)xy. (1.6.4)Define the local time up to time T at x ∈ V byLxT =∫ T01X(S)=x dS. (1.6.5)211.6. Self-interacting walksWe define the intersection local timeIT =∫ T0∫ T01X(S1)=X(S2) dS1dS2 =∑x∈V(LxT )2 (1.6.6)and the contact self-attractionCT =∫ T0∫ T01X(S1)∼X(S2) dS1dS2 =∑x∈V∑y∼xLxTLyT (1.6.7)up to time T .Given a parameter g > 0, and γ ∈ R, letUg,γ(f) = g∑x∈Vf2x −γ2d∑x∈V∑y∼xfxfy (1.6.8)for f : V → R. The weakly self-avoiding walk with self-attraction (WSAW-SA) is defined by theHamiltonianUg,γ,T = Ug,γ ◦ LT = gIT − γ2dCT (1.6.9)which induces a Gibbs measure with respect to the measure of X. We will refer to the caseγ = 0 as the weakly self-avoiding walk (WSAW).Figure 1.3: Monte Carlo simulation of discrete-time WSAW with 100 stepsWe letcT = E0(e−Ug,γ,T), cT (x) = E0(e−Ug,γ,T1XT=x). (1.6.10)The two-point function and susceptibility are defined as the Laplace transforms of these weights:Gx(g, γ, ν) =∫ ∞0cT (x)e−νT dT (1.6.11)221.6. Self-interacting walksandχ(g, γ, ν) =∫ ∞0cT e−νT dT =∑x∈ZdGx(g, γ, ν). (1.6.12)Note that (1.6.12) is more-or-less consistent with (1.4.31) We will establish an exact analogueof (1.4.31) in Proposition 1.8.4. The relationship between the two-point function for walks andspin systems will be discussed in Section 1.8.2.We also have a version of the correlation length of order p:ξp(g, γ, ν) =(∑x∈Zd |x|pGx(g, γ, ν)χ(g, γ, ν))1/p. (1.6.13)Note thatξpp(g, γ, ν) =∫∞0 〈|X(T )|p〉cT e−νT dT∫∞0 cT e−νT dT, (1.6.14)where〈F (X)〉 = 1cTE0(F (X)e−Ug,γ,T ) (1.6.15)is the expectation induced by the weights (1.6.10). Thus, ξp is related to the Laplace transformof the mean p-th displacement 〈|X(T )|p〉. On Zd, a version of the correlation length can alsobe defined exactly as in (1.5.2).Remark 1.6.2. The discrete-time version of the WSAW-SA is straightforward to define interms of discrete-time simple random walk; when γ = 0, it is known as the Domb-Joyce modelor discrete-time weakly self-avoiding walk. A sample of the Domb-Joyce model with 100 stepsis shown in Figure 1.3. The SAW can be recovered as an appropriate limit of the Domb-Joycemodel or the continuous-time WSAW [18].Alternative representationFor f : Zd → R, let|∇fx|2 =∑|e|=1|∇efx|2, |∇f |2 =∑x∈Zd|∇fx|2. (1.6.16)Then by (1.3.9) ∑x∈Zdfx∆fx = −12|∇f |2. (1.6.17)It follows that∑x∈Zd∑e∈Ufxfx+e = 2d∑x∈Zdf2x +∑x∈Zdfx∆fx = 2d∑x∈Zdf2x −12∑x∈Zd|∇fx|2 (1.6.18)and so we get the useful representation:Ug,γ(f) = (g − γ)∑x∈Zdf2x +γ4d∑x∈Zd∑e∈U|∇efx|2. (1.6.19)231.6. Self-interacting walksIn particular,Ug,γ,T = (g − γ)IT + γ4d|∇LT |2. (1.6.20)1.6.2 Predicted behaviourWe can view the susceptibility (1.6.12) as the partition function for a measure on walks of anylength (sometimes called a grand ensemble). When ν reaches the critical pointνc = νc(g, γ) = inf{ν : χ(g, γ, ν) <∞}, (1.6.21)we expect the susceptibility to diverge. The susceptibility is a partition function for walkswith ν playing the role of an external field, so this divergence would be indicative of a phasetransition as discussed in Section 1.4.3. In a certain sense [42, 61], paths in the ν > νc phaseshould scale as geodesics while paths for ν < νc should be space-filling.In fact, it is not clear how to show that χ(g, γ, νc) = ∞ in general, although this can beestablished for γ = 0 (see [9, Lemma A.1]). For γ sufficiently small, this divergence will be partof our main result.The two-point function, susceptibility, and correlations lengths of the self-avoiding walk and(discrete- or continuous-time) WSAW-SA (with γ small depending on g) on Zd are all expectedto scale according to analogues of (1.5.7)–(1.5.10). Moreover, the discussion in Section 1.1suggests thatcT ∼ C5e−νcTT γ¯−1, (1.6.22)〈|XT |2〉 ∼ C6T 2ν¯ . (1.6.23)The critical exponents γ¯, ν¯, η are expected to be universal; in particular, they should onlydepend on d in this context.Below, we discuss the predicted values of the exponents for γ small before turning ourattention to the case of large γ. A more detailed reference is [90]. The values of ν¯ were firstpredicted7 by the chemist Paul Flory [53], who later won the 1974 Nobel Prize in Chemistryfor his work on polymers.Dimension d = 1For the SAW, dimension 1 is trivial: the only self-avoiding walks are straight lines. This is notthe case for the WSAW, see e.g. the survey [73].7Flory’s prediction for d = 3 is no longer generally accepted.241.6. Self-interacting walksDimension d = 2In d = 2, the predicted values of the critical exponents areν¯ = 3/4, γ¯ = 43/32, η = 5/24. (1.6.24)It was shown in [84] that the scaling limit of SAW, if it exists and is conformally invariant, isgiven by SLE8/3, which is consistent with the predicted exponents given above.It is not immediately clear how to make sense of the supercritical regime (ν < νc for WSAW-SA). However, the authors of [42] considered SAW on a discretized bounded planar domain.They showed that the scaling limit of supercritical SAW conditioned to start and end on theboundary of the domain is space-filling (their results extend to all dimensions d ≥ 2).Dimension d = 3In d = 3, again very little is known rigorously. Approximate values of the self-avoiding walkcritical exponents (assuming their existence) have been obtained by running simulations, seee.g. [35, 36].Dimension d > 4The upper-critical dimensions for these models is d = 4 and the mean-field exponents are thesame as for models in the Ising universality class, namely γ¯ = 1, ν¯ = 1/2, η = 0. In otherwords, self-avoiding walk is expected to scale like simple random walk in dimensions above 4.Brydges and Spencer introduced the lace expansion in [32] and used it to show that ν¯ = 1/2if d > 4 for the discrete-time WSAW. By vastly extending this method, Hara and Slade [67,68]showed for the SAW that γ¯ = 1, ν¯ = 1/2 (for the mean-squared displacement, correlationlength, and correlation length of order 2), η = 0, and the scaling limit is Brownian motion.Even above the upper-critical dimension, very little is known about WSAW-SA with γ 6= 0.Exceptions include [65,111].Remark 1.6.3. Define the free bubble diagram8 to be the `2(Zd) norm of the massive Greenfunction x 7→ C0x. Thus, by (1.3.13), if z = 2d/(m2 + 2d), thenBm2 = ‖C‖`2(Zd) = (m2 + 2d)−2∞∑m,n=0zm+n P(Xm = Yn), (1.6.25)where X and Y are independent simple random walks started at 0. In other words, Bm2 isproportional to the expected number of intersections between the traces of two such randomwalks killed at rate 1− z. The upper-critical dimension can be “guessed” as follows. First, we8This is sometimes represented by a graph consisting of two edges (forming a “bubble”) joining a vertexlabeled 0 (denoting the origin) to an unlabeled vertex (denoting an arbitrary point x that is summed over Zd).251.6. Self-interacting walksmake the convenient definitionsBm2 = (n+ 8)Bm2 , b =n+ 816pi2. (1.6.26)Then it is an exercise in Fourier analysis to show that in the limit m2 ↓ 0 (equivalently, z ↑ 1)Bm2 ∼b logm−2, d = 4B0, d > 4, (1.6.27)which suggests the value dc = 4 of the upper-critical dimension.Dimension d = 4Weakly self-avoiding walk on a hierarchical lattice was studied by a renormalization groupmethod in [20,24,25,61]. As discussed in Section 1.5.4, Bauerschmidt, Brydges, and Slade haverecently made great strides in the case d = 4 on the Euclidean lattice using a new renormal-ization group method. This method was first applied to walks in [8,9], where the susceptibilityand two-point function were studied.Phase diagramLet d ≥ 2. The predicted behaviour of self-avoiding walk with attraction is discussed in [74,112]. The predicted phase diagram and critical exponents are shown in Figure 1.4. Generallyspeaking, one might expect the self-attraction to dominate when γ > g so that the walk typicallyremains in a bounded region, i.e. ν¯ = 0. A proof of this fact appears in [71] for a related modelin d = 1. The authors of [71] also conjecture that ν¯ = 1/(d+1) for g = γ and this is establishedfor d = 1 in [72].While it is natural to expect the self-avoidance to dominate when γ < g, a collapse transitionis believed to occur as γ crosses the θ-curve g 7→ γθ(g). The value of the critical exponent ν¯ forγ = γθ is predicted to be given by ν¯θ = 4/7 if d = 2 and ν¯θ = 1/2 if d ≥ 3 (with a logarithmiccorrection in d = 3). However, very little is known rigorously about the θ-curve. Only recentlyhas a collapse transition been shown to exist for a model of prudent self-avoiding walk withself-attraction [101].Remark 1.6.4. De Gennes [38] related the behaviour of polymers in poor solvents to the tri-critical behaviour of certain spin systems. Such systems typically possess two phase transitions:one corresponding to each critical point on a line of critical points and one corresponding to atricritical point given by an endpoint of this line. For the WSAW-SA, the tricritical point shouldbe given by (g, γθ, νc). Moreover, the upper-critical dimension for such tricritical behaviour isd = 3, consistent with the predicted values of ν¯θ.261.7. Main resultν¯SAWν¯θν¯ = 1/dν¯ = 1/(1 + d)ν¯ = 0gγFigure 1.4: Phase diagram for the WSAW-SA1.7 Main resultFor any integer n ≥ 1, let Gx(g, γ, ν;n) denote the two-point point function for the versionof the |ϕ|4 model defined by (1.4.30). We let Gx(g, γ, ν; 0) denote the two-point function ofthe WSAW-SA, defined in (1.6.11); this notation will be explained in Section 1.8. We employsimilar conventions for the susceptibility, correlation length of order p, and critical point, whichwe denote by χ(g, γ, ν;n), ξp(g, γ, ν;n), νc(g, γ;n), respectively, with n ≥ 0 an integer. Whenn = 0, these correspond to the WSAW-SA, whereas for n ≥ 1 they correspond to the |ϕ|4model. The following theorem is the main result of this thesis.Theorem 1.7.1. Let d = 4 and n ≥ 0. For L sufficiently large (depending on n), there existsg∗ > 0 and a positive function γ∗ : (0, g∗)→ R such that whenever 0 < g < g∗ and |γ| < γ∗(g),there are constants Ag,γ,n and Bg,γ,n such that the following hold:(i) The critical two-point function decays asGx(g, γ, νc;n) = Ag,γ,n|x|−2(1 +O((log |x|)−1)) as |x| → ∞, (1.7.1)with Ag,γ,n = (4pi)−2(1 +O(g)) as g ↓ 0.(ii) The susceptibility diverges asχ(g, γ, νc + ε;n) ∼ Bg,γ,nε−1(log ε−1)(n+2)/(n+8), ε ↓ 0 (1.7.2)with Bg,γ,n = ((n+ 8)g/16pi2)(n+2)/(n+8)(1 +O(g)) as g ↓ 0.(iii) For any p > 0, if L is chosen large and g∗ small (both depending on p), then the correlation271.8. Relations between modelslength of order p diverges asξp(g, γ, νc + ε;n) ∼ B1/2g,γ,ncpε−1/2(log ε−1)(n+2)/2(n+8), ε ↓ 0 (1.7.3)withcpp =∫R4|x|p(−∆R4 + 1)−10x dx. (1.7.4)The γ = 0 cases of (i) and (ii) were proved by Bauerschmidt, Brydges, and Slade in [8, 9];in fact, the n = 1 case of their results was first obtained in [48,57,66,69]. The n > 0 case withγ 6= 0 is a new result in this thesis. We will only discuss the proof of the γ ≥ 0 case, whichis of primary interest. The proof of the γ < 0 case with n = 0 can be found in [13] and theextension to n ≥ 1 is straightforward.Remark 1.7.2.1. The behaviour (1.7.3) is consistent with predictions for the correlation length ξ and shouldbe understood as a step towards a rigorous understanding of ξ in four dimensions (evenin the case γ = 0).2. The statement of Theorem 1.7.1 does not provide a quantitative upper bound on γ.However, it should be possible to extend this result to all |γ| ≤ Cg3 for some constantC. We did not pursue this extension here as we expect that the results of Theorem 1.7.1may well hold for γ larger than O(g3). Indeed, these results should hold for all γ belowthe θ-curve and we know of no particular reason to expect the θ-curve to scale like g3.3. We expect that the results of [108] on higher-order correlation functions can be extendedto the γ-dependent case considered here by the methods used to prove Theorem 1.7.1.For instance, it should be possible to show that1n〈|ϕ0|2; |ϕx|2〉g,γ,νc ∼ Cg,γ,n|x|−4(log |x|)−2(n+2n+8) (1.7.5)1.8 Relations between modelsOne way to understand universality is via representation theorems that relate different models.For instance, the Kac-Siegert transformation can be used to write the partition function ofthe O(n) model as a partition function for a perturbation of the |ϕ|4 model (we will discussthis further in Section 6.1.2). In the other direction, the Simon-Griffiths construction [106] canbe used to approximate the 1-component |ϕ|4 model as a suitable limit of Ising models. Suchtheorems do not necessarily imply universality (in the sense that models related in this wayhave the same critical exponents or scaling limit), but tend to be suggestive of it and may insome cases be used as the basis for the proof of a universality-type result.We have already noted in Example 1.5.2 the close relationship between the simple randomwalk and the Gaussian free field, which ultimately stems from the representation of matrix281.8. Relations between modelspowers in terms of walks and which is familiar to anyone who has studied Markov chains.Namely, if M is a V × V matrix, thenMnab =∑x1,...,xn∈VMax1Mx1x2 . . .Mxnb. (1.8.1)When M is indexed by E , the sum above can be replaced by a sum over n-step walks from a to bon G. When the entries of M are nonnegative, such a sum acquires a probabilistic interpretationas an expectation with respect to the random walk whose steps are weighted by the entries ofM .It was discovered by Symanzik [110] that certain spin systems could be represented as modelsof interacting walks in a background of interacting loops. Symanzik used this insight to studyquantum field theories in terms of walks. Such representations were also studied, e.g. in [21,43].A comprehensive reference is [49].In the opposite direction, one can consider studying walks by looking for corresponding spinsystems. In [37], de Gennes argued that the self-avoiding walk corresponds to an n→ 0 “limit”of the O(n) spin model and used this to predict the values of its critical exponents. Since nis the number of components of the spins, the O(n) model is only well-defined for n a positiveinteger and it is not clear how to make sense of such a limit.Parisi and Sourlas [99] and McKane [91] discovered an alternative approach to the pre-dictions of de Gennes. They argued that the weakly self-avoiding walk two-point functioncould be represented as the two-point function for a version of the |ϕ|4 model, involving bosonand fermion fields (we discuss these below). The formal appearance of n = 0 quantities wasthen explained as a consequence of a symmetry between the bosons and fermions known assupersymmetry.In Section 1.8.1, we provide a brief description of the heuristic relation between spin systemsand self-avoiding walk. Then in Section 1.8.2, we describe the rigorous representation of WSAW-SA as a supersymmetric field theory.1.8.1 The n→ 0 limitThe heuristic relation between self-avoiding walk and spin systems is most easily treated onfinite graphs G of degree 3 so we restrict our attention to this case. In addition, we consider aversion of the O(n) model with spins normalized to lie on the sphere of radius√n, which weequip with the uniform measure. We denote the product measure on the resulting configurationspace bydσ =∏x∈Vdσx. (1.8.2)Remark 1.8.1. This normalization of the spins was in fact used when the O(n) model was orig-inally introduced in [109]. Moreover, in [78], it was shown that this normalization is necessaryfor the study of the n→∞ limit.291.8. Relations between modelsThe high-temperature expansion of a spin system is based on the expansion of the Boltzmannweight e−βH about β = 0. For the O(n) spin model with interaction Jxy = 1x∼y, neglectinghigher-order terms in the high-temperature expansion yieldsZ =∫dσ∏xy∈Eeβσx·σy≈∫dσ∏xy∈E(1 + βσx · σy)=∑E⊂Eβ|E|∫dσ∏xy∈Eσx · σy (1.8.3)By reflection-invariance, the last integral above is non-zero if and only if every vertex in theproduct over E appears an even number of times. On a graph of degree 3, this is only possible ifE is a (possibly empty) collection of mutually avoiding (i.e. disjoint) self-avoiding loops (walksfrom a vertex to itself that are self-avoiding everywhere except this vertex).Moreover, for any loop L, invariance under orthogonal transformations and the fact thatspins have radius√n implies that∫dσ∏xy∈Lσx · σy =n∑i=1∏x∈V(L)∫dσx (σix)2 = n, (1.8.4)where V(L) is the set of vertices in L. Thus,Z ≈ 1 +∑N≥1nNN !∑L1,...,LNβ|L1|+···+|LN |, (1.8.5)where the inner sum is over all collections of disjoint loops L1, . . . , LN and permutations of theseloops are accounted for by the 1/N ! factor. Notice that the final expression on the right-handside of (1.8.5) makes sense for any N and equals 1 when n = 0.The two-point function for the O(n) model can be defined analogously to (1.4.30) and(1.4.22). By a similar expansion as was used to study the partition function above, the numer-ator in the two-point function becomesn−1∫dσ(σa · σb)∏xy∈Eeβσx·σy ≈ n−1∑E⊂Eβ|E|∫dσ(σa · σb)∏xy∈Eσx · σy. (1.8.6)Once again, every vertex must appear twice on the right-hand side in order to make a non-zerocontribution to the sum. Due to the presence of the factor σa · σb, this means (unless a = b)that the sum can be replaced by a sum over subsets E containing a self-avoiding walk from ato b together with with a (possibly empty) family of mutually avoiding self-avoiding loops thatalso avoid this walk. (As a very simple example, if a ∼ b, then there is a non-zero contribution301.8. Relations between modelsfrom E = {a, b}.) For any such configuration E containing N loops,∫dσ (σa · σb)∏xy∈Eσx · σy = n1+N . (1.8.7)The extra factor of n arises from the walk in E but is cancelled by the normalization in (1.8.6).Thus, after formally setting n = 0 (so that Z = 1), the two-point function is approximatelygiven by1 +∑ω∈Sn(a,b)β|ω|, (1.8.8)which is the two-point function, i.e. the generating function for all self-avoiding walks from ato b.1.8.2 Self-avoiding walk representationIn this section we describe an integral representation of the of WSAW-SA on the discrete torusΛ. We begin with the necessary background on Grassmann integration, which was introducedin [15]. However, we follow the treatment of [26] in terms of differential forms.Boson and fermion fieldsLet φx, φ¯x denote complex variables indexed by x ∈ Λ. We refer to (φ, φ¯) as a boson field. Letux, vx denote the real and imaginary parts of φx and define the differentials dφx = dux + idvxand likewise for dφ¯x. We multiply differential forms in the usual way via the anticommutativewedge product ∧ but drop this in our notation; in particular,dφ¯xdφx = 2iduxdvx. (1.8.9)Example 1.8.2. Let C be a positive-definite symmetric Λ×Λ matrix. The complex Gaussianmeasure with covariance C is the probability measure on R2Λ given bydµC(φ, φ¯) =dφ¯dφdet(2piiC)e−φ·Aφ¯ (1.8.10)where A = C−1 anddφ¯dφ :=∏x∈Λdφ¯xdφx (1.8.11)The order in which the product over x ∈ Λ is taken does not matter since the dφ¯xdφx commute.The complex Gaussian satisfies a version of Wick’s theorem. In particular,∫φ¯xφy dµC(φ, φ¯) = Cxy. (1.8.12)311.8. Relations between modelsLetψx =1√2piidφx, ψ¯x =1√2piidφ¯x, (1.8.13)where we fix a choice of complex square root. We refer to (ψx, ψ¯x)x∈Λ as a fermion field. Adifferential form that is the product of a function of (φ, φ¯) with p differentials is said to havedegree p. A sum of forms of even degree is said to be even.We introduce a copy Λ¯ of Λ and we denote the copy of X ⊂ Λ by X¯ ⊂ Λ¯. We also denotethe copy of x ∈ Λ by x¯ ∈ Λ¯ and define φx¯ = φ¯x and ψx¯ = ψ¯x. Then any differential form F canbe writtenF =∑~yF~y(φ, φ¯)ψ~y (1.8.14)where the sum is over finite sequences ~y over Λunionsq Λ¯, and ψ~y = ψy1 . . . ψyp when ~y = (y1, . . . , yp).Here, we take the sequences to be ordered in some fixed but arbitrary fashion. We let F 0 denotethe 0-degree (bosonic) part of F , given by the coefficient F~y with ~y = ∅ the empty sequence.In order to apply the results of [8, 9, 12], we require smoothness of the coefficients F~y ofF . For Theorem 1.7.1(i,ii), we need these coefficients to be C10, and for Theorem 1.7.1(iii) werequire a p-dependent number of derivatives for the analysis of ξp. In either case, we let pNdenote the desired degree of smoothness. We will discuss this further in Section 4.2.2.We letN∅ be the algebra of even forms (i.e. differential forms of even degree) with sufficientlysmooth coefficients and we let N∅(X) ⊂ N∅ be the sub-algebra of even forms only dependingon fields in X. Thus, for F ∈ N∅(X), the sum in (1.8.14) runs over sequences ~y over X unionsq X¯.Now let F = (Fj)j∈J be a finite collection of even forms indexed by a set J and writeF 0 = (F 0j )j∈J . Given a C∞ function f : RJ → C, we define f(F ) by its Taylor expansion aboutF 0:f(F ) =∑α1α!f (α)(F 0)(F − F 0)α. (1.8.15)The summation terminates as a finite sum, since ψ2x = ψ¯2x = 0 by anticommutativity.We define the integral∫F of a differential form F in the usual way as the Riemann integralof its top-degree part (which may be regarded as a function of the boson field). In particular,given a positive-definite symmetric Λ × Λ matrix C with inverse A = C−1, we define theGaussian expectation (or super-expectation) of F byECF =∫e−SAF, (1.8.16)whereSA =∑x∈Λ(φx(Aφ¯)x + ψx(Aψ¯)x). (1.8.17)The super-expectation has the following self-normalizing property:EC1 =∫e−SA = 1. (1.8.18)321.8. Relations between modelsMoreover, if F is a degree-0 form, thenECF =∫F dµC . (1.8.19)There is also a version of Wick’s theorem for fermions. In particular,∫e−SAψ¯xψx = Cxx. (1.8.20)Proofs of the statements (1.8.18)–(1.8.20) can be found in [26].For F = f(φ, φ¯)ψ~y, we letθF = f(φ+ ξ, φ¯+ ξ¯)(ψ + η)~y, (1.8.21)where ξ is a new boson field, η = (2pii)−1/2dξ a new fermion field, and ξ¯, η¯ are the correspondingconjugate fields. We extend θ to all F ∈ N∅ by linearity and define the convolution operatorECθ by letting ECθF ∈ N∅ denote the Gaussian expectation of θF with respect to (ξ, ξ¯, η, η¯),with φ, φ¯, ψ, ψ¯ held fixed.Integral representation of the two-point functionAn integral representation formula applying to general local time functionals is given in [20,26].We state the result we need in the proposition below. A direct proof can be obtained by a smallmodification to the proof in [108, Appendix A].We define the differential forms:τx = φxφ¯x + ψxψ¯x (1.8.22)τ∆,x =12(φx(−∆φ¯)x + (−∆φ)xφ¯x + ψx(−∆ψ¯)x + (−∆ψ)xψ¯x)(1.8.23)|∇τx|2 =∑|e|=1(∇eτ)2x. (1.8.24)The forms τx are special due to the following remarkable property of the super-expectation(see [26]): ∫e−SAF (τ) = F (0). (1.8.25)Note that (1.8.18) is an immediate consequence of this fact. Recall (1.6.19) and defineVg,γ,ν,N = Ug,γ(τ) +∑x∈ΛN(ντx + τ∆,x)(1.8.26)Proposition 1.8.3. Let d > 0 and g > 0. For γ < g and ν ∈ R,Gx,N (g, γ, ν; 0) =∫e−Vg,γ,ν,N φ¯0φx. (1.8.27)331.9. OutlineFinite-volume approximationIn order to make use of the integral representation above, we must approximate the WSAW-SAon Zd by a model on ΛN .Let XLNdenote the simple random walk on ΛN . For FT = FT (X) any one of the functionsLxT , IT , CT of X defined in (1.6.5)–(1.6.7), we write FN,T = FT (XLN ). For instance, withn = LN ,LxN,T =∫ T01Xnt = xdt, IN,T =∑x∈ΛN(LxN,T )2. (1.8.28)As before, we identify the vertices of ΛN with nested subsets of Zd, centred at the origin (ap-proximately if L is even), with ΛN+1 paved by Ld translates of ΛN . We denote the expectationof XLNstarted from 0 ∈ ΛN by EΛN0 and definecN,T (x) = EΛN0(e−Ug,γ,T1X(T )=x), x ∈ ΛN (1.8.29)cN,T = EΛN0(e−Ug,γ,T). (1.8.30)The finite-volume two-point function and susceptibility are defined byGx,N (g, γ, ν; 0) =∫ ∞0cN,T (x)e−νT dT, (1.8.31)χN (g, γ, ν; 0) =∫ ∞0cN,T e−νT dT. (1.8.32)The proof of the following proposition is given in Appendix A.Proposition 1.8.4. Let d > 0, g > 0 and γ < g. For all ν ∈ R,limN→∞Gx,N (g, γ, ν; 0) = Gx(g, γ, ν; 0) (1.8.33)andlimN→∞χN (g, γ, ν; 0) = χ(g, γ, ν; 0). (1.8.34)In fact, χN and χ are analytic in Reν > νc and χN → χ uniformly on compact subsets of thisdomain.1.9 OutlineChapter 2 introduces the elements and formalism of the renormalization group method devel-oped in [10,28–31]. However, we proceed differently from these papers in two regards.Firstly, in Section 2.4.3 we employ a different choice of norm weights from that used in [31],where the renormalization group map was constructed. However, these new weights cannotbe used with the same norms as in [31]. In Chapter 4, we explain how to overcome thisobstacle by a new choice of norm and we provide a detailed verification that the estimates341.9. Outlineon the renormalization group map are improved by this choice. The result is summarized asTheorem 2.7.1, which is the first main technical achievement of this thesis. The improvedestimates that we obtain are required for the proof of Theorem 1.7.1(iii), even when we takeγ = 0. This result first appeared in [12].Secondly, the initial coordinates for the renormalization group that we define in Section 2.6.1involve a non-trivial error coordinate that captures the self-attraction term in the WSAW-SA and the γ(∇|φx|2)2 term in the generalized |ϕ|4 model. This error coordinate is coupledto a coordinate capturing the relevant and marginal directions and a version of the implicitfunction theorem is consequently required for the identification of critical parameters suchthat the renormalization group can be initialized on its stable manifold when γ 6= 0. Theconstruction of these critical parameters is carried out in Chapter 5 and the result is summarizedas Theorem 2.8.1. This is the second main technical achievement of this thesis and is requiredfor the proof of Theorem 1.7.1 with γ 6= 0. This result first appeared in [13] for n = 0; here, wehave extended it to all n ≥ 0. However, we restrict our attention to the more interesting caseof γ ≥ 0 for simplicity.Prior to proving Theorems 2.7.1 and 2.8.1, we show in Chapter 3 how to obtain Theo-rem 1.7.1 as a consequence of these results. The proof of Theorem 1.7.1(iii) is a novel contri-bution even in the case γ = 0, which first appeared in [12].The proof of Theorem 1.7.1(i)–(ii) was previously obtained for γ = 0 by Bauerschmidt,Brydges, and Slade in [7–9] and Slade and Tomberg in [108]. The extension to γ 6= 0 isan adaptation of the proofs found in those papers but involves (in addition to the proof ofTheorem 2.8.1) a change of variables result stated and proved in Section 3.1.1.We conclude in Chapter 6 with a discussion of some open problems.35Chapter 2Renormalization group methodThis chapter introduces the elements of the renormalization group method developed in theseries of papers [10, 28–31] and applied in [7–9, 108]. We will often state results from thesepapers without proof.The main contribution of this thesis, which is based on the work in [12,13], is the improve-ment of the estimates in Theorem 2.7.1 and the extension to γ0 6= 0 in Theorem 2.8.1.2.1 NotationTo unify our treatment of the two models, we define the forms τx, τ∆,x, |∇τx|2 according to(1.8.22)–(1.8.24) if n = 0 andτx =12 |ϕx|2, τ∆,x = 12ϕx · (−∆ϕ)x, |∇τx|2 =∑|e|=1(∇e|ϕx|2)2 (2.1.1)if n ≥ 1. Then by (1.4.27), (1.6.19), and (1.8.26),Vg,γ,ν,N =∑x∈ΛN((g − γ)τ2x + ντx + τ∆,x + 14dγ|∇τx|2)(2.1.2)for any choice of n. We write〈F 〉g,γ,ν,N =∫Fe−Ug,γ,ν,N , n = 01Zg,γ,ν,N∫F (ϕ)e−Ug,γ,ν,N dϕ, n ≥ 1.(2.1.3)The action SA is defined by (1.8.17) if n = 0 andSA =12∑x∈Λϕx · (Aϕ)x (2.1.4)if n ≥ 1. In either case, if A = −∆ +m2, thenSA =∑x∈Λ(τ∆,x +m2τx). (2.1.5)362.2. Reformulation of the problemThus, if ECθ is the super-expectation (1.8.16) for n = 0 and Gaussian integration over (Rn)Λif n ≥ 1, then for ν > 0,〈F 〉0,0,m2,N = ECF, C = (−∆ +m2)−1. (2.1.6)By (1.4.30), (1.8.33), and (1.8.27),Gx(g, γ, ν;n) = limN→∞Gx,N (g, γ, ν;n), (2.1.7)whereGx,N (g, γ, ν;n) =〈φ¯0φx〉g,γ,ν,N , n = 0〈ϕ0 · ϕx〉g,γ,ν,N n ≥ 1. (2.1.8)By Proposition 1.8.4 and (1.4.31), for any integer n ≥ 0,χ(g, γ, ν;n) = limN→∞χN (g, γ, ν;n) (2.1.9)χN (g, γ, ν;n) =∑x∈ΛNGx,N (g, γ, ν;n). (2.1.10)ξp(g, γ, ν;n) =(∑x∈Zd |x|pGx(g, γ, ν;n)χ(g, γ, ν;n))1/p(2.1.11)νc = νc(g, γ;n) = inf{ν : χ(g, γ, ν;n) <∞}. (2.1.12)2.2 Reformulation of the problemIn preparation for our application of the renormalization group, we write the two-point functionand susceptibility in terms of appropriate perturbations of Gaussian measures.Given m2 > 0 and z0 > −1, letg0 = (g − γ)(1 + z0)2, ν0 = ν(1 + z0)−m2, γ0 = 14dγ(1 + z0)2. (2.2.1)We discuss the role of (m2, z0) some more in Remark 2.3.2.We fix two points 0, x ∈ Λ and introduce observable fields σ0, σx ∈ R. We distinguish thesefrom the bulk fields ϕ, φ, φ¯, ψ, ψ¯. We also make a distinction between bosonic fields ϕ, φ, φ¯,σ, and fermionic fields ψ, ψ¯.For any y ∈ Λ, we define the polynomialsV +0,y = g0τ2y + ν0τy + z0τ∆,y − f0σ01y=0 − fxσx1y=x, U+y = |∇τy|2 (2.2.2)372.2. Reformulation of the problemwherefu =φ¯0, n = 0, u = 0φx, n = 0, u = xϕ1u, n ≥ 1.(2.2.3)These are examples of local polynomials, which are polynomials in the fields and their derivativesat a point y ∈ Λ. For any such local polynomial Vy, we will usually writeV (X) =∑y∈XVy. (2.2.4)LetZ0 =∏y∈Λe−(V+0,y+γ0U+y ) (2.2.5)andZN = ECθZ0 (2.2.6)where the covariance is given by C = (−∆ +m2)−1 as in (2.1.6). In particular,ECZ0 = Z0N (0). (2.2.7)Recall here that Z0N denotes the 0-degree part of ZN (when n ≥ 1, Z0N = ZN ). This is afunction of the bulk bosonic fields, which we have set to 0 on the right-hand side of (2.2.7).Recall that the Gaussian convolution operator ECθ was defined in Section 1.8.2. We definea test function 1 : ΛN → R by 1y = 1 for all y. If F is a sufficiently smooth function of thebosonic fields (i.e. F = F (φ, φ¯) if n = 0 and F = F (ϕ) if n ≥ 1), letD2F (0;1,1) =∂2∂s∂t∣∣∣0F (s1, t1), n = 0F (s1+ t1), n ≥ 1 (2.2.8)where the derivative is evaluated with all fields (bulk and observable) and s, t set to 0. Let F (0)denote F evaluated at 0 bulk field. We denote by D2σ0σxF (0) the second partial derivative ofF (0) with respect to the observable fields σ0, σx evaluated at σ0 = σx = 0.Proposition 2.2.1. Let d > 0, γ, ν ∈ R, g > 0 and γ < g. If the relations (2.2.1) hold, thenGx,N (g, γ, ν;n) = (1 + z0)D2σ0σx logECZ0 (2.2.9)andχN (g, γ, ν;n) = (1 + z0)χˆN (m2, g0, γ0, ν0, z0;n), (2.2.10)withχˆN (m2, g0, γ0, ν0, z0;n) =1m2+1m41|Λ|D2Z0N (0;1,1)Z0N (0). (2.2.11)382.2. Reformulation of the problemProof. We prove the case n = 0 and drop the parameter n from the notation. Note that by(1.8.25), ZN (0)∣∣σ0=σx=0= 1 in this case. The proof for n ≥ 1 is similar and involves onlyordinary integration with respect to real boson fields.We make the change of variables φx 7→ (1 + z0)1/2φx and likewise for φ¯x, ψx, ψ¯x in (1.8.27),and obtainGx,N (g, γ, ν) = (1 + z0)∫e−∑x∈Λ(g0τ2x+γ0|∇τx|2+ν(1+z0)τx+(1+z0)τ∆,x)φ¯aφb. (2.2.12)Note here that the Jacobian factor is automatically accounted for by the change of variables inthe fermionic fields. For any m2 ∈ R, it follows thatGx,N (g, γ, ν) = (1 + z0)∫e−∑x∈Λ(τ∆,x+m2τx)Z0φ¯0φx (2.2.13)(m2 simply cancels with ν0 on the right-hand side). We use this with m2 > 0, so that theinverse matrix C = (−∆ +m2)−1 exists andGx,N (g, γ, ν) = (1 + z0)EC(Z0φ¯0φx) (2.2.14)by (2.1.6). The identity (2.2.9) follows by the standard procedure of writing the moments ofan integral as a derivative of a moment-generating function.Summation of (2.2.14) over x ∈ ΛN gives the formula χN (g, γ, ν) = (1+z0)∑x∈Λ EC(Z0φ¯0φx).Call the right-hand side χˆN (g, γ, ν). To show that this is consistent with (2.2.11), begin by not-ing thatχˆN (g, γ, ν) = |Λ|−1D2Σ(0;1,1)ZN (0), (2.2.15)whereΣ(J, J¯) = EC(Z0eJ ·φ¯+φ·J¯). (2.2.16)Completing the square yieldsΣ(J, J¯) = eJ ·CJ¯Z0N (CJ,CJ¯) (2.2.17)and differentiating this expression givesD2Σ(0;1,1) = (1, C1) +D2Z0N (0;C1, C1) (2.2.18)The result then follows from the fact thatC1 = A−11 = m−21. (2.2.19)392.3. Progressive integration2.3 Progressive integrationBy Proposition 2.2.1, our task is to understand the Gaussian expectation ZN = ECZ0 and itsderivatives to leading order, uniformly in the volume ΛN and the mass m2 near 0.We proceed using the covariance decompositionC = C1 + · · ·+ CN−1 + CN,N (2.3.1)constructed in [5]; a similar decomposition was also constructed in [22]. The covariancesC1, . . . , CN−1 are independent of the volume ΛN . The final covariance CN,N does dependon the volume; so, for instance, CN,N 6= CN,N+1. Nevertheless, we will often write CN := CN,Nwhen the volume is implicit.The covariances Cj have the following important finite-range property :Cj;xy = 0 if |x− y| ≥ 12Lj . (2.3.2)Thus, if ζ is a Gaussian field with covariance Cj , then ζx is independent of ζy whenever |x−y| ≥12Lj . In particular, if Fx, Fy are functions of the fields at x, y, respectively, thenECj+1(FxFy) = (ECj+1Fx)(ECj+1Fy). (2.3.3)In addition, we have the following covariance bounds (this is a restatement of [10, Proposi-tion 6.1(a)]).Proposition 2.3.1. Let d > 2, L ≥ 2, j ≥ 1, m¯2 > 0. For multi-indices α, β with `1 norms|α|1, |β|1 at most some fixed value p, for any k, and for m2 ∈ [0, m¯2],|∇αx∇βyCj;x,y| ≤ c(1 +m2L2(j−1))−kL−(j−1)(d−2+|α|1+|β|1), (2.3.4)where c = c(p, k, m¯2) is independent of m2, j, L. The same bound holds for CN,N if m2L2(N−1) ≥ε for some ε > 0, with c depending on ε but independent of N .It is a basic property of the Gaussian distribution that a sum of independent Gaussianrandom variables with covariances C ′ and C ′′ is itself Gaussian with covariance C ′ + C ′′. Itfollows that for any boson field F ,EC′+C′′θF = EC′θ ◦ EC′′θF. (2.3.5)This extends to any sufficiently smooth form F (see [28]). It follows thatZN = ECN θ ◦ ECN−1θ ◦ . . . ◦ EC1θZ0. (2.3.6)402.4. The space of field functionalsWe define the renormalization group map Zj 7→ Zj+1 byZj+1 = ECj+1θZj , j < N. (2.3.7)Remark 2.3.2. The key to understanding ZN for large N is the careful choice of critical initialconditions (m2, z0) in (2.2.1). Viewed as functions of (g, γ, ν), these define a stable manifold forthe dynamical system induced by the renormalization group map and the fixed point for thisstable manifold is the Gaussian measure with covariance (1+z0)(−∆+m2)−1. However, we havescaled out the factor 1+z0 in the change of variables performed in the proof of Proposition 2.2.1.Indeed, for n ≥ 1 the exponent in (2.2.12) contains the term −12(1+z0)∑x∈Λ ϕx ·[(−∆+m2)ϕ]x.The construction of the critical parameters for γ 6= 0 will be carried out in Section 3.1.1and is a key step in the proof of Theorem 1.7.1.2.4 The space of field functionalsFor the analysis of the dynamical system (2.3.7), we require a suitable space on which thissystem evolves.Let N∅ be defined as in Section 1.8.2 if n = 0 andN∅ = N∅(Λ) = CpN ((Rn)Λ,R) (2.4.1)if n ≥ 1. Recall that pN is the smoothness parameter discussed in Section 1.8.2.We extend N∅ to a space N that includes functions of the observable fields σ0 and σx, whichwe identify to order 1, σ0, σx, σ0σx (this is sufficient for computing the derivative in (2.2.9)).Formally, we let N ′ denote the extension of N∅ whose elements may depend smoothly on σ0,σx. In other words, if n ≥ 1, then N ′ consists of functions of (ϕ, σ0, σx) that are CpN in ϕand C∞ in σ0, σx. Likewise, for n = 0, a similar statement is true of the coefficients F~y in(1.8.14). Letting I ⊂ N ′ be the ideal consisting of elements whose formal expansion to order1, σ0, σx, σ0σx is 0, we define N = N ′/I. Then N has the direct sum decompositionN = N∅ ⊕N a ⊕N b ⊕N ab, (2.4.2)where N a consists of elements of the form σaF with F ∈ N∅ and a similar statement is trueof N b,N ab. Thus, every F ∈ N has the formF = F∅ + σ0F0 + σxFx + σ0σxF0x, F∅, F0, Fx, F0x ∈ N∅. (2.4.3)There are natural projections piα : N → Nα with α = ∅, 0, x, 0x such that piαF = Fα. ForX ⊂ Λ, we let N (X) denote the subspace of N consisting of field functionals that only dependon fields in X.In order to control the evolution of Zj on N , we make use of a family ‖ · ‖Tφ,j(hj) of scale-412.4. The space of field functionalsdependent seminorms defined in terms of a sequence of weights hj > 0; the field φ lies in CΛif n = 0 and (Rn)Λ if n ≥ 1. For convenience, we will simply write ‖ · ‖Tφ(hj) with the scale jimplied by the choice of parameter hj .We given the precise definitions below for n = 0. The case n ≥ 1 involves only minorchanges, which we describe in Remark 2.4.1.2.4.1 Test functionsRecall the notation introduced in Section 1.8.2. A test function g is defined to be a function(~x, ~y) 7→ g~x,~y, where ~x and ~y are finite sequences of elements in Λ unionsq Λ¯. When ~x or ~y is theempty sequence ∅, we drop it from the notation as long as this causes no confusion; e.g., wemay write g~x = g~x,∅. The length of a sequence ~x is denoted |~x|. Gradients of test functions aredefined component-wise. Thus, if ~x = (x1, . . . , xm) and α = (α1, . . . , αm) with each αi ∈ NU0 ,and similarly for ~y = (y1, . . . , yn) and β = (β1, . . . , βn), then∇α,β~x,~y g~x,~y = ∇α1x1 . . .∇αmxm∇β1y1 . . .∇βnyn gx1,...,xm,y1,...,yn . (2.4.4)We fix a positive constant pΦ ≥ 4 and restrict our attention to test functions that vanishwhen |~x|+ |~y| > pN . The Φj = Φ(hj) norm on such test functions is defined by‖g‖Φj = sup~x,~yh−(|~x|+|~y|)j supα,β:|α|1+|β|1≤pΦLj(|α|1+|β|1)|∇α,βg~x,~y|, (2.4.5)where |α|1 denotes the total order of the differential operator ∇α. Thus, for any test functiong and for sequences ~x, ~y with |~x|+ |~y| ≤ pN and corresponding α, β with |α|1 + |β|1 ≤ pΦ,|∇α,βg~x,~y| ≤ h|~x|+|~y|j L−j(|α|1+|β|1)‖g‖Φj . (2.4.6)2.4.2 The Tφ seminormIf n = 0, then for any F ∈ N∅, there are unique functions F~y of (φ, φ¯) that are anti-symmetricunder permutations of ~y, such thatF =∑~y1|~y|!F~y(φ, φ¯)ψ~y. (2.4.7)Given a sequence ~x with |~x| = m, we defineF~x,~y =∂mF~y∂φx1 . . . ∂φxm. (2.4.8)We define a φ-dependent pairing of elements of N with test functions by〈F, g〉φ =∑~x,~y1|~x|!|~y|!F~x,~y(φ, φ¯)g~x,~y. (2.4.9)422.4. The space of field functionalsLet B(Φ) denote the unit Φ-ball in the space of test functions. Then the Tφ = Tφ(hj)seminorm on N∅ is defined by‖F‖Tφ = supg∈B(Φj)|〈F, g〉φ|. (2.4.10)Remark 2.4.1. If n ≥ 1, a test function is a function g on sequences over Λ× {1, . . . , n}. Forany such sequence ~x = ((x1, i1), . . . , (xm, im)), we write |~x| = m and setF~x =∂mF∂ϕi1x1 . . . ∂ϕimxm(2.4.11)and〈F, g〉ϕ =∑|~x|≤pN1|~x|!F~x(ϕ)g~x. (2.4.12)Then the Tϕ seminorm can be defined as in (2.4.10).To extend the Tφ seminorm to N , we make use of an additional sequence of parametershσ,j . For any F ∈ N of the form (2.4.3), we let‖F‖Tφ = ‖F∅‖Tφ + (‖F0‖Tφ + ‖Fx‖Tφ)hσ + ‖F0x‖Tφh2σ. (2.4.13)By its definition, the Tφ seminorm controls the values of F and its derivatives (up to orderpN ) at φ. For instance, we will make use of the following facts.Lemma 2.4.2. If F ∈ N∅, then |F 0(0)| ≤ ‖F‖T0. For F ∈ N ,|D2F 0(0;1,1)| ≤ 2‖F‖T0(hj)‖1‖2ΦN (hj) = 2‖F‖T0(hj)h−1j (2.4.14)and|D2σ0σxF 0(0)| ≤ h−2σ,j‖F‖T0 . (2.4.15)An essential property of the Tφ seminorm is the following product property, which is essentialto fully take advantage the factorization property (2.3.3) that follows from the finite-rangeproperty of the covariance decomposition.Proposition 2.4.3. If F,G ∈ N , then ‖FG‖Tφ ≤ ‖F‖Tφ‖G‖Tφ.Remark 2.4.4. This follows essentially from the fact that the series expansion of the productof two functions is the product of their respective series expansions (see [28]). This is part ofthe reason the Tφ seminorm was defined in terms of the pairing (2.4.9).2.4.3 Norm weightsControl of the Tφ seminorm is needed for all values of φ in order to obtain control of theconvolution (2.3.7) sufficient for iteration of the renormalization group map. This will be432.4. The space of field functionalsdiscussed further in Section 4.2.2.For now, we turn our attention to the special case of the T0 seminorm. Recalling (2.2.7),it is natural to choose the weights hj so that ECj+1F is of order ‖F‖T0(hj). By Wick’s theorem(1.4.9), for a 1-component field ϕ,ECj+1ϕ2px = (2p− 1)!!Cpj+1;00 (2.4.16)and similar statements hold for complex and fermionic fields by the analogues of Wick’s theoremfor such fields. On the other hand, by definition of the T0 seminorm,‖ϕ2px ‖T0(hj) h2pj . (2.4.17)This suggests defining hj so that |Cj+1;00| ≤ O(h2j ).The key to our analysis of the correlation length is that we make a choice of norm weightsthat takes full advantage of the k-dependence in the covariance bounds (2.3.4). With k = s+1,this estimate together with the elementary bound(1 +m2L2j)−k ≤ cLL−2(s+1)(j−jm)+ (2.4.18)imply that|Cj;xy| ≤ O(L−j(d−2)−s(j−jm)+), (2.4.19)where jm is the mass scale, defined byjm = blogLm−1c. (2.4.20)Based on this, when d = 4, we define the following weights:`j = `0L−j−s(j−jm)+ , `σ,j = `−1j∧jx2(j−jx)+ g˜j , (2.4.21)wherejx = max{0, blogL(2|x|)c} (2.4.22)is the coalescence scale and the sequence g˜j = g˜j(m2, g0) will be discussed in Section 2.5.3. Theorigin of the definition of `σ,j is discussed in [30, Remark 3.3].We will set hj = `j to estimate “small” fields. These are fields which are assumed not todeviate too much from their expected value. A different norm parameter hj = hj will be usedto control “large” fields. This will be discussed in Section 4.2.2.Remark 2.4.5. The parameter g˜j is used to overcome what [1] refers to as the “fibred normproblem”. Briefly, the norms used to control the renormalization group trajectory must bedecoupled from the initial parameter g0. Ultimately, we will set g˜0 = g0 (see Remark 5.2.3).442.5. Perturbative coordinate2.4.4 SymmetriesIt is useful to restrict our attention to field functionals F ∈ N that obey certain symmetryconditions preserved by Gaussian expectation (and which are obeyed by V +0 ).We let any automorphism E of Λ act on N by EF (ϕ) = F (Eϕ) with (Eϕ)x = ϕEx. Wesay that F ∈ N is Euclidean-invariant if EF = F for all such automorphisms.If n = 0, we define the gauge flow (q, q¯) 7→ (e−2piitq, 22piitq¯), where q = φx, ψx, σ with σ0 = σand σx = σ¯ for all x ∈ Λ. A form F ∈ N is said to be gauge-invariant if it is invariant underthe gauge flow. We also define the supersymmetry generatorQ = (2pii)1/2∑x∈Λ(ψx∂∂φx+ ψ¯x∂∂φ¯x− φx ∂∂ψx+ φ¯x∂∂ψ¯x.)(2.4.23)A form F ∈ N is said to be supersymmetric if QF = 0.If n ≥ 1, we let an n × n matrix T act on N by TF (ϕ) = F (Tϕ), where (Tϕ)x = T (ϕx).We say that F ∈ N is O(n)-invariant if TF = F for all orthogonal matrices T .2.5 Perturbative coordinateAs mentioned in Section 1.5.4, one of Wilson’s key insights was that the renormalization groupcould be well-approximated by a finite-dimensional dynamical system. In this section, wereformulate Wilson’s insights in terms of the covariance decomposition and define a subspaceon which this finite-dimensional system will evolve.2.5.1 Dimensional analysisWe call Mx ∈ N a local monomial if it is a monomial in ϕx and its (discrete) gradients. Forinstance, for a 1-component field, such Mx has the formMx = (∇α1ϕx) . . . (∇αpϕx). (2.5.1)The T0 seminorm of a local monomial Mx essentially just counts the number of fields andderivatives in Mx. For instance, for Mx as above,‖Mx‖T0(`j) = O(L−j(|α|+p[ϕ]))(2.5.2)where |α| = |α1|+ · · ·+ |αp| and[ϕ] =d− 22(2.5.3)is the scaling dimension of the field. Based on this observation, we define the dimension of Mxby[Mx] = |α|+ p[ϕ]. (2.5.4)452.5. Perturbative coordinateNote here that we have neglected the rapid decay of fields above the mass scale.By (2.3.4), ϕ is approximately constant on blocks of side Lj . In a sense, the fields on a blockB act as a unit and this contributes to a volume factor |B| = Ljd. This leads us to comparethe dimension of a monomial with the dimension d of the lattice. We say that Mx is relevantif [Mx] < d, marginal if [Mx] = d, and irrelevant if [Mx] > d.Remark 2.5.1. Note that the self-attraction term |∇τx|2 is irrelevant in the above sense.However, this does not mean that the inclusion of this term should not have an effect on thecritical behaviour of the model under consideration (indeed, this term is responsible for thephase diagram given by Figure 1.4). Rather, the notion of irrelevance is an asymptotic one:irrelevant terms are only “unimportant” at very large scales j. At scale j = 0 there is littledifference between a relevant and an irrelevant term, which is why we must choose the coefficientγ of |∇τx|2 to be small in Theorem 1.7.1.2.5.2 Local field polynomialsFor y ∈ Λ, we supplement (1.8.22)–(1.8.24) and (2.1.1) by definingτ∇∇,y =12∑e∈U((∇eφ)y(∇eφ¯)y + (∇eψ)y(∇eψ¯)y), n = 014∑|e|=1∇eϕy · ∇eϕy, n ≥ 1.(2.5.5)When n = 0, it can be shown that the only marginal and relevant local monomials that areEuclidean-invariant and supersymmetric are constant multiples of1, τx, τ2x , τ∆,x, τ∇∇,x. (2.5.6)When n ≥ 1, these are the only marginal and relevant monomials that are Euclidean-invariantand O(n)-invariant (see [10]).The marginal and relevant contributions to the evolution of the renormalization group willbe tracked by a local polynomial (a sum of local monomials) of the form∑y∈Λ Uy, where (recall(2.2.3))Uy = gτ2y + ντy + zτ∆,y + u− 1y=0λ0f0σ0 − 1y=xλxfxσx− 12(1y=0q0 + 1y=xqx)σ0σx. (2.5.7)We have omitted τ∇∇ as (1.3.9) gives∑x∈Λτ∇∇,x =∑x∈Λτ∆,x. (2.5.8)462.5. Perturbative coordinateRemark 2.5.2. When n = 0, we can also omit u since constant terms are not produced by theGaussian super-expectation. For example, ECθτx has constant part 0 by (1.8.12) and (1.8.20).More generally, this is a consequence of the supersymmetry identity (1.8.25).We define U to be the space of all polynomials of the form Uy. Given X ⊂ Λ, we letU(X) = {U(X) : U ∈ U}, (2.5.9)where U(X) is defined as in (2.2.4). We also make use of the subspace V of polynomials withu = y = q0 = qx = 0. We will usually denote an element of V as V . For U ∈ U , we define themap U 7→ U (0) ∈ V, which sets u = q0 = qx = 0.We define the U = Uj norm by‖U‖U = max{|g|, L2j |ν|, |z|, L4j |u|, `j`σ,j(|λ0| ∨ |λx|), `2σ,j(|q0| ∨ |qx|)}(2.5.10)on U ∈ U , which depends on the parameters `j and `σ,j . The U = Uj norm is equivalent to theT0(`j) seminorm on U(B) when |B| = Ljd:‖U‖U ‖U(B)‖T0(`j) = Ljd‖Uy‖T0(`j). (2.5.11)2.5.3 Perturbative flowHere we discuss how to maintain the form Zj ≈ e−Vj(Λ) to second order with Vj ∈ V. The basicidea begins with the cumulant expansionECθe−V (Λ) ≈ e−ECθV (Λ)+ 12EC(θV (Λ);θV (Λ)), (2.5.12)whereEC(F ;G) = EC(FG)− (ECF )(ECG) (2.5.13)is the truncated expectation. In [29] an operator Locx is defined so that Locx F is an approxi-mation of F by a local polynomial at x. We make the split12EC(θV (Λ); θV (Λ)) =12Locx EC(θV (Λ); θV (Λ)) +12(1− Locx)EC(θV (Λ); θV (Λ)) (2.5.14)With eF ≈ 1 + F , we getECθe−V (Λ) ≈ e−ECθV (Λ)+ 12 Locx EC(θV (Λ);θV (Λ))(1 + 12(1− Locx)EC(θV (Λ); θV (Λ))). (2.5.15)Based on this idea, in [10] a map1 Upt : V → U of the formUpt(V ) = ECθV − P (2.5.16)1In [10], Upt maps into a larger space including τ∇∇. Here, following (2.5.8), we define Upt by composing thatmap with the map that replaces zτ∆ + yτ∇∇ by (z + y)τ∆.472.5. Perturbative coordinatewith P a local polynomial quadratic in V chosen so that the approximationZj ≈ e−Uj(Λ)(1 +Wj) (2.5.17)can be maintained with Wj = Wj(V ) a non-local remainder. Precisely,Px = Locx ECθWj(V, x) +12Locx[ECθ(VxV (Λ))− (ECθVx)(ECθV (Λ))] (2.5.18)andWj(V, x) =12(1− Locx)[Ewjθ(VxV (Λ))− (EwjθVx)(EwjθV (Λ))]. (2.5.19)By [30, (4.57)],‖Wj‖T0(`j) ≤ O(ϑj)‖V ‖2V , (2.5.20)where ϑj is a parameter that decays exponentially above the mass scale and will be discussedin Section 2.7. We will elaborate on the meaning of (2.5.17) in Section 2.6.The map Upt depends on the covariance C and in practice we set C = Cj+1 and obtaina sequence Upt = Upt,j+1. By successively iterating these maps, we generate a sequence ofcoupling constants that we refer to as the perturbative flow. The equations defining this flowcan be computed exactly by way of Feynman diagrams or with a computer program [6]. In [10],these flow equations are summarized and it is shown that a change of variables can be used totriangularize the resulting system of equations up to third-order errors. Below, we summarizethese transformed flow equations for g, λ, and q.The flow of gThe (transformed) perturbative flow of g takes the formg¯j+1 = g¯j − βj g¯2j , g¯0 = g0 (2.5.21)whereβj = (n+ 8)∑x∈Zd(w2j+1;0x − w2j;0x), wj =j∑i=1Ci. (2.5.22)The sequence βj is closely related to the free bubble diagram (1.6.26). Indeed, using thetelescope nature of∑j βj , we can show that∞∑j=1βj = Bm2 . (2.5.23)The logarithmic divergence of the bubble diagram in (1.6.27) is reflected in the behaviour of gand, ultimately, in the appearance of logarithmic corrections in Theorem 1.7.1. Precisely, the482.6. Non-perturbative coordinateresults of [11], were used to show in [9, Proposition 6.1] thatg¯j = O((logm−1)−1) for j ≥ jm, g¯jx = O((log |x|)−1) for jx ≤ jm. (2.5.24)Remark 2.5.3. A heuristic argument is as follows: Using Proposition 2.3.1, it is straightforwardto show thatβj = O(L−j(d−4)−s(j−jm)+). (2.5.25)Thus, a crude approximation to the flow of g¯ is the recursionyj+1 = yj − c1j≤jmy2j , c > 0. (2.5.26)Comparing this to the differential equation y˙ = −cy2, which has solutions of the form y(t) =(C + ct)−1, it is reasonable to expect that yj ≈ (cj)−1 for j ≤ jm. By definition, yj = yjm forj > jm. Thus, yj ≈ (c logm−1)−1 for j ≥ jm. A similar argument can be used to study yjx .Following [9, (6.15)], we define the parameter g˜j in (2.4.21) as a function of two variables(m˜2, g˜0) byg˜j(m˜2, g˜0) = g¯j(0, g˜0)1j≤jm˜ + g¯jm˜(0, g˜0)1j>jm˜ . (2.5.27)These parameters play an important role in Section 2.7 and in the proof of Theorem 2.8.1.The flow of λ and qIt was shown in [10, (3.34)–(3.35)] (for n = 0) and [108, Proposition 3.2] (for n ≥ 1) that, withC = Cj+1 and u = 0, x,λu,pt =(1− δ[νw(1)])λu, j + 1 < jxλu, j + 1 ≥ jx (2.5.28)qpt = q + λ0λxC0x, (2.5.29)whereδ[νw(1)] = (ν + 2gC00)w(1)j+1 − νw(1)j , w(1)j =∑x∈Λj∑i=1Ci;0x. (2.5.30)Note that qpt = q for j + 1 < jx.2.6 Non-perturbative coordinateLet V denote either ΛN or Zd. We allow N to depend on V. If V = Λ, then N = N (Λ) wasdefined in Section 2.4. Otherwise, we setN (Zd) =⋃finite X⊂VN (X). (2.6.1)492.6. Non-perturbative coordinateWe set N(V) = N if V = ΛN and N(V) = ∞ if V = Zd. For j ≤ N(V) (meaning j < ∞if N(V) = ∞), we partition V into disjoint scale-j blocks of side length Lj , each of which is atranslate of the block {x ∈ Λ : 0 ≤ xi < Lj , i = 1, . . . , d}. A scale-j polymer is a union of scale-jblocks. Given a scale-j polymer X and k ≤ j, we let Bk(X) (respectively, Pk(X)) denote theset of all scale-k blocks (respectively, scale-k polymers) in X. We sometimes write Bj = Bj(V)and Pj = Pj(V) when the volume V is implicit.Any map F : Bj → N can be extended to a map on Pj by block-factorization:F (X) = FX :=∏B∈Bj(X)F (B). (2.6.2)Given maps F,G : Pj(Λ) → N (sometimes called polymer activities), we define the circleproduct F ◦G : Pj(Λ)→ N by(F ◦G)(X) =∑Y ∈Pj(X)F (X \ Y )G(Y ). (2.6.3)The circle product is commutative, associative, and has the identity element1∅(X) =1, X = ∅0, X 6= ∅ . (2.6.4)We track Zj using renormalization group coordinates uj , q0,j , qx,j ∈ R, Ij ,Kj : Pj → N suchthatZj = eζj (Ij ◦Kj)(Λ), ζj = −uj |Λ|+ 12(q0,j + qx,j)σ0σx. (2.6.5)The coordinate Ij = Ij(V, ·) is defined by settingIj(V,B) = e−V (B)(1 +Wj(B, V )), X ∈ Pj , V ∈ V (2.6.6)and extending this by block-factorization. Thus, (2.6.5) gives a rigorous implementation of(2.5.17).Before defining the space in which Kj lies, we need the following notions:• We call a nonempty polymer X ∈ Pj connected if for any x, x′ ∈ X, there is a sequencex = x0, . . . , xn = x′ ∈ X such that |xi+1 − xi|∞ = 1 for i = 0, . . . , n− 1. Let C0 = C0(V)denote the set of connected polymers.• For X ∈ Pj , let |X|j denote the number of scale-j blocks in X. We call a connectedpolymer X ∈ Cj a small set if |X|j ≤ 2d. Let Sj = Sj(V) denote the collection of smallsets. The small set neighbourhood X of a polymer X is defined byX =⋃Y ∈Sj :Y ∩X 6=∅Y. (2.6.7)502.6. Non-perturbative coordinate• Two polymers X,Y do not touch if min(|x− y|∞ : x ∈ X, y ∈ Y ) > 1. We let Comp(X)denote the set of maximal connected components that do not touch in X.We say that a map F : Pj → N is Euclidean-covariant if E(F (X)) = F (EX) for all X ∈ Pjand all automorphisms E of V. We also say that F is gauge-invariant, supersymmetric, orO(n)-invariant if F (X) is gauge-invariant, supersymmetric, or O(n)-invariant, respectively.Definition 2.6.1. For j ≤ N(V), let CKj = CKj(V) denote the real vector space of mapsK : Cj(V)→ N (V) satisfying the following properties:• Field Locality: If X ∈ Cj , then K(X) ∈ N (X). Also: (i) piαK(X) = 0 unless α ∈ X forα = 0, x; (ii) pi0xK(X) = 0 unless a ∈ X and x ∈ X or vice versa; and (iii) pi0xK(X) = 0if X ∈ Sj and j < jx.• Symmetry: (i) pi∅K is Euclidean-covariant; (ii) if n = 0, then K is gauge-invariant andpi∅K is supersymmetric and has no constant part; if n ≥ 1, then pi∅K is O(n)-invariant.We let Kj = Kj(V) denote the real vector space of functions K ∈ CKj with the followingadditional property:• Component factorization: If X ∈ Pj , then K(X) =∏Y ∈Comp(X)K(Y ).Addition in CKj is defined by (F1 + F2)(X) = F1(X) + F2(X). We extend any F ∈ CKj toKj by defining F (X) =∏Y ∈Comp(X) F (Y ).2.6.1 Initial coordinatesAt scale j = 0, we are given V +0 as defined in (2.2.2) and we set ζ0 = 0. In particular, theinitial values of u, q0, qx are zero, and the initial values of λ0, λx are 1. By definition, W0 = 0.For X ⊂ Λ, we defineI+0 (X) = I0(V+0 , X) =∏y∈Xe−V+0,y , K+0 (X) =∏y∈XI+0,y(e−γ0U+y − 1), (2.6.8)where I+0,y = I+0 ({y}). It is straightforward to verify that K0 ∈ K0. Moreover, by (2.2.5),Z0 =∏y∈Λ(I+0,y + I+0,y(e−γ0U+y − 1))= (I+0 ◦K+0 )(Λ). (2.6.9)The second equality here follows from the binomial expansion formula∏y∈Λ(Fy +Gy) =∑X⊂Λ( ∏y∈Λ\XFy)( ∏z∈XGz). (2.6.10)Thus, Z0 takes the form (2.6.5) and we seek (uj , q0,j , qx,j , Vj ,Kj) such that this continues tohold as the scale advances.512.7. Renormalization group stepEquivalently, given (Vj ,Kj), we must define (δuj+1, δq0,j+1, δqx,j+1, Vj+1,Kj+1) so thatEj+1θ(Ij ◦Kj)(Λ) = e−δζj+1(Ij+1 ◦Kj+1)(Λ). (2.6.11)Moreover, we need Kj to contract as the scale advances, under an appropriate norm. Theconstruction of (scale-dependent) maps U+ and K+ such that (2.6.11) holds with(δuj+1, δq0,j+1, δqx,j+1, Vj+1) = U+(Vj ,Kj), Kj+1 = K+(Vj ,Kj) (2.6.12)is the main accomplishment of [31].2.7 Renormalization group stepIn [31, Section 1.7.3], a sequence of norms ‖ · ‖Wj = ‖ · ‖Wj(m˜2,g˜j ,Λ) parameterized by (m˜2, g˜j) isdefined on Kj . These are defined in terms of the Tφ(hj) norms with parameters hj = `j , hj . Inorder to make use of the improved norm parameters with s > 1, we must modify the definitionof Wj when j is above the mass scale. The precise definition of the Wj norm adapted to ourcurrent setting will be discussed in Section 5.1.3. We note here only the fact that the Wj(Λ)norm dominates the T0(`j) norm in the following sense:‖F (Λ)‖T0(`j) ≤ ‖F‖Wj . (2.7.1)We let Wj = Wj(V) denote the space of K ∈ Kj(V) with finite Wj norm and denote the ballof radius r in the normed space Wj by BWj (r).LetjΩ = jΩ(m2) = inf{k ≥ 0 : |βj | ≤ 2−(j−k)‖β‖∞ for all j} (2.7.2)and note that, by (2.5.25), jΩ <∞ for m2 > 0. We defineϑj = ϑj(m2) = 2−(j−jΩ)+ (2.7.3)and write ϑ˜j = ϑj(m˜2). Given constants α > 0 and CD > 0, we define the (finite-volume)renormalization group domainsDj = {V ∈ V : g > C−1D g˜j , ‖V ‖U < CDg˜j}, (2.7.4)Dj = Dj(V) = Dj ×BWj (αϑ˜j g˜3j ). (2.7.5)The domain Dj is independent of the volume V while Dj depends on V through Wj .In the statement of the following theorem, we fix the scale j and consider maps U+ = Uj+1and K+ = Kj+1 that act on the domain Dj and map into U , Kj+1, respectively. We will dropthe scale j from the notation for objects at scale j and replace j + 1 with +. When V = Λ, we522.7. Renormalization group steptake j < N . The deviation of the map U+ from the perturbative map Upt is denoted by R+:R+(V,K) = U+(V,K)− Upt(V ). (2.7.6)The renormalization group map depends also on the mass m2 through its dependence onthe covariance Cj+1. We let I˜j(m˜2) be the neighbourhood of m˜2 defined byI˜j = I˜j(m˜2) =[12m˜2, 2m˜2] ∩ Ij (m˜2 6= 0)[0, L−2(j−1)] ∩ Ij (m˜2 = 0) , (2.7.7)where Ij = [0, δ] if j < N and IN = [δL−2(N−1), δ].Theorem 2.7.1. Let d = 4, n ≥ 0, and V = Λ or Zd. Fix s > 1 (or s = 0). Let CD and Lbe sufficiently large. There exist M > 0, δ > 0, and κ = O(L−1) such that for g˜ ∈ (0, δ) andm˜2 ∈ I+, and with the domain D defined using any α > M , the mapsR+ : D× I˜+ → U , K+ : D× I˜+ →W+ (2.7.8)are analytic in (V,K) and satisfy the estimates‖R+‖U ≤Mϑ˜+g˜3+, ‖K+‖W+ ≤Mϑ˜+g˜3+ (2.7.9)and‖DKK+‖L(W,W+) ≤ κ. (2.7.10)When V = Λ, these maps define (V,K) 7→ (U+,K+) obeying (2.6.11).Remark 2.7.2. When the improved weights are used, a new norm must be employed above themass scale. This will be discussed in Section 4.2.2. A technical requirement of this new normis that we set s > 1 rather than s > 0 as Lemma 4.2.2 fails with s ∈ (0, 1). This issue is absentwhen s = 0 as we do not change the norms in this case. Since we are ultimately interested onlyin the cases s = 0 and s large, we have not attempted to handle the case s ∈ (0, 1).With s = 0 in the choice of weights `j and `σ,j , this theorem was the main achievementof [31]. The statement in [31] with s = 0 additionally contains bounds on the derivatives of themaps R+ and K+. Our improvements apply to these bounds as well, but we do not state themhere as we will not make direct use of these bounds. One of the main novelties in this thesisis the case s > 1, for which the bounds on the observables derived from (2.7.9) are greatlyimproved beyond the mass scale.Note that the maps R+ and K+ themselves are independent of s. The proof of Theorem 2.7.1involves showing that the inductive estimates (2.7.9) hold for any s. In some cases, we will makeuse of these estimates both with s > 1 and s = 0. The proof for s > 1 is an adaptation of532.8. Renormalization group flowthe proof of the s = 0 case contained in2 [30, 31]. Some steps in this proof continue to holdunchanged whereas others require some modification. As mentioned above, a major changethat is required is a new definition of Wj above the mass scale. A detailed verification that theproof holds for s > 1 is carried out in Chapter 4.2.8 Renormalization group flowTheorem 2.7.1 allows us to perform a single renormalization group step. The fact that K+is a contraction, as expressed by the estimate (2.7.10), was used in [9, Proposition 7.1] toconstruct critical initial conditions νc0, zc0 depending on (m2, g0, n) such that the renormalizationgroup map can be iterated indefinitely (this was shown for n = 0 in [9] but extends withoutdifficulty to n ≥ 1 as discussed in [7]). This results in a sequence (Uj ,Kj) generated by therenormalization group map, hence whose elements lie in the domains Dj . This was proved withs = 0, but the sequence itself is independent of s and continues to exist in our setting. Inparticular, Theorem 2.7.1 shows that this sequence satisfies improved estimates. Thus, there isno difficulty in extending [9, Proposition 7.1] to the s-dependent domains used here.However, in order to study the WSAW-SA and the generalized |ϕ|4 model, we must extend[9, Proposition 7.1] to γ0 6= 0. We state this extension as Theorem 2.8.1 below, which is one ofthe main contributions of this thesis. Its proof, which depends on the results of [11] togetherwith a specially tailored version of the implicit function theorem, is the subject of Chapter 5.We note that, for n = 0, this proof first appeared in [13]; the proof for n ≥ 1 is new to thisthesis.Let δ > 0 and suppose r : [0, δ] → [0,∞) is a continuous positive-definite function; by thiswe mean3 that r(x) > 0 if x > 0 and r(0) = 0. We defineD(δ, r) = {(w, x, y) ∈ [0, δ]3 : y ≤ r(x)} (2.8.1)and we let C0,1,+(D(δ, r)) denote the space of continuous functions f = f(w, x, y) on D(δ, r)that are C1 in (x, y) away from y = 0, C1 in x everywhere, and whose right-derivative in y aty = 0 exists. In our applications, we take w = m2, x = g0 or g, and y = γ0 or γ.Theorem 2.8.1. There exists a domain D(δ, rˆ) (with δ > 0 and rˆ positive-definite) andfunctions νˆc0, zˆc0 ∈ C0,1,+(D(δ, rˆ)) such that for any (m2, g0, γ0) ∈ D(δ, rˆ) with g0 > 0 andm2 ∈ [δL−2(N−1), δ), the following holds: if (U0,K0) = (V +0 ,K+0 ) with (ν0, z0) = (νˆc0, zˆc0), thenfor any N ∈ N, there exists a sequence (Uj ,Kj) ∈ Dj(m2, g0) such that(Uj+1,Kj+1) = (Uj+1(Vj ,Kj),Kj+1(Vj ,Kj)) for all j < N (2.8.2)2For n ≥ 1, there is an additional step to deal with observables. This is dealt with in the proof of [108,Theorem 5.1] and is unchanged in the present context.3Note that our usage of this term is different from that in the theory of quadratic forms.542.9. Bibliographic remarksand (2.6.11) is satisfied. Moreover, the sequence Uj , j = 1, . . . , N is independent of the volumeΛ andνˆc0 = O(g0), zˆc0 = O(g0) (2.8.3)uniformly in (m2, γ0).Note that in the statement of Theorem 2.8.1 flow, we have evaluated the domains Dj at(m˜2, g˜0) = (m2, g0), where m2 is the mass in the covariance C and g0 = g(1 + zc0)2.2.9 Bibliographic remarksThe notion of a polymer used in Section 2.6 was introduced in [63]. The utility of multi-scaledecompositions of a singular covariance as a sum of regular covariances in the context of therenormalization group was probably first clearly articulated in [14]. The use of expansionsof the form (2.6.3) together with carefully weighted norms to achieve rigorous control of therenormalization group map goes back to Brydges and Yau [33]. This method was extended byDimock and Hurd, see e.g. [39,40]. Finite-range decompositions were first used with this methodto study a continuum model in [94], following a suggestion of Brydges. Lattice covariancedecompositions were constructed in [22] and used in [93] to study the renormalization group flowfor the supersymmetric field theory corresponding to WSAW; however, critical exponents werenot computed. Critical exponents for a version of weakly self-avoiding walk on a hierarchicallattice were computed by a renormalization group method in [24,25] (such hierarchical modelsgo back to [44]).55Chapter 3Analysis of critical behaviourIn this chapter, we prove Theorem 1.7.1 using Theorem 2.8.1. For simplicity, we drop theparameter n from the notation. In order to employ Theorem 2.8.1, we fixν0 = νˆc0(m2, g0, γ0), z0 = zˆc0(m2, g0, γ0). (3.0.1)Then Theorem 2.8.1 defines a sequence(Uj ,Kj) ∈ Dj , 0 ≤ j ≤ N (3.0.2)for any N . Moreover, Uj is independent of the volume, so we actually have an infinite sequenceUj ∈ Dj j ≥ 0. (3.0.3)Throughout this section we write Uj asUj;x = gjτ2y +νjτy+zjτ∆,y+uj−λ0,jf01y=0−λx,jfx1y=x− 12(1y=0q0,j +1y=xqx,j)σ0σx. (3.0.4)3.1 SusceptibilityThe proof of Theorem 1.7.1(ii) involves some small changes to the proof of the γ0 = 0 casein [9]. Rather than specifying the individual changes that need to be made, here we sketch thecomplete argument.Since the only scale-N blocks are the empty set and Λ, at scale j = N the representation(2.6.5) becomesZN = eζN (IN (Λ) +KN (Λ)). (3.1.1)In particular, (2.7.4)–(2.7.5), (2.5.10), (2.5.20), and (2.7.1) imply thatuN |ΛN | = O(1), νN = O(L−2NgN ), (3.1.2)‖WN‖T0(`N ) ≤ O(ϑNg2N ), ‖KN‖T0(`N ) ≤ O(ϑNg3N ). (3.1.3)Now by (3.1.1) and (2.2.11) together with the definitions of IN and VN ,χˆN (m2, g0, γ0, νc0, zc0) =1m2+1m4|Λ|−νN |Λ|+D2W 0N (0;1,1) +D2K0N (0;1,1)(1 +W 0N (0) +K0N (0)). (3.1.4)563.1. SusceptibilityRemark 3.1.1. In fact, with a bit more work it can be shown that D2W 0N (0;1,1) = 0.However, we will not need this here.Using Lemma 2.4.2 with s = 0 (recall (2.4.21)) together with (3.1.3), we see that the lastterm vanishes as N →∞ leavingχˆ(m2, g0, γ0, ν0, z0) = limN→∞χˆN (m2, g0, γ0, ν0, z0) =1m2. (3.1.5)In order to identify the asymptotics of m2 as ν approaches the critical point, we will needinformation about the derivative of χˆ with respect to ν0. Let us denote by F′ the derivativeof a function F with respect to ν0. By (3.1.4), the derivative χˆ′N will contain a term −ν ′N/m4.An argument using Lemma 2.4.2 shows that the remaining terms are of strictly higher order.Together with a careful analysis of the derivatives of the renormalization group flow with respectto the initial condition ν0 (as in [9, Section 8] for γ = 0), we getχˆ′(m2, g0, γ0, νc0, zc0) ∼ −1m4c(gˆ0, γ0)(gˆ0Bm2)(n+2)/(n+8)as (m2, g0, γ0)→ (0, gˆ0, γˆ0), (3.1.6)where c is a continuous function. The bubble diagram Bm2 was defined in (1.6.26) and itslogarithmic divergence as m2 ↓ 0 is ultimately the source of the logarithmic corrections inTheorem 1.7.1.Remark 3.1.2. There is one aspect of the proof of (3.1.6) that must be modified when γ0 = 0:This is the verification of the third bound in the base case (j = 0) of the inductive hypothesis [9,(8.34)]. This will be done in Section 5.1.4 (see Remark 5.1.7).3.1.1 Change of parametersWe wish to recover the asymptotics of χ from (3.1.5) and (3.1.6). By (2.2.10),χN (g, γ, ν) = (1 + z0)χˆN (m2, g0, γ0, ν0, z0), (3.1.7)whenever the variables on the left- and right-hand sides satisfyg0 = (g0 − γ)(1 + z0)2, ν0 = ν(1 + z0)−m2, γ0 = 14dγ(1 + z0)2. (3.1.8)On the other hand, (3.1.5) is contingent on the initialization of the renormalization group withthe critical parametersν0 = νˆc0(m2, g0, γ0), z0 = zˆc0(m2, g0, γ0). (3.1.9)Given g, γ, ν, the relations (3.1.8) leave free two of the variables (m2, g0, γ0, ν0, z0). Moregenerally, if any three of the variables (g, γ, ν,m2, g0, γ0, ν0, z0) are fixed, then two of the re-maining variables are free. In the following two propositions, which together form an extension573.1. Susceptibilityof [9, Proposition 4.2], we fix three variables and show that the addition of the constraints(3.1.9) allows us to uniquely specify the two remaining variables. For this, we make use of thefollowing version of the implicit function theorem, which we prove in Appendix C.Proposition 3.1.3. Let δ > 0, and let r1, r2 be continuous positive-definite functions on [0, δ].Recalling (2.8.1), setD(δ, r1, r2) = {(w, x, y, z) ∈ D(δ, r1)× Rn : |z| ≤ r2(x)}, (3.1.10)and let F be a continuous function on D(δ, r1, r2) that is C1 in (x, z). Suppose that for all(w¯, x¯) ∈ [0, δ]2 there exists z¯ such that both F (w¯, x¯, 0, z¯) = 0 and DY F (w¯, x¯, 0, z¯) is invert-ible. Then there is a continuous positive-definite function r on [0, δ] and a continuous mapf : D(δ, r) → Rn that is C1 in x and such that F (w, x, y, f(w, x, y)) = 0 for all (w, x, y) ∈D(δ, r). Moreover, if F is left-differentiable (respectively, right-differentiable) in y at somepoint (w, x, y, z), then f is left-differentiable (respectively, right-differentiable) at (w, x, y).Our first application of this result is Proposition 3.1.4, in which the three fixed variablesare (m2, g0, γ).Proposition 3.1.4. There exist δ∗ > 0, a continuous positive-definite function r∗ : [0, δ∗] →[0,∞), and continuous functions (ν∗, g∗0, γ∗0 , ν∗0 , z∗0) defined for (m2, g, γ) ∈ D(δ∗, r∗), such that(3.1.8) and (3.1.9) hold with ν = ν∗ and (g0, γ0, ν0, z0) = (g∗0, γ∗0 , ν∗0 , z∗0). Moreover,g∗0 = g0 +O(g20), ν∗0 = O(g0), z∗0 = O(g0). (3.1.11)Proof. Suppose we have found the desired continuous functions (g∗0, γ∗0) and that g∗0 satisfiesthe first bound in (3.1.11). Then the functions defined byν∗0 = νˆc0(m2, g∗0, γ∗0), z∗0 = zˆc0(m2, g∗0, γ∗0), ν∗ =ν∗0 +m21 + z∗0(3.1.12)are continuous, satisfy (3.1.8), and satisfy the remaining bounds in (3.1.11) by (2.8.3).In order to construct (g∗0, γ∗0), we first solve the third equation of (3.1.8), and then solve thefirst equation of (3.1.8). To this end, we begin by definingf1(m2, g0, γ, γ0) = γ0 − (4d)−1γ(1 + zˆc0(m2, g0, γ0))2 (3.1.13)for (m2, g0, γ0) ∈ D(δ, rˆ) and |γ| ≤ rˆ(g0). Although f1 is well-defined for any γ ∈ R, we restrictthe domain in preparation for our application of Proposition 3.1.3. Note that f1 is C1 in γand f1(·, ·, γ, ·) ∈ C0,1,+(D(δ, rˆ)) for any γ. The equation f1(m2, g0, γ, γ0) = 0 has the solutionγ0 = 0 when γ = 0 and, for any γ0 6= 0,∂f1∂γ0= 1− (2d)−1γ(1 + zˆc0(m2, g0, γ0))∂zˆc0∂γ0. (3.1.14)583.1. SusceptibilityBy Theorem 2.8.1, the one-sided γ0 derivatives of zˆc0 exist at γ0 = 0. Thus, the γ0 derivativeof f1 is well-defined and equal to 1 when γ = 0 for any small γ0 (including γ0 = 0). It followsby Proposition 3.1.3 (with w = m2, x = g0, y = γ, z = γ0 and r1 = r2 = rˆ) that there exists acontinuous function γ(1)0 (m2, g0, γ) on D(δ, r(1)) (for some continuous positive-definite functionr(1) on [0, δ]) such that f1(m2, g0, γ, γ(1)0 ) = 0. Moreover, γ(1)0 is C1 in (g0, γ).Next, we definef2(m2, g, γ, g0) = g0 − (g − γ)(1 + zˆc0(m2, g0, γ(1)0 (m2, g, γ)))2 (3.1.15)for (m2, g0, γ) ∈ D(δ, r(1)) and g ∈ [0, δ∗], where δ∗ > 0 will be made sufficiently small below.Then f2(m2, g, γ, g0) = 0 is solved by (γ, g0) = (0, g∗0(m2, g, 0)), where g∗0(m2, g, 0) was con-structed in [9, (4.35)]. By [9, (4.37)], g∗0 = g + O(g2), so we may restrict the domain of f2 sothat |g0| ≤ 2g. Moreover,∂f2∂g0= 1− 2(g − γ)(1 + zˆc0(m2, g0, γ(1)0 ))(∂zˆc0∂g0+∂zˆc0∂γ0∂γ(1)0∂g0). (3.1.16)Differentiating both sides ofγ(1)0 =14dγ(1 + zˆc0(m2, g0, γ(1)0 ))2, (3.1.17)and solving for∂γ(1)0∂g0, gives∂γ(1)0∂g0=γ(1 + zˆc0)∂zˆc0∂g02d− γ(1 + zˆc0) ∂zˆc0∂γ0, (3.1.18)where zˆc0 and its derivatives are evaluated at (m2, g0, γ(1)0 ). Thus,∂γ(1)0∂g0= 0 when γ = 0. Itfollows that ∂f2/∂g0 is well-defined when (γ, g0) = (0, g∗0(m2, g, 0)) and equals1− 2g(1 + zˆc0(m2, g∗0, 0))∂zˆc0∂g0(m2, g∗0, 0), (3.1.19)which is positive when δ∗ is small, by (2.8.3). Thus, by Proposition 3.1.3 (with w = m2, x = g,y = γ, z = g0 and r1 = r(1), r2(g) = 2g), there exists a function g∗0(m2, g, γ) ∈ C0,1,+(D(δ∗, r(2)))(for some continuous positive-definite function r(2) on [0, δ∗]) such that f2(m2, g, γ, g∗0) = 0.By the fact that g∗0 solves f2 = 0,g∗0 = (g − γ) +O((g − γ)2). (3.1.20)Since |γ| ≤ r(2)(g0) and r(2)(g0) can be taken as small as desired, this implies the first estimatein (3.1.11). Thus, by taking r∗ sufficiently small, if |γ| ≤ r∗(g0), then |γ| ≤ r(2)(g∗0(m2, g, γ)).593.1. SusceptibilityThus, for g < δ∗ and |γ| ≤ r∗(g), we can defineγ∗0(m2, g, γ) = γ(1)0 (m2, g∗0(m2, g, γ), γ), (3.1.21)which completes the proof.Using Proposition 3.1.4, it is possible to identify the critical point νc, as follows. By (3.1.5),(3.1.7), Proposition 1.8.4, and Proposition 3.1.4,χ(g, γ, ν∗) =1 + z∗0m2=1 +O(g)m2. (3.1.22)Thus, with ν = ν∗, we see that χ < ∞ when m2 > 0, and χ = ∞ when m2 = 0. By (2.1.12),this implies thatνc(g, γ) = ν∗(0, g, γ) = O(g), νc(g, γ) < ν∗(m2, g, γ) (m2 > 0). (3.1.23)It follows thatχ(g, γ, νc) =∞, (3.1.24)which is a fact that cannot be concluded immediately from the definition (2.1.12).In (3.1.22), χ is evaluated at ν∗ = ν∗(m2, g, γ). However, in the setting of Theorem 1.7.1,we need to evaluate χ at a given value of ν and then take ν ↓ νc. To do so, we must determinea choice of m2 in terms of ν such that (3.1.8) is satisfied and this choice must approach 0 (asit should by (3.1.23)) right-continuously as ν ↓ νc. The following proposition carries out thisconstruction. In the following, the functions m˜2, g˜0 should not be confused with the parameterm˜2, g˜0 that appeared previously in the Wj norms (these norms are not used in this chapter).Proposition 3.1.5. Write ν = νc + ε. There exist functions m˜2, g˜0, γ˜0, ν˜0, z˜0 of (ε, g, γ) ∈D(δ∗, r∗) (all right-continuous as ε ↓ 0) such that (3.1.8) and (3.1.9) hold with(m2, g0, γ0, ν0, z0) = (m˜2, g˜0, γ˜0, ν˜0, z˜0). (3.1.25)Moreover,m˜2(0, g, γ) = 0, m˜2(ε, g, γ) > 0 (ε > 0) (3.1.26)g˜0 = g +O(g2), ν˜0 = O(g), z˜0 = O(g). (3.1.27)Proof. The proof is a minor modification of the proof in [9], using Proposition 3.1.4. Definem˜2 = m˜2(ε, g, γ) = inf{m2 > 0 : ν∗(m2, g, γ) = νc(g, γ) + ε}, (3.1.28)603.2. Two-point functionon D(δ∗, r∗). By continuity of ν∗, the infimum is attained andνc(g, γ) + ε = ν∗(m˜2(ε, g, γ), g, γ). (3.1.29)From the above expression, continuity of ν∗, and (3.1.23), it follows that m˜2 is right-continuousas ε ↓ 0. It is immediate that (3.1.26) holds. Also, the functions of (ε, g, γ) defined byν˜0 = ν∗0(m˜2, g, γ), z˜0 = z∗0(m˜2, g, γ), (3.1.30)g˜0 = (g − γ)(1 + z˜0)2, γ˜0 = 14dγ(1 + z˜0)2 (3.1.31)are right-continuous as ε ↓ 0 and satisfy (3.1.8). The bounds (3.1.27) follow from the definitionsand (3.1.11), and the proof is complete.3.1.2 Conclusion of the argumentWe sketch the remainder of the argument, which follows as in [9, Section 4]. By Proposi-tion 1.8.4, (3.1.7), (3.1.5), and Proposition 3.1.5,χ(g, γ, ν) =1 + z˜0m˜2. (3.1.32)Similarly, from (3.1.6) (using (3.1.32)), we getχ′(g, γ, ν) ∼ −χ2(g, γ, ν) c0(g, γ)(g˜0Bm˜2)n+2n+8(3.1.33)with c0(g, γ) = limε↓0 c(g˜0, γ˜0). By exactly the same argument as in [9, Section 4.3], thedifferential relation (3.1.33) can be solved, which gives the result of Theorem 1.7.1(ii).Remark 3.1.6. It is a consequence of (1.7.2) and (3.1.32) thatm˜2 ∼ A˜−1g,nε(log ε−1)−n+2n+8 as ε ↓ 0. (3.1.34)3.2 Two-point functionOur analysis of the two-point function and finite-order correlation length is based on the fol-lowing proposition.Proposition 3.2.1. Let d = 4, n ≥ 0, ε ∈ (0, δ) with δ sufficiently small, and ν = νc + ε. Letx ∈ Z4 with x 6= 0. Fix s = 0 or s > 1. For L sufficiently large and for g > 0 sufficiently small(depending on s),11 + z˜0Gx(g, γ, ν) = (1 +O(g¯jx))Gx(0, 0, m˜2) +Rx(m˜2) (3.2.1)613.2. Two-point functionand the remainder Rx satisfies the bound|Rx(m2)| ≤ O(g¯jx)|x|2 ×1, (m|x| ≤ 1)(m|x|)−2s, (m|x| ≥ 1) (3.2.2)with the constant depending on L and s.Proof. Let Dσ0 and Dσx denote differentiation with respect to σ0 and σx, respectively, evaluatedwith all fields set to 0. By (2.2.9), (3.1.1), and (2.6.5),11 + z˜0Gx,N (g, γ, ν) =12(q0,N + qx,N ) +D2σ0σxK0N1 +K0N−(Dσ0K0N) (DσxK0N)(1 +K0N )2, (3.2.3)where the quantities on the right-hand side are evaluated at (m˜2, g˜0, γ˜0, ν˜0, z˜0). No WN termappears on the right-hand side since WN is quadratic in V and V has no constant part. By(3.1.3) and Lemma 2.4.2, the last two terms vanish as N →∞ leaving11 + z˜0Gx(g, γ, ν) =12(q0,∞ + qx,∞). (3.2.4)Now it is a straightforward computation using (2.5.28)–(2.5.30) and (2.7.6) to show thatqu,∞ = λ0,jxλx,jxGx(0, 0, m˜2) +∞∑i=jxRqui , u = 0, x (3.2.5)where Rqui is the coefficient of 1y=uσ0σx (recall (2.5.7)) in R+,i. Moreover, as in [108, (5.30)]and [108, Corollary 6.4],λu,jx = 1 +O(ϑjx g¯jx). (3.2.6)It follows that11 + z˜0Gx(g, γ, ν) = (1 +O(g¯jx))Gx(0, m˜2) +Rx (3.2.7)withRx =12∞∑i=jx(Rq0i +Rqxi ). (3.2.8)By the first bound of (2.7.9) and the definition (2.5.10) of the V norm,|Rqu+,i| ≤ O(`−2σ,iϑig¯3i ). (3.2.9)We insert the definition of `σ,j from (2.4.21) into (3.2.9). We also use g˜−2j = O(g¯−2j ), ϑi ≤ 1,`20 ≤ O(1), as well as g¯j ≤ O(g¯jx) for j ≥ jx. The definitions of the coalescence scale jx and the623.3. Finite-order correlation lengthmass scale jm imply that L−2jx ≤ O(|x|−2) and L−(jx−jm)+ ≤ O((m|x|)−1). All this leads to∞∑j=jx|Rquj | ≤ L−2jx−2s(jx−jm)+∞∑j=jxO(g¯j)4−(j−jx)≤ |x|−2(m|x|)−2sO(g¯jx). (3.2.10)This gives the desired estimate (3.2.2).A version of this result with s = 0 and γ = 0 was obtained in [8, 108]. This version issufficient for studying the critical two-point function with γ = 0. With the extension to γ 6= 0,we can complete the proof of the first part of Theorem 1.7.1.Proof of Theorem 1.7.1(i). We apply Proposition 3.2.1 with s = 0 to get11 + z˜0Gx(g, γ, ν) = (1 +O(g¯jx))Gx(0, 0, m˜2) +Rx(m˜2). (3.2.11)By Proposition 3.1.5, m˜2 = 0 when ν = νc. Since Rx(0) = O(g¯jx)Gx(0, 0, 0),11 + z˜0Gx(g, γ, νc) = (1 +O(g¯jx))Gx(0, 0, 0) (3.2.12)and the result follows from (2.5.24).3.3 Finite-order correlation lengthAn elementary ingredient in the proof of Theorem 1.7.1(iii) is the following result for the g = 0case, which is independent of n ≥ 0. For simplicity, we restrict attention to dimensions d > 2,as only d = 4 is used here. A proof is provided in Appendix B.Proposition 3.3.1. Let cp be the constant defined by (1.7.4). For all dimensions d > 2 andall p > 0, as m2 ↓ 0, ∑x∈Zd|x|pGx(0, 0,m2) = cppm−(p+2)(1 +O(m)). (3.3.1)In particular, ξp(0, 0, ε) = cpε−1/2(1 +O(ε1/2)) as ε ↓ 0.Proof of Theorem 1.7.1. We multiply (3.2.1) by |x|p, sum over x ∈ Z4, and use (3.1.22) toobtainξpp(g, γ, ν) =∑x∈Z4|x|pGx(g, γ, ν)χ(g, γ, ν)= m˜2∑x∈Z4|x|p(Gx(0, 0,m2) + rx(m˜2)), (3.3.2)633.3. Finite-order correlation lengthwithrx(m2) = O(g¯jx)Gx(0, 0,m2) +Rx(m2). (3.3.3)By Proposition 3.3.1, this gives (as m˜2 ↓ 0)ξpp(g, γ, ν) ∼ cppm˜−p + m˜2∑x∈Z4|x|prx(m˜2). (3.3.4)By (3.1.34), it suffices to prove that the first term on the right-hand side of (3.3.4) is dominant.For the term O(g¯jx)Gx(0, 0,m2) in (3.3.3), we apply (2.5.24) to obtain∑x∈Z4g¯jx |x|pGx(0, 0, m˜2)≤∑x:0<jx≤jm˜c|x|plog |x|Gx(0, 0, m˜2) +clog m˜−1∑x:jx>jm˜|x|pGx(0, 0, m˜2). (3.3.5)In the first term, we use Gx(0, 0,m2) ≤ Gx(0, 0, 0) ≤ O(|x|−2). The restriction jx ≤ jm˜ensures that |x| ≤ O(m˜−1). Therefore the first term is bounded above by a multiple of(m˜−1)d+p−2(log m˜−1)−1, which suffices. For the term with jx > jm˜, we extend the sum tox ∈ Z4 and apply Proposition 3.3.1 to obtain a bound of the same form as for the first term.For the term Rx of (3.3.3), we use Proposition 3.2.1 to see that|Rx(m˜2)| = O(g¯jx)L−2jx−2s(jx−jm˜)+ . (3.3.6)We divide Z4 into shells S1 = {x : |x| < 12L} and, for j ≥ 2, Sj = {x : 12Lj−1 ≤ |x| < 12Lj}.The number of points in Sj is bounded by O(L4j). Note that jx is the unique scale so thatx ∈ Sjx+1. (3.3.7)By (3.3.6) with s > 12(p+ 2) and (3.3.7),∑x∈Z4|x|p|Rx(m˜2)| =∞∑j=1∑x∈Sj|x|p|Rx(m˜2)| =∞∑j=1L4j+pj−2j−2s(j−jm˜)+O(g¯j), (3.3.8)with an L-dependent constant. By Lemma 3.3.2 below (with a = p+ 2 and b = 1), we obtainm˜2∑x∈Z4|x|p|Rx(m˜2)| = O(m˜−p(log m˜−1)−1). (3.3.9)The first term on the right-hand side of (3.3.4) therefore dominates, and the proof is complete.The estimate used to obtain (3.3.9) is given by the following lemma, which is stated more643.3. Finite-order correlation lengthgenerally for use in the proof of Proposition 3.3.1.Lemma 3.3.2. Let L > 1, 2s > a > 0, b ≥ 0, and let g¯0 > 0 be sufficiently small. Then∞∑j=1Laj−2s(j−jm)+ g¯bj = O(m−ag¯bjm) = O(m−a(logm−1)−b). (3.3.10)Proof. We divide the sum at the mass scale as∞∑j=1Laj−2s(j−jm)+ g¯bj =jm∑j=1Laj g¯bj +∞∑j=jm+1Laj−2s(j−jm)g¯bj . (3.3.11)For the second sum on the right-hand side, we use g¯j = O(g¯jm) for j > jm by (2.5.24), andobtain a bound consistent with the first equality of (3.3.10). For the first term, we use thecrude bound g¯i/g¯i+1 = 1 +O(g0) (by [11, Lemma 2.1]), and findjm∑j=1Laj g¯bj ≤ Lajm g¯bjmjm∑j=1((1 +O(g¯0))L−a)jm−j = O(Lajm g¯bjm), (3.3.12)for sufficiently small g¯0 > 0. This proves the first equality in (3.3.10). The second equality thenfollows since g¯jm = O(logm−1) by (2.5.24).65Chapter 4The renormalization group stepIn this chapter we discuss the proof of Theorem 2.7.1 with s > 1 in the definitions (2.4.21) ofthe weights `, `σ. The proof involves numerous changes to results in [29–31]. Consequently, thearguments presented here will not be completely self-contained; for instance, we will not detailthe construction of the renormalization group map, which makes up the bulk of [31]. However,we will begin in Section 4.1 with an informal overview of some of the key ideas used in thesepapers.The details of the proof begin in Section 4.2, where we define new norms above the massscale and show that they satisfy two key hypotheses required by results in [30]. The use of thesenew norms is essential: the old norms fail to work with the new weights for technical reasonsdiscussed briefly in Remark 4.2.3. In Section 4.3, we discuss the changes that must be madein [29,30] with the new weights. Of particular importance is the adaptation of the proof of thecrucial contraction to these new weights, which is discussed in Section 4.3.4.Throughout this chapter, we use the notation appropriate for the spin field ϕ ∈ (Rn)Λ forn ≥ 1; only notational modifications are needed for n = 0. Since we are dealing with a singlerenormalization group step, we will often drop the index j of the current scale and write asubscript + to indicate objects at the next scale j + 1.4.1 Simplified renormalization group stepFor this discussion, let us drop λ, q, u from the notation and write U = V . In this setting ourgoal, given (V,K), is to construct (V+,K+) such that, with I = I(V ) and I+ = I(V+),E+θ(I ◦K)(Λ) = (I+ ◦K+)(Λ) (4.1.1)with K 7→ K+ contractive in some sense; this is needed not only to control the error producedby K in the computation of critical exponents (e.g. recall the use of Lemma 2.4.2 in Section 3)but also so that the map (V,K) 7→ (V+,K+) can be iterated an arbitrary number of times asin Theorem 2.8.1. The algebraic problem (4.1.1) admits a multitude of solutions and the maindifficulty is the construction of a solution with good analytic properties.A possible definition of the map (V,K) 7→ V+ is suggested by perturbation theory, asdiscussed in Section 2.5.3. For now, let us suppose that V+ = Vpt is the correct definition andset I+ = Ipt = I(Vpt). Then we have the following procedure for the construction of K+ interms of Ipt such that (4.1.1) holds.664.1. Simplified renormalization group stepRecall (2.6.2). For B ∈ B, let δI(B) = θI(B)−Ipt(B) and extend this to X ∈ P by imposingblock-factorization. Then by (2.6.10) and associativity of the circle product,(I ◦K)(Λ) =∑X∈P(Ipt + δI)Λ\XK(X)= [(Ipt ◦ δI) ◦K](Λ)= [Ipt ◦ (δI ◦K)](Λ). (4.1.2)Note that the fluctuation fields at scale j have been integrated out in the definition of Vpt.Thus,E+θ(I ◦K)(Λ) = [Ipt ◦ K˜](Λ) (4.1.3)withK˜ = E+(δI ◦ θK). (4.1.4)This has the form (4.1.1) but with the circle product on the right-hand side at the wrong scale.This is remedied by a simple resummation:E+θ(I ◦K)(Λ) =∑Y ∈PIΛ\Ypt K˜(Y ) =∑X∈P+IΛ\Xpt K+(X) = (Ipt ◦K+)(Λ) (4.1.5)whereK+(X) =∑Y ∈P¯(X)IX\Ypt K˜(Y ). (4.1.6)Here, P¯(X) is the collection of polymers Y ∈ P(X) whose polymer closure is X, i.e. for whichX is the smallest polymer in P+ containing Y .4.1.1 Main contributions to K+By expanding the circle product in the definition of K˜, we can writeK+(X) = IXpt [h(X) + k(X) +R(X)] (4.1.7)where (letting C¯(X) = C ∩ P¯(X))h(X) =∑Y ∈P¯(X)I−Ypt E+δI(Y ) (4.1.8)k(X) =∑Y ∈C¯(X)I−Ypt E+θK(Y ) (4.1.9)and R is the remainder. If δI and K are sufficiently small (in an appropriate sense), then itis reasonable to view the terms h and k as first-order contributions to K+. Note that the sumdefining k is restricted to connected polymers; by component-factorization, terms involving the674.1. Simplified renormalization group stepvalues of K on disconnected polymers should be higher-order in this sense. Our main task thenis to bound h and k.Perturbative contribution and covariance boundBy the definition (2.6.6) of I, we expect that δI = O(δV ) where δV = θV−Vpt. Indeed, a versionof this statement is true by [30, Proposition 2.7]. Thus, in order to bound the contribution hdefined in (4.2.13), we must consider the size of δV . By definition,‖δV (B)‖T0(`) ≤ ‖θV (B)− V (B)‖T0(`) + ‖V (B)− Vpt(B)‖T0(`). (4.1.10)By (2.5.16), Vpt − V = (EθV − V ) − P (V ) and the first term on the right-hand side canbe bounded term-by-term. For instance, the difference between a single quartic term and itsexpectation is given byϕ4x − Eθϕ4x = 6C00ϕ2x + 3C200. (4.1.11)Covariance terms such as those above can be estimated using Proposition 2.3.1. The methodof [30] is more flexible, however, and does not require the precise bounds in (2.3.4). Rather,necessary bounds on the covariance and its derivatives are encoded in the hypothesis [30,(1.73)] on the Φ(`) norm estimate of the covariance. This constraint naturally ensures thatthe T0(`) seminorm estimates properly reflect the size of the expectation of a field as discussedin Section 2.4.3. Our main bound on the covariance, which extends [30, (1.73)], will be statedin Section 4.2.1.Remark 4.1.1. The generality provided by predicating the results of [30] on a norm estimateon the covariance is very useful, e.g. as in [107].Extraction and contractionThe term k in (4.1.9) is the contribution to K+ that is linear in K. Thus, its control is essentialto obtaining the contractivity estimate (2.7.10).In the simple case that K = 0, we will have K+ = Ipth, which is a Taylor remainder thatcontains terms at all orders in the fields. Thus, it includes relevant and marginal terms as wellas non-local irrelevant terms. The size and number of such terms will prevent K from shrinkingunder the action of the renormalization group unless they are somehow dealt with. This is doneby using the operator Loc mentioned in Section 2.5.3 to extract a marginal/relevant part fromK prior to integration.Thus, the true definition of the renormalization group map constructed in [31] involvesseveral more steps than (4.1.6). In fact, the definition of the map (V,K) 7→ (V+,K+) is acomposition of 6 maps, called Maps 1–6. In Maps 1–2, the operator Loc is used to performextraction. Map 3 then implements the expectation and change of scale in (4.1.6).684.2. Improved normOnce a sufficiently large portion of the marginal and relevant terms have been extractedfrom K in Maps 1–2, we expect by the discussion in Section 2.5.1 that the expectation in Map3 should cause K to contract. The fact that irrelevant terms shrink under expectation andchange of scale is formally captured by [30, Proposition 2.8], which we refer to as the crucialcontraction. In Section 4.3.4, we prove that the crucial contraction continues to hold whens > 1.Remark 4.1.2. In order to maintain the form (4.1.1), the extraction step in Maps 1–2 mustmake a corresponding adjustment to V . This adjustment results in a map V 7→ V+, given byV+ = Vpt(V −Q), Q(B) =∑B⊂Y ∈SLocY,B I−YK(Y ), (4.1.12)where LocY,B F is the restriction to B ⊂ Y of the polynomial on Y determined by LocY F .The map V+ is a perturbation of Vpt whose size is determined by the norm of K. Inparticular, for (V,K) ∈ D we get the first bound of (2.7.9) (recall (2.7.6)), which ensures thatthe flow of coupling constants exhibits the same qualitative behaviour as the perturbative flow.On the other hand, if K is large, then the resulting perturbation may have a non-trivial effecton the flow of coupling constants, resulting in different critical behaviour than that predictedby the perturbative flow.4.2 Improved normIn this section, we prove an improved covariance estimate, which indicates why it is possible touse the improved weights (2.4.21). This leads to a discussion of simplified norm pairs beyond themass scale. A lemma concerning the fluctuation-field regulator indicates why the simplificationis possible.4.2.1 Covariance boundsRecall from (2.4.21) that`j = `0L−j−s(j−jm)+ , `σ,j = `−1j∧jx2(j−jx)+ g˜j . (4.2.1)The analysis of [30, 31] uses the norm parameters `j and `σ,j with s = 0. To distinguish thesefrom our new choice (4.2.1) of `j and `σ,j , we write`oldj = `0L−j , `oldσ,j = (`oldj∧jx)−12(j−jx)+ g˜j . (4.2.2)We may regard a covariance Cj in the decomposition (2.3.1) as a test function depending694.2. Improved normon two arguments x, y, and with this identification its Φj(`j) norm is‖Cj‖Φj(`j) = `−2j supx,y∈Λsup|α|1+|β|1≤pΦL(|α|1+|β|1)j |∇αx∇βyCj;x,y|. (4.2.3)The following lemma justifies our choice of `j in (4.2.1), by showing that the bound [30,(1.73)], proved there only for the s = 0 version `oldj of (4.2.2), remains true with the strongerchoice of norm parameter `j that permits arbitrary s ≥ 0. Recall that the sequence ϑj wasdefined in (2.7.3).Lemma 4.2.1 (Extension of [30, (1.73)]). Given c ∈ (0, 1], `0 can be chosen large (dependingon L, c, s) so that‖Cj‖Φj(`j) ≤ min(c, ϑj). (4.2.4)Proof of Lemma 4.2.1. For d = 4, insertion of (2.3.4) into (4.2.3) gives‖Cj‖Φj(`j) ≤ cLpΦ`−2j (1 +m2L2(j−1))−kL−2(j−1). (4.2.5)With s = 0 in (4.2.1), (4.2.5) gives ‖Cj‖Φj(`j) ≤ cL`−20 (1 + m2L2(j−1))−k for an L-dependentconstant cL (whose value may now change from line to line). We insert (2.4.18) and the definition`j = `0L−j−s(j−jm)+ from (4.2.1) into (4.2.5), to conclude that there exists c0 = c0(s, L) suchthat‖Cj‖Φj(`j) ≤ c0`−20 L−2(j−jm)+ . (4.2.6)By definition of ϑj , L−2(j−jm)+ is bounded by a multiple of ϑj . It thus suffices to choose `0large enough that `20 ≥ c0c−1.4.2.2 New choice of norm beyond the mass scaleA field ϕ can be viewed as a test function supported on sequences with |~x| = 1 and |~y| = 0. Inparticular,‖ϕ‖Φj(`j) = `−1j supx∈Λsup1≤i≤nsup|α|1≤pΦLj|α|1 |∇αϕix|, (4.2.7)As in [30, (1.36)], we use the localized version of (4.2.7), defined for subsets X ⊂ Λ by‖ϕ‖Φj(X) = inf{‖ϕ− f‖Φj : f ∈ CΛ such that fx = 0 ∀x ∈ X}. (4.2.8)A similar definition is given for general test functions. Given X ⊂ Λ and ϕ ∈ (Rn)Λ, we recallfrom [30, (1.38)] that the fluctuation-field regulator Gj is defined byGj(X,ϕ) =∏x∈Xexp(|Bx|−1‖ϕ‖2Φj(Bx ,`j)), (4.2.9)704.2. Improved normwhere Bx ∈ Bj is the unique block that contains x, and hence |Bx| = Ldj . The large-fieldregulator is defined in [30, (1.41)] byG˜j(X,ϕ) =∏x∈Xexp(12|Bx|−1‖ϕ‖2Φ˜j(Bx ,`j)). (4.2.10)The Φ˜j seminorm appearing on the right-hand side of (4.2.10) will be defined in Section 4.3.4.The two regulators serve as weights in the regulator norms of [30, Definition 1.1]. The regulatornorms are defined, with t ∈ (0, 1] and for F ∈ N (X) by‖F‖Gj(`j) = supϕ∈(Rn)Λ‖F‖Tϕ,j(`j)Gj(X,ϕ), (4.2.11)‖F‖G˜tj(hj) = supϕ∈(Rn)Λ‖F‖Tϕ,j(hj)G˜tj(X,ϕ). (4.2.12)The parameter `j that appears in the regulators (4.2.9)–(4.2.10) and in the numerator of (4.2.11)was taken to be `oldj in [30], but now we use `j instead. As in [30], the parameter hj and itsobservable counterpart hσ,j are given byhj = k0g˜−1/4j L−j , hσ,j = (`oldj∧jx)−12(j−jx)+ g˜1/4j . (4.2.13)In [30], estimates on ‖ · ‖j+1 are given in terms of ‖ · ‖j , where the pair (‖ · ‖j , ‖ · ‖j+1) refersto either of the norm pairs‖F‖j = ‖F‖Gj(`oldj ) and ‖F‖j+1 = ‖F‖T0,j+1(`oldj+1), (4.2.14)or‖F‖j = ‖F‖G˜j(hj) and ‖F‖j+1 = ‖F‖G˜tj+1(hj+1). (4.2.15)We will show that, above the mass scale jm, the results of [30] hold with both norm pairs in(4.2.14) and (4.2.15) replaced by the single new norm pair‖F‖j = ‖F‖Gj(`j) and ‖F‖j+1 = ‖F‖Gj+1(`j+1), (4.2.16)with the improved `j of (4.2.1) with s > 1 fixed as large as desired.The use of two norm pairs adds intricacy to [30,31]. The pair (4.2.14) is insufficient, on itsown, because the scale-(j+ 1) norm is the T0 seminorm which controls only small fields, and anestimate in this norm does not imply an estimate for the Gj+1 norm. The norm pair (4.2.15)is used to supplement the norm pair (4.2.14), and estimates in both of the scale-(j + 1) normscan be combined to provide an estimate for the Gj+1 norm. This then sets the stage for thenext renormalization group step. Above the mass scale, the use of (4.2.16) now bypasses manyissues. For example, for j > jm the Wj norm of [31, (1.45)] is replaced simply by the Fj(G)714.2. Improved normnorm, and there is no need for the Yj norm of [31, (2.12)] nor for [31, Lemma 2.4].The need for both norm pairs (4.2.14)–(4.2.15) is discussed in [30, Section 1.2.1] and isrelated to the so-called large-field problem. Roughly speaking, the norm pair (4.2.15) is usedto take advantage of the quartic term in the interaction to suppress the effects of large valuesof the fields. This approach relies on the fact that the interaction polynomial is dominated bythe quartic term in the h-norm, as expressed by [30, (1.91)], together with the lower bound [30,(1.90)] on the quartic term. However, above the mass scale, large fields are naturally suppressedby the rapid decay of the covariance. This idea is captured in Lemma 4.2.2 below, whichreplaces [30, Lemma 1.2] above the mass scale. The regulators in its statement are defined by(4.2.9) with the s-dependent `j of (4.2.1).Lemma 4.2.2 (Replacement for [30, Lemma 1.2]). Let X ⊂ Λ and assume that s > 1. Forany q > 0, if L is sufficiently large depending on q, then for jm ≤ j < N ,Gj(X,ϕ)q ≤ Gj+1(X,ϕ). (4.2.17)Proof. By (4.2.9), it suffices to show that, for any scale-j block Bj and any scale-(j + 1) blockBj+1 containing Bj ,q‖ϕ‖2Φj(Bj ,`j)≤ L−4‖ϕ‖2Φj+1(Bj+1,`j+1). (4.2.18)In fact, since ‖ϕ‖Φj(Bj ,`j) ≤ ‖ϕ‖Φj(Bj+1,`j) by definition, it suffices to prove the above boundwith Bj replaced by Bj+1 on the left-hand side. According to the definition of the norm in(4.2.8), to show this it suffices to prove thatq‖ϕ‖2Φj(`j) ≤ L−4‖ϕ‖2Φj+1(`j+1) (4.2.19)(then we replace ϕ by ϕ− f in the above and take the infimum).By definition,‖ϕ‖Φj(`j) ≤ `−1j `j+1 supx∈Λsup|α|≤pΦ`−1j+1L(j+1)|α||∇αϕx|, (4.2.20)with the inequality due to replacement of Lj|α| on the left-hand side by L(j+1)|α| on the right-hand side. Since `−1j `j+1 = L−1−s1j≥jm ,‖ϕ‖Φj(`j) ≤ L−1−s1j≥jm‖ϕ‖Φj+1(`j+1). (4.2.21)Thus,q‖ϕ‖2Φj(`j) ≤ qL−4L2−2s1j≥jm‖ϕ‖2Φj+1(`j+1), (4.2.22)and then (4.2.19) follows once L is large enough that qL2−2s ≤ 1.Remark 4.2.3. The elimination of the h-norm after the mass scale is more than a convenience.It becomes a necessity when we improve the `-norm. Briefly, the reason is as follows. In the724.3. Proof of Theorem 2.7.1proof of [31, Lemma 2.4], the ratio `σ/hσ must be bounded. For this, we would need to increasehσ beyond the mass scale (since `σ has been increased). This forces a compensating decreasein h beyond jm, to keep the product hhσ bounded for stability (as in Section 4.3.2 below). Butif we do this, we lose the lower bound required on gτ2 required for stability in the h-norm(see [30, (3.8)]).4.3 Proof of Theorem 2.7.1In this section, we show that Theorem 2.7.1 holds, thereby completing the proof of Proposi-tion 3.2.1. The key steps in the proof of the s = 0 case of Theorem 2.7.1 are contained in [30,31].Our main objective in this section is to show that the results in [30, 31] continue to hold withthe new norm parameters `j , `σ,j . To this end, we may and do use the fact that the estimatesof [30] have already been established with the old norm parameters.In the following, we indicate the changes in the analysis of [30,31] that arise due to the newchoice of norm parameters (4.2.1) beyond the mass scale, and due to the reduction from twonorm pairs to one. This requires repeated reference to previous papers. In such references, wewill sometimes use the notation from those papers without defining it here.4.3.1 Norm parameter ratiosThe analysis of [30] assumes that the norm parameters hj , hσ,j , for h = ` or h = h, satisfy theestimates [30, (1.79)]; these assert thathj ≥ `j , hj+1hj≤ 2L−1, hσ,j+1hσ,j≤ constL (j < jx)1 (j ≥ jx). (4.3.1)We do not change hj or hσ,j for j below the mass scale, so there can be no difficulty until abovethe mass scale. Above the mass scale, the parameters hj , hσ,j are eliminated, and requirementsinvolving them become vacuous. Thus, for (4.3.1), we need only verify the second and thirdinequalities for the case h = `. By definition,`j+1`j= L−(1+s1j≥jm ),`σ,j+1`σ,j=g˜j+1g˜j×L1+s1j≥jm (j < jx)2 (j ≥ jx). (4.3.2)According to [30, (1.77)], 12 g˜j+1 ≤ g˜j ≤ 2g˜j+1. Thus, the second estimate of (4.3.1) is satisfied(the ratio being improved when j ≥ jm), while the third is not when s > 1 and jm < jx.This potentially dangerous third estimate in (4.3.1) is used to prove the scale monotonicitylemma [30, Lemma 3.2], as well as the crucial contraction. We discuss [30, Lemma 3.2] next,and return to the crucial contraction in Section 4.3.4 below.734.3. Proof of Theorem 2.7.1[30, Lemma 3.2] There is actually no problem with the scale monotonicity lemma. Indeed,for the case α = ab of the proof of [30, Lemma 3.2], the hypothesis that pi0xF = 0 for j < jxensures that this case only relies on the dangerous estimate for j ≥ jx where the danger isabsent in (4.3.2). For the cases α = a and α = b of the proof of [30, Lemma 3.2], what isimportant is the inequality `σ,j+1`j+1 ≤ const `σ,j`j , which continues to hold with (4.2.1) forall scales j, both above and below the mass scale, since the products in this inequality are thesame for the new and the old choices of `. So [30, Lemma 3.2] continues to hold with the choice(4.2.1). In addition,‖F‖Tϕ(`j) ≤ ‖F‖Tϕ(`oldj ). (4.3.3)This strengthened special case of the first inequality of [30, (3.6)] (strengthened due to theconstant 1 on the right-hand side of (4.3.3) compared to the generic constant in [30, (3.6)]) canbe seen from an examination of the proof of the α = a, b case of [30, Lemma 3.2], together withthe observation that `σ,j`j = `oldσ,j`oldj by definition.4.3.2 Stability domainsIn [30, (1.83)], an extension of the domain (2.7.4) is defined. By some abuse of notation, wewill also denote this extended domain by Dj . We modify Dj only for the coupling constant q,by replacing rq in [30, (1.84)] byL2jx+2s(jx−jm)+22(j−jx)rq,j =0 j < jxCD j ≥ jx. (4.3.4)[30, Proposition 1.5] With (4.3.4), [30, Proposition 1.5] as it pertains to h = ` (omitting allreference to h = h) continues to hold beyond the mass scale by the same proof. In particular,with the smaller choice for the domain of q, [30, (3.14)] holds with the larger s-dependent `σ,j .Note that we do not need to change the domain of λ. This is because the bound [30, (3.13)]continues to hold with the new norm parameters. Indeed, while `j and `σ,j have been modified,their product `j`σ,j has not. This guarantees that the T0 seminorm ‖σϕ¯a‖T0 = `σ` remainsidentical to what it was with the old norm parameters, and therefore there is no new stabilityrequirement arising from this.The choice (4.3.4) places a more stringent requirement on the domain than does the s = 0version. To see that this requirement is actually met by the renormalization group flow, wenote a minor improvement to the proof of [31, Lemma 6.2(ii)], where the bound |δq| ≤ cL−2jis used to show that v(X) (defined there) satisfies‖v(X)‖ ≤ cL−2j(`oldσ,j )2 ≤ c′. (4.3.5)Here the factor L−2j arises as a bound on the covariance Cj+1;00 in the perturbative flow [30,(3.35)] of q and it can therefore be improved to L−2j−2s(j−jm)+ by Lemma 4.2.1. Thus also with744.3. Proof of Theorem 2.7.1`old, `oldσ replaced by `, `σ, the required bound ‖v(X)‖ ≤ c′ remains valid.4.3.3 Extension of stability analysisIn this and the next section, we verify that the results of [30, Section 2] remain valid with`old replaced by `. In this section, we deal with the results whose proofs need only minormodification.First, we note that the supporting results of [30, Section 4] hold with the new norms. Indeed,it is immediate from (4.3.3) that analogues of [30, Proposition 4.1] and [30, Lemmas 3.4, 4.11–4.12] hold with the new `j . Moreover, [30, Lemma 4.7] and [30, Proposition 4.10] hold forgeneral values of the parameters hj (which are implicit in the T0,j-norm). We discuss [30,Proposition 4.9] in Section 4.3.4 below, and the remaining results of [30, Section 4] do not makeuse of norms.[30, Proposition 2.1] With h = `, [30, (2.1)] continues to hold with the same proof; in factthe proof does not depend on the explicit choice of h. We do not need [30, (2.2)] as it is onlyapplied with h = h.[30, Proposition 2.2] The only change to the proof is for the case j∗ = j + 1. To get [30,(2.9)], we proceed as previously in the case h = h but applying Lemma 4.2.2 rather than [30,Lemma 1.2] following [30, (5.22)]. In the same way, we get [30, (2.10)] and the remaining partsof the proposition follow without changes to the proof.[30, Proposition 2.3] Again the only required change in the proof is the use of Lemma 4.2.2in the case j∗ = j + 1, for which as previously we use Lemma 4.2.2 instead of [30, Lemma 1.2].[30, Proposition 2.4] No changes need to be made to the proof. In fact, it is necessary notto use the h = ` case of the estimate [30, (5.32)]. Instead, the h = `old case of this estimateshould be used for gQ. This is possible since the renormalization group map, and in particularthe coupling constants, are independent of the choice of norm.[30, Proposition 2.5] Using (4.3.3), we see that the proof continues to hold above the massscale. The only change to the proof is that in the application of [30, Proposition 2.2], j shouldbe replaced by j + 1 in [30, (2.9)] with j∗ = j + 1 (corresponding to the Gj+1 norm). Thisyields [30, (6.6)] with a Gj+1 norm on the left-hand side.[30, Proposition 2.6] A version of [30, Lemma 6.1] with the new ` continues to hold. Thislemma makes use of ˆ`, which superficially depends on the choice of ` in its definition [30, (3.17)].However, brief scrutiny of [30, (3.17)] reveals that the apparent dependence on ` actually cancelsand there is in fact no dependence. Similarly, [30, Lemma 3.4] continues to hold without anychanges to its proof. The proof of [30, Proposition 2.6] then applies without change.754.3. Proof of Theorem 2.7.1[30, Proposition 2.7] With the new choice of ` (and G = G), [30, Lemma 7.1] continues tohold with no changes to its proof. Thus, by [30, (3.6)] and [30, Lemma 7.1],‖Ej+1δIXθF (Y )‖Tϕ,j+1(`j+1)≤ ‖Ej+1δIXθF (Y )‖Tϕ,j(`j)≤ α|X|j+|Y |jE (CδV ¯)|X|j‖F (Y )‖Gj(`j)Gj(X ∪ Y, ϕ)5. (4.3.6)By Lemma 4.2.2, Gj(X ∪Y, ϕ)5 ≤ Gj+1(X ∪Y, ϕ). Now we divide both sides by Gj+1(X ∪Y, ϕ)and take the supremum over ϕ to complete the proof.4.3.4 Extension of the crucial contractionThe proof of the “crucial contraction” [30, Proposition 2.8] makes use of the third estimate in(4.3.1), which is now violated above the mass scale due to our new choice of `j . On the otherhand, the second estimate of (4.3.1) is improved by the new choice and compensates for thedegraded third estimate, as we explain in this section.The operator LocThere is a certain kind of duality between the space Φ of test functions and the space N offield functionals induced by the pairing (2.4.9). By exploiting this, in [30] the operator Locis defined as a kind of adjoint to an operator Taya, which replaces test functions by a latticeTaylor expansion at a ∈ Λ. Non-constant polynomials are not well-defined on the whole torusΛ, but such a Taylor expansion can nevertheless be defined for test functions supported onsequences whose components lie in a sufficiently “small” subset of Λ. These are referred toin [30] as coordinate patches. By definition, they are nonempty and any small set is containedin a coordinate patch. (In this chapter, we are ultimately only concerned with the case of smallsets).Suppose we fix a coordinate patch Λ′ ⊂ Λ. By definition, it can be identified with a rectanglein Zd. Then given a local monomial M of the form (2.5.1), we define pM ∈ Φ bypM (x1, . . . , xp) = xα11 . . . xαpp , x1, . . . , xp ∈ Λ′ (4.3.7)and set pM (~x) = 0 if |~x| 6= p or if the lattice points in ~x do not all lie in Λ′. Following (2.5.4),we define the dimension of such a monomial to be the dimension of M , i.e. |α|+ p[ϕ]. We letd+ ≥ 0 and let Π = Π[Λ′] denote the span of the monomials of this form with dimension atmost d+. For X ⊂ Λ′, we can also define Π(X) = Π[Λ′](X) as the space of test functions thatagree with an element of Π on X. Thus, Π(X) ⊃ Π.For a ∈ Λ′, in [29] the operator Taya is defined as a map Taya : Φ→ Π by a lattice analogueof Taylor expansion. Although the monomials pM form an obvious basis with respect to which764.3. Proof of Theorem 2.7.1this expansion can be performed, a different basis1 is used in [29]. The operator Tayag satisfieslattice analogues of the usual properties of Taylor polynomials. We will not discuss this furtheras we will only use the fact that Tayag ∈ Π here.The following, which is a restatement of [30, Proposition 1.5] specialized to small sets,defines LocX F as the unique element of V(X) that agrees with F to order d+ in the sense ofthe pairing.Proposition 4.3.1. Let X ∈ S be a small set and let F ∈ N (X). Then there is a uniquepolynomial V ∈ V such that〈F, g〉0 = 〈V (X), g〉0 (4.3.8)for all g ∈ Π. Moreover, V is independent of the choice of coordinate patch used to define Π.We write LocX F = V (X).Define the seminorm‖g‖Φ˜(X) = inf{‖g − f‖Φ : f ∈ Π(X)} (4.3.9)on Φ (this is used in the definition of the large-field regulator (4.2.10)). We will need thefollowing lemma, which is a restatement of [29, Lemma 2.6].Lemma 4.3.2. Let X ∈ S be a small set and let g ∈ Φ. There exists f ∈ Π(X) such that, withh = g − f , we have ‖g‖Φ˜(X) ≤ ‖h‖Φ ≤ (1 + )‖g‖Φ˜(X) and ‖f‖Φ ≤ (2 + )‖g‖Φ.Proof of the crucial contractionBelow the mass scale, we continue to use the crucial contraction as stated in [30, Proposition2.8] in terms of two norm pairs. Next, we state a version of the crucial contraction for use abovethe mass scale using the new norm pair (4.2.16). Throughout this section, we sometimes writethe dimension as d for emphasis, although we only consider d = 4. We defineI˜pt(B) = e−Vpt(B)(1 +Wj+1(Vpt, B)), B ∈ Bj (4.3.10)and extend this to I˜pt(X) = I˜Xpt for X ∈ Pj+1 by block-factorization.Proposition 4.3.3 (Improvement of [30, Proposition 2.8]). Let jm ≤ j < N and V ∈ Dj. LetX ∈ Sj and let U be the polymer closure of X. Let F (X) ∈ N (X) be such that piαF (X) = 0when α /∈ X (α = 0, x) and such that pi0xF (X) = 0 unless j ≥ jx. There is a constant C(independent of L) such that‖I˜U\Xpt ECj+1θF (X)‖Gj+1(`j+1) ≤ C((L−d−1 + L−11X∩{0,x}6=∅)κF + κLocF), (4.3.11)with κF = ‖F (X)‖Gj(`j) and κLocF = ‖I˜XptLocX I˜−Xpt F (X)‖Gj(`j).1This basis is similar to the Newton polynomial basis used in the calculus of finite differences.774.3. Proof of Theorem 2.7.1An ingredient in the proof of Proposition 4.3.3 is [29, Lemma 3.6], which is the s = 0version of the following lemma. The proof of Lemma 3.6 with s = 0 is based on the assumption`j+1/`j ≤ cL−1 (we take [ϕi] = 1; the parameters `σ,j are not used). For our new values of `,the stronger assumption `j+1/`j ≤ L−1−s1j≥jm holds. The unique change to the proof occursin the transition from [29, (3.42)] to [29, (3.43)], where the ratio `j+1/`j is used.In the following, we let Φ = Φj(hj) and Φ′ = Φj+1(hj+1). We employ similar conventionsfor Φ(X) and Φ˜(X). The constant d′+ is defined in [29, (1.38)] and in this context becomesd′+ = d+ + 1. The enlargement X+ of a polymer X ∈ Pj is defined by replacing each blockB ∈ Bj(X) by a cube of twice the side length of B (minus 1 if Lj is odd) that is centred at B.Lemma 4.3.4 (Improvement of [29, Lemma 3.6]). With the same hypotheses and notation asin [29, Lemma 3.6],‖g‖Φ˜(X) ≤ C¯3L−(1+s1j≥jm )d′+‖g‖Φ˜′(X+). (4.3.12)Proof. Assume without loss of generality that X is connected. Let f ∈ Π(X) and h ∈ Φ be asin Lemma 4.3.2. Thus, g = f + h and so we have g − (h−Tayah) = f + Tayah ∈ Π(X), wherea is the largest point which is lexicographically no larger than any point in X. By definition ofthe Φ˜(X) seminorm,‖g‖Φ˜(X) = ‖h− Tayah‖Φ˜(X) ≤ ‖h− Tayah‖Φ(X). (4.3.13)By the bound on h (from Lemma 4.3.2), it suffices to show that‖h− Tayah‖Φ(X) ≤12C¯3L−(1+s1j≥jm )d′+‖g‖Φ˜′(X+). (4.3.14)To this end, let r = h− Tayah. By [29, Lemma 3.3] with t = 1/2, there exists K > 1 suchthat‖r‖Φ(X) ≤ sup~x∈X+(K`−1j )~x sup|β|∞≤pΦLj|β|1 |∇βr~x| (4.3.15)where A~x = A|~x| and X+ is the set of sequences whose components lie in X+. In other words,we can estimate the Φ(X) norm of r in terms of the values of r and its derivatives in theenlargement X+ of X. With the new ratio (4.3.2), we can rewrite this as‖r‖Φ(X) ≤ sup~x∈X+(K`−1j+1)~x sup|β|∞≤pΦL−(|~x|+|~x|s1j≥jm+|β|1)L(j+1)|β|1 |∇βr~x|, (4.3.16)replacing [29, (3.43)].By definition, for the empty sequence ∅, (Tayah)∅ = h∅, and thus r∅ = 0. It follows thatwe can take |~x| ≥ 1 in the supremum over ~x ∈ X+ in (4.3.16). Thus,‖r‖Φ(X) ≤ L−s1j≥jm sup~x∈X+(K`−1j+1)~x sup|β|∞≤pΦL−(|~x|+|β|1)L(j+1)|β|1 |∇βr~x|. (4.3.17)784.3. Proof of Theorem 2.7.1The quantitysup~x∈X+(K`−1j+1)~x sup|β|∞≤pΦL−(|~x|+|β|1)L(j+1)|β|1 |∇βr~x| (4.3.18)is identical to the right-hand side of [29, (3.43)] when [ϕi] = 1 and is bounded in the same way.Namely, it is shown in [29] that this quantity can be bounded by a constant timesL−d′+‖h‖Φ′(X+), (4.3.19)which completes the proof.Roughly speaking, the L-dependent factor in (4.3.12) implements the dimensional gain forirrelevant directions in a renormalization group step when passing from one scale to the next.In other words, we may regard the dimension of the field as improving from 1 below the massscale to 1 + s above the mass scale. The s = 0 version of Lemma 4.3.4 is adapted to thescaling at the critical point, where m2 = 0. In the noncritical case m2 > 0, the dimensionalgain improves greatly for j > jm as apparent from (2.3.4), and is captured more accuratelyby the general-s version of (4.3.12). As a consequence of the former improvement we have thefollowing two further improvements.[29, Proposition 1.19] The improvement in Lemma 4.3.4 propagates to [29, Proposi-tion 1.19], which now holds as stated except with γα,β improved toγα,β =(L−(d′α+s1j≥jm ) + L−(A+1))(`σ,j+1`σ,j)|α∪β|. (4.3.20)The right-hand side can be estimated as follows. By (4.3.2),`σ,j+1`σ,j≤ 4L1+s1j≥jm j < jx1 j ≥ jx, (4.3.21)and henceγα,β ≤ C ′′(L−(d′α+s1j≥jm ) + L−(A+1))×L(1+s1j≥jm )(|α∪β|) j < jx1 j ≥ jx. (4.3.22)[30, Proposition 4.9] As we explain next, using (4.3.20) and identical notation to thatdefined in and around [30, Proposition 4.9], the proposition holds as stated also for the improvednorms, provided we take A ≥ 5 + s. For this, what is required is to show that under the794.3. Proof of Theorem 2.7.1hypotheses of [30, Proposition 4.9], the γα,β that arise in its proof obeyγα,β ≤ CL−5 |α ∪ β| = 0L−1 |α ∪ β| = 1, 2. (4.3.23)For |α∪β| = 0, the first term of (4.3.22) obeys the bound of (4.3.23), since d′∅ = d+ 1. For theremaining cases, d′α = 2 for j < jx and d′α = 1 for j ≥ jx. For |α ∪ β| = 2, the assumption thatF1, F2, F1F2 have no component in N0x unless j ≥ jx means that we are in the case with nogrowth due the ratio `σ,j+1/`σ,j in (4.3.22), and its first term again obeys the bound (4.3.23)with room to spare. Finally, when |α∪β| = 1, the first term of (4.3.22) also obeys the estimate(4.3.23), and again with room to spare. Concerning the second term of (4.3.22), given our choiceof A and the fact that we need only consider the growing factor in (4.3.22) for |α ∪ β| = 1, itsuffices to observe thatL−(A+1)L1+s1j≥jm ≤ L−5. (4.3.24)This completes the proof of the improved version of [30, Proposition 4.9].Proof of Proposition 4.3.3. We complete the proof of Proposition 4.3.3 by modifying the proofof [30, Proposition 2.8] above the mass scale. The estimate [30, (7.22)] follows from [30, Propo-sition 2.7] as an estimate in terms of the modified norm pair (4.2.16), for which [30, Proposi-tion 2.7] was verified in Section 4.3.3. The bound [30, (7.25)] with improved γ is obtainedby applying the improved version of [30, Proposition 4.9]. In the remainder of the proofof [30, Proposition 2.8], we specialize each occurrence of G to the case G = G and we con-clude by obtaining an analogue of [30, (7.31)] with G˜ replaced by G by applying Lemma 4.2.2rather than [30, Lemma 1.2].An additional detail is that it is required that we choose the parameter defining the spaceN to obey pN > A. Since we have changed A (depending on s), we must make a correspondingchange to pN . This does not pose problems (beyond the previously discussed requirement thatg needs to be chosen small depending on p), as this parameter may be fixed to be an arbitraryand sufficiently large integer (see [108, Section 7.1.3] where this point is addressed in a differentcontext). Similarly, the value of A is immaterial and can be any fixed number in the proofof [30, Proposition 2.8].80Chapter 5Critical initial conditionsIn this chapter, we prove Theorem 2.8.1. We begin in Section 5.1.1 by showing that K+0 satisfiesthe inductive assumption required by Theorem 2.7.1. In Section 5.2, we discuss a general versionof this theorem for a parameter K0 that is independent of the coupling constants. This theoremis then applied with K0 = K+0 by solving a set of implicit equations in Section 5.3.Throughout this chapter, we take n ≥ 1, drop n from the notation, and denote fields byϕ. The n = 0 case is dealt with in [13]. We also deal only with the bulk flow (so we setσ0 = σx = 0). The construction of the observable flow follows as in [108] (with the same criticalinitial conditions) once the bulk flow has been constructed.5.1 Initial coordinates for the renormalization groupWe establish norm estimates on K+0 in Sections 5.1.1–5.1.3. The initial coordinate K+0 dependson the coupling constants (g0, γ0, ν0, z0) of (2.2.1) and regularity of K0 as a function of thesevariables is shown in Section 5.1.4.5.1.1 Properties of the Tϕ seminormWe will need several properties of the Tϕ seminorm, whose proofs can be found in [28]. Wehave already mentioned the product property in Proposition 2.4.3. An immediate consequenceis that ‖e−F ‖Tϕ ≤ e‖F‖Tϕ . This is improved in [28, Proposition 3.8], which states that‖e−F ‖Tϕ ≤ e−2F (ϕ)+‖F‖Tϕ . (5.1.1)We will also use [28, Proposition 3.10], which states that if F ∈ N is a polynomial in ϕ oftotal degree A ≤ pN , then‖F‖Tϕ ≤ ‖F‖T0(1 + ‖ϕ‖Φ)A. (5.1.2)Let x denote the small set neighbourhood of a singleton {x} and recall that the Φx ≡ Φ(x)norm of ϕ ∈ (Rn)Λ was defined in (4.2.8). By taking the infimum in (5.1.2) over all possiblere-definitions of ϕy for y /∈ x, we get‖F‖Tϕ ≤ ‖F‖T0(1 + ‖ϕ‖Φx)A (5.1.3)when F ∈ N (x).815.1. Initial coordinates for the renormalization group5.1.2 Bounds on K0The main estimate on K+0,x is given by the following proposition. Recall that Dj was defined in(2.7.4).Proposition 5.1.1. Suppose that V +0 ∈ D0, with g˜0 sufficiently small. If 0 ≤ γ0 ≤ g˜0, then(with constants that may depend on L)‖K+0,x‖G0 = O(γ0), ‖K+0,x‖G˜0 = O(γ0/g0). (5.1.4)The form of the estimates (5.1.4) can be anticipated from the definition of K+0 . The upperbound arises from the small size of e−γ0U+x − 1. For small fields, hence small U+x , this is oforder γ0, as reflected by the G0 norm estimate of (5.1.4). For large fields, namely fields of size|ϕ| ≈ g˜−1/40 , the difference e−γ0U+x − 1 is roughly of size γ0 |ϕ|4 ≈ γ0/g˜0. This effect is measuredby the G˜0 norm.Before proving the proposition, we writeK+0,x = I+0,xJ+x (5.1.5)where, by the fundamental theorem of calculus,I+0,x = e−V +0,x (5.1.6)J+x = e−γ0U+x − 1 = −∫ 10γ0U+x e−tγ0U+x dt. (5.1.7)Let F ∈ N (x) be a polynomial of degree at most pN . Then the stability estimates [30,(2.1)–(2.2)] imply that there exists c3 > 0 and, for any c1 ≥ 0, there exist positive constantsC, c2 such that if V+0 ∈ D0 then‖I+0,xF‖Tϕ(h0) ≤ C‖F‖T0(h0)ec3g0(1+‖ϕ‖2Φx(`0))h0 = `0e−c1k40‖ϕ‖2Φx(h0)ec2k40‖ϕ‖2Φ˜x(`0) h0 = h0.(5.1.8)This essentially reduces our task to estimating J+x . The next lemma is an ingredient for this.Lemma 5.1.2. There is a universal constant C˜ such that‖U+x ‖Tϕ(h0) ≤ 2U+x + C˜h40(1 + ‖ϕ‖2Φx(h0)). (5.1.9)Proof. LetM+ = M+e = (∇eτx)2 (5.1.10)so that U+x =∑e∈UM+e . It suffices to prove (5.1.9) with U+x replaced by M+ on both sides.In addition, we can replace the Φx norm by the Φ norm; the bound with the Φx norm thenfollows in the same way that (5.1.3) is a consequence of (5.1.2), since M+ ∈ N (x).825.1. Initial coordinates for the renormalization groupBy definition of the Tϕ seminorm,‖∇e|ϕx|2‖Tϕ ≤ ∇e|ϕx|2 + 2h0(|ϕx|+ |ϕx+e|) + 2h20. (5.1.11)With the product property and (2.4.6), this implies that‖M+‖Tϕ ≤M+ + 2|∇e|ϕx|2|(2h0(|ϕx|+ |ϕx+e|)) +O(h40)(1 + ‖ϕ‖2Φ). (5.1.12)By the inequality2|ab| ≤ |a|2 + |b|2 (5.1.13)and another application of (2.4.6),2|∇e|ϕx|2|(2h0(|ϕx|+ |ϕx+e|)) ≤M+ +O(h20‖ϕ‖2Φ), (5.1.14)and the bound on M+ follows.An immediate consequence of Lemma 5.1.2, using (5.1.1), is that for any s ≥ 0,‖e−sU+x ‖Tϕ(h0) ≤ eC˜sh40(1+‖ϕ‖2Φx(h0)). (5.1.15)Proof of Proposition 5.1.1. According to the definition of the regulator norms in (4.2.11)–(4.2.12), it suffices to prove that, under the hypothesis on γ0,‖K+0,x‖Tϕ(h0) = O(γ0h40)e‖ϕ‖2Φx (h0 = `0)et2‖ϕ‖Φ˜ (h0 = h0).(5.1.16)For t ∈ [0, 1], let I˜+x (t) = e−tγ0U+x . By (5.1.5), (5.1.7), and the product property,‖K+0,x‖Tϕ(h0) ≤ γ0‖I+0,xU+x ‖Tϕ(h0) supt∈[0,1]‖I˜+x (t)‖Tϕ(h0). (5.1.17)By (5.1.8) and Lemma 5.1.2, there exists c3 > 0, and, for any c1 ≥ 0 there exists c2 > 0, suchthat‖I+0,xU+x ‖Tϕ(h0) ≤ O(h40)ec3g0‖ϕ‖2Φx(`0) h0 = `0e−c1k40‖ϕ‖2Φx(h0)ec2k40‖ϕ‖2Φ˜x(`0) h0 = h0.(5.1.18)The constant in O(γ0h40) may depend on c1, but this is unimportant. Also, by (5.1.15),supt∈[0,1]‖I˜+x (t)‖Tϕ(h0) ≤ eC˜γ0h40(1+‖ϕ‖2Φx(h0)). (5.1.19)835.1. Initial coordinates for the renormalization groupThus, for h0 = `0, the total exponent in our estimate for the right-hand side of (5.1.17) isC˜γ0`40 + (c3g0 + C˜γ0`40)‖ϕ‖2Φx(`0). (5.1.20)This gives the h0 = `0 version of (5.1.16) provided that g0 is small and γ0 is small dependingon L.For h0 = h0, the total exponent in our estimate for the right-hand side of (5.1.17) isC˜γ0k40 g˜−10 + (C˜γ0k40 g˜−10 − c1k40)‖ϕ‖2Φx(h0) + c2k40‖ϕ‖2Φ˜x(`0). (5.1.21)This gives the h0 = h0 version of (5.1.16) provided that γ0 ≤ g˜0, c1 ≥ C˜, and c2k40 ≤ t/2.All the provisos are satisfied if we choose c1 ≥ C˜, k0 small depending on c1 and g˜0 small.Remark 5.1.3. By a small modification to the proof of Proposition 5.1.1, it can be shown thatif Mx ∈ N (x) is a monomial of degree r ≤ pN − 4 (so that MxU+x has degree at most pN ),then‖MxK+0,x‖G0 = O(γ0h4+r0 ). (5.1.22)5.1.3 Unified bound on K0We begin by recalling the definition of the Wj norm from [31]. It follows from the productproperty of the Tϕ seminorm that the regulator norms obey the following version of the productproperty:‖F1F2‖Gj ≤ ‖F1‖Gj‖F2‖Gj for Fi ∈ N (Xi ) with X1 ∩X2 = ∅. (5.1.23)Given a map K ∈ Kj , we define the Fj(G) norms (for G = G, G˜) by‖K‖Fj(G) = supX∈Cjg˜−fj(a,X)j ‖K(X)‖Gj (5.1.24)‖K‖Fj(G˜) = supX∈Cjg˜−fj(a,X)j ‖K(X)‖G˜tj , (5.1.25)withfj(a,X) = a(|X|j − 2d)+ =a(|X|j − 2d) if |X|j > 2d0 otherwise (5.1.26)(recall that |X|j is the number of j-blocks in X). Here a is a small constant; its value isdiscussed below [31, (1.46)]. The Wj =Wj(g˜j ,V) norm is then defined by‖K‖Wj = max{‖K‖Fj(G), g˜9/4j ‖K‖Fj(G˜)}. (5.1.27)845.1. Initial coordinates for the renormalization groupProposition 5.1.4. If V +0 ∈ D0 with g˜0 sufficiently small (depending on L), and if γ0 ≤O(g˜1+a′0 ) for some a′ > a, then ‖K+0 ‖W0 ≤ O(γ0), where all constants may depend on L.Proof. Let X ∈ C0. By the product property and Proposition 5.1.1,‖K+0 (X)‖G0 ≤ (cγ0h40)|X| = (cγ0h40)|X|∧2d(cγ0h40)(|X|−2d)+ . (5.1.28)For G0 = G0, we use h0 = `0, (cγ0h40)|X|∧2d ≤ O(γ0), and(cγ0h40)(|X|−2d)+ ≤ (c′g˜0)(1+a′)(|X|−2d)+ ≤ g˜f0(a,X)0 . (5.1.29)For G0 = G˜0, we use h0 = h0 = O(g˜−1/40 ) and, since a′ > a,(cγ0h40)(|X|−2d)+ ≤ (c′g˜0)a′(|X|−2d)+ ≤ g˜f0(a,X)0 . (5.1.30)Since γ0 ≤ g˜0, it follows from (5.1.28) thatg˜9/40 ‖K+0 ‖F0(G˜) ≤ g˜9/40 O(γ0g˜−10 ) ≤ γ0, (5.1.31)and the proof is complete.5.1.4 Smoothness of K0Given any map F : D →W0(g˜0,Zd) for D ⊂ R an open interval, let us write FX : D → N (X)and FϕX : D → R for the maps defined by partial evaluation of F atX and at (X,ϕ), respectively.Lemma 5.1.5. Let D ⊂ R be open and F : D → W0(g˜0,Zd) be a map. Suppose that FϕX isC2 for all X ∈ C0 and ϕ ∈ (Rn)Λ, and define F (i) : D →W0(g˜0,Zd) by (F (i)(t))ϕX = (FϕX)(i)(t)for i = 1, 2, where the right-hand side denotes the ith derivative of FϕX . If ‖F (i)(t)‖W0 <∞ fori = 1, 2 and t ∈ D, then F (1) is the (Fre´chet) derivative of F .Proof. For t, t+ s ∈ D, define R(t, s) ∈ W0 byRϕX(t, s) = FϕX(t+ s)− FϕX(t)− s(FϕX)′(t). (5.1.32)By Taylor’s theorem, for any ϕ and X,RϕX(t, s) = s2∫ 10(FϕX)′′(t+ us)(1− u) du. (5.1.33)It follows that‖R(t, s)‖W0 ≤ |s|2 supu∈[0,1]‖F ′′(t+ us)‖W0 ≤ O(|s|2), (5.1.34)855.1. Initial coordinates for the renormalization groupso F is differentiable and its derivative satisfies (F ′)ϕX = (FϕX)′. Continuity of F ′ followssimilarly, since, by the fundamental theorem of calculus,‖F ′(t+ s)− F ′(t)‖W0 ≤ |s| supu∈[t,t+s]‖F ′′(u)‖W0 ≤ O(|s|), (5.1.35)which suffices.Let us view K+0 as a map(g0, γ0, ν0, z0) 7→ K+0 ∈ W0(g˜0,Zd). (5.1.36)for (g0, γ0, ν0, z0) satisfying the hypotheses of Proposition 5.1.4. The map K0 is in fact analyticaway from γ0 = 0. However, we only prove the following, which is what we need later.Proposition 5.1.6. Suppose that V +0 ∈ D0, with g˜0 sufficiently small (depending on L) andγ0 ≤ O(g˜1+a′0 ) for some a′ > a. Then the map K+0 (g0, γ0, ν0, z0) is jointly continuous in its fourvariables, is C1 in (g0, ν0, z0), and (when γ0 6= 0) is C1 in (g0, γ0, ν0, z0), with partial derivativeswith respect to t = g0, ν0, and z0 satisfying‖∂K+0 /∂t‖W0 = O(γ0). (5.1.37)Moreover, K+0 is right-differentiable in γ0 at γ0 = 0.Proof. Let t denote any one of the coupling constants g0, γ0, ν0 or z0. We drop the subscript0 and superscript +, and let K(t) denote K+0 viewed as a function of t, with the remainingcoupling constants fixed. Then KϕX is smooth for any ϕ,X. If t is g0, ν0 or z0, then(Kϕx )′ = −Mx(ϕ)Kϕx , (Kϕx )′′ = M2x(ϕ)Kϕx , (5.1.38)where Mx is τ2x , τx or τ∆,x, respectively. The maximal degree of Mx is 4, so (5.1.22) impliesthat‖K ′x‖G0 ≤ O(γ0h80), ‖K ′′x‖G0 ≤ O(γ0h120 ). (5.1.39)For t denoting γ0, we write U = U+ and V0 = V+0 . Then(Kϕx )′ = −Ux(ϕ)e−Vx(ϕ)−γ0Ux(ϕ), (Kϕx )′′ = U2x(ϕ)e−Vx(ϕ)−γ0Ux(ϕ), (5.1.40)and (5.1.8) and (5.1.15) imply that‖K ′x‖G0 ≤ O(h40), ‖K ′′x‖G0 ≤ O(h80). (5.1.41)By definition, KX =∏x∈X Kx so, for derivatives with respect to any one of the four variables865.1. Initial coordinates for the renormalization group(with γ0 6= 0 when differentiating with respect to γ0),(KϕX)′ =∑x∈X(Kϕx )′KϕX\x, (KϕX)′′ =∑x∈X((Kϕx )′′KϕX\x + (Kϕx )′(KϕX\x)′). (5.1.42)Thus, by the product property, (5.1.39), and Proposition 5.1.1,‖K ′X‖G0 ≤ O(|X|)γ0h80(γ0h40)|X|−1. (5.1.43)when differentiating with respect to g0, ν0, or z0. The bound (5.1.37) then follows from thehypothesis on γ0. Similarly, using (5.1.41),‖K ′X‖G0 ≤ O(|X|)h40(γ0h40)|X|−1 (5.1.44)when differentiating with respect to γ0 away from γ0 = 0. In both cases, we have‖K ′′X‖G0 ≤ O(|X|2)h80(γ0h40)(|X|−2)∧0. (5.1.45)Thus, by Lemma 5.1.5, K is C1 in any of its variables. Therefore, K is C1 in (g0, ν0, z0) on thewhole domain and in all the variables when γ0 6= 0.To show right-continuity in γ0 at γ0 = 0, fix (g0, ν0, z0) and define F ∈ W0 byF (X) =−Uxe−V0,x X = {x}0 |X| > 1, (5.1.46)where Ux, V0,x are defined above. Let K′(γ0) denote the γ0 derivative of K evaluated at γ0 > 0.Then (5.1.40) and (5.1.42) imply thatF (X)−K ′X(γ0) =UxKx(γ0) X = {x}∑x∈X K′x(γ0)KX\x(γ0) |X| > 1.(5.1.47)Thus, by (5.1.22), (5.1.41), and Proposition 5.1.1,‖F (X)−K ′X(γ0)‖G0 ≤O(γ0h80) X = {x}O(|X|)h40(γ0h40)|X|−1 |X| > 1. (5.1.48)It follows thatlimγ0↓0‖F −K ′(γ0)‖W0 = 0, (5.1.49)i.e., F is the right-derivative of K in γ0 at γ0 = 0.875.2. Renormalization group flowRemark 5.1.7. With γ0 sufficiently small, the bound (5.1.37) verifies the condition‖∂K+0 /∂ν0‖W0 ≤ O(g30) (5.1.50)required in the proof of [9, Lemma 8.6] (see [9, (8.34)]), which is in turn needed in Section 3.1.2.5.2 Renormalization group flowThe following theorem is an extension of [9, Proposition 7.1] to non-trivial K0. This extensionis possible with only minor modifications to the proof of the K0 = 1∅ case, due to the generalityallowed by the main result of [11].The theorem provides, for any N ≥ 1 and for initial error coordinate K0 in a specified do-main, a choice of initial condition (νc0, zc0) for which there exists a finite-volume renormalizationgroup flow (Vj ,Kj) ∈ Dj for 0 ≤ j ≤ N . In order to ensure a degree of consistency amongstthe sequences (Vj ,Kj), which depend on the volume ΛN , a notion of consistency must be im-posed upon the collection of initial error coordinates K0,Λ ∈ K0(Λ) for varying Λ. Specifically,the family K0,Λ is required to satisfy the property (Zd) of [31, Definition 1.15]. We refer toany such family as a Λ-family. As discussed in [31, Definition 1.15], any Λ-family induces aninfinite-volume error coordinate K0,Zd ∈ K0(Zd) in a natural way.Remark 5.2.1. Roughly, the requirement that the (KΛ) form a Λ-family is that if Λ ⊂ Λ′,then KΛ and KΛ′ agree on polymers X ∈ Pj(Λ). However, some care must be taken with thiswhen the polymer X “wraps around” the torus. This issue is handled using coordinate patches,as was done for discussing “torus polynomials” in (4.3.7).Theorem 5.2.2. Let d = 4. There exists a constant a∗ > 0 and continuous functions νc0, zc0of (m2, g0,K0), defined for (m2, g0) ∈ [0, δ]2 (for some δ > 0 sufficiently small) and for anyK0 ∈ W0(m2, g0,Zd) with ‖K0‖W0(m2,g0,Zd) ≤ a∗g30, such that the following holds for g0 > 0: ifK0,Λ ∈ K0(Λ) is a Λ-family that induces the infinite-volume coordinate K0, and ifV0 = Vc0 (m2, g0,K0) = (g0, νc0(m2, g0,K0), zc0(m2, g0,K0)), (5.2.1)then for any N ∈ N and m2 ∈ [δL−2(N−1), δ], there exists a sequence (Vj ,Kj) ∈ Dj(m2, g0,Λ)such that(Vj+1,Kj+1) = (Vj+1(Vj ,Kj),Kj+1(Vj ,Kj)) for all j < N (5.2.2)and (2.6.5) is satisfied. Moreover, νc0, zc0 are continuously differentiable in g0 ∈ (0, δ) andK0 ∈ BW0(m2,g0,Λ)(a∗g30), andνc0(m2, 0, 0) = zc0(m2, 0, 0) = 0,∂νc0∂g0= O(1),∂zc0∂g0= O(1), (5.2.3)where the estimates above hold uniformly in m2 ∈ [0, δ].885.3. Critical parametersProof. The proof results from small modifications to the proofs of [9, Proposition 7.1] and thento [9, Proposition 8.1], where (in both cases) we relax the requirement that K0 = 1∅, which waschosen in [9] due to the fact that K0 = 1∅ when γ = 0. The more general condition that K0 ∈BW0(m2,g0,Λ)(a∗g30) comes from the hypothesis of [11, Theorem 1.4] when (m2, g0) = (m˜2, g˜0).By [11, Remark 1.5], no major changes to the proof result from this choice of K0. The followingparagraph outlines in more detail the modifications to the proof of [9, Proposition 7.1].By [11, Theorem 1.4] and [11, Corollary 1.8], for any (m˜2, g˜0) ∈ (0, δ)2 and for any K˜0 ∈BW0(m˜2,g˜0,Zd)(a∗g˜30), there is a neighbourhood N(g˜0, K˜0) of (g˜0, K˜0) such that for all (m2, g0,K0) ∈I˜(m˜2)× N(g˜0, K˜0), there is an infinite-volume renormalization group flow(Vˇj ,Kj) = xˇdj (m˜2, g˜0, K˜0;m2, g0,K0) (5.2.4)in transformed variables (Vˇj ,Kj). The transformed variables are defined in [9, Section 6.6] and aflow in the original variables can be recovered from the transformed flow. The global solution isdefined by xˇcj(m2, g0,K0) = xˇdj (m2, g0,K0;m2, g0,K0) (or xˇc ≡ 0 if g0 = 0). By [11, Remark 1.5],the proof of regularity of xˇc can proceed as in [9]. The functions (νc0, zc0) are given by the (ν0, z0)components of xˇc0 = (Vˇ0,K0) = (V0,K0).Remark 5.2.3. The proof of [9, Proposition 7.1], hence of Theorem 5.2.2, makes importantuse of the parameter g˜0 in order to prove regularity of the renormalization group flow in g0.However, once the flow has been constructed, we can and do set g˜0 = g0.We wish to apply this theorem with (g0,K0) = (g0,K+0 ). We have already remarked thatK+0 ∈ K0. It also follows from the definition that the finite-volume coordinates K+0,Λ form aΛ-family.Moreover, by Proposition 5.1.4, if γ0 is sufficiently small (depending on g0; we now takeg˜0 = g0) then K0 = K+0 satisfies the bound required by Theorem 5.2.2. However, we cannotapply the theorem immediately with this choice of K0, due to the fact that K+0 depends on(g0, ν0, z0). We resolve this issue in the next section.5.3 Critical parametersWe wish to initialize the renormalization group with (ν0, z0) a solution to the system of equationsν0 = νc0(m2, g0,K+0 (g0, γ0, ν0, z0)), (5.3.1)z0 = zc0(m2, g0,K+0 (g0, γ0, ν0, z0)). (5.3.2)Such a choice of (ν0, z0) will be critical for K+0 , where K+0 is itself evaluated at this same choiceof (ν0, z0).When γ0 = 0, we get K+0 = 1∅, so K+0 no longer depends on (ν0, z0) and this systemis solved by (νc0(m2, g0, 0), zc0(m2, g0, 0)) for any (small) m2, g0 ≥ 0. Local solutions for γ0 6=895.3. Critical parameters0 can then be constructed using a version of the implicit function theorem from [89] thatallows for the continuous but non-smooth behaviour of K+0 in γ0. In order to obtain globalsolutions with certain desired regularity properties (needed in the next section), we make useof Proposition 3.1.3, which is based on the implicit function theorem from [89].Recall that D(δ, r) was defined in (2.8.1) byD(δ, r) = {(w, x, y) ∈ [0, δ]3 : y ≤ r(x)}. (5.3.3)Proposition 5.3.1. There exists a continuous positive-definite function rˆ : [0, δ]→ [0,∞) andcontinuous functions νˆc0, zˆc0 ∈ C0,1,+(D(δ, rˆ)) such that the system (5.3.1)–(5.3.2) is solved by(ν0, z0) = (νˆc0, zˆc0) whenever (m2, g0, γ0) ∈ D(δ, rˆ). Moreover, these functions satisfy the boundsνˆc0 = O(g0), zˆc0 = O(g0) (5.3.4)uniformly in (m2, γ0).Proof. LetF (m2, g0, γ0, ν0, z0) = (ν0, z0)− (νc0(m2, g0,K0), zc0(m2, g0,K0)), (5.3.5)where K0 = K+0 (g0, γ0, ν0, z0). Then for δ > 0 small and an appropriate constant c > 0(depending on a∗), F is well-defined on{(m2, g0, γ0, ν0, z0) : (m2, g0, γ0) ∈ D(δ, cg30), |ν0|, |z0| ≤ CDg0}. (5.3.6)Indeed, for (m2, g0, γ0, ν0, z0) in this domain, Proposition 5.1.4 (with g˜0 = g0) implies that(m2, g0,K0) is in the domain of (νc0, zc0). By Theorem 5.2.2 and Proposition 5.1.6, F is C1 in(g0, ν0, z0) and also in γ0 away from γ0 = 0, continuous in m2, and is right-differentiable in γ0at γ0 = 0.For fixed (m¯2, g¯0) ∈ [0, δ]2, set (ν¯0, z¯0) = (νc0(m¯2, g¯0, 0), zc0(m¯2, g¯0, 0)) so thatF (m¯2, g¯0, 0, ν¯0, z¯0) = (0, 0). (5.3.7)By (5.1.37), at (g¯0, 0, ν¯0, z¯0),∂K0,x∂ν0=∂K0,x∂z0= 0. (5.3.8)It follows that Dν0,z0F (m¯2, g¯0, 0, ν¯0, z¯0) is the identity map on R2. The existence of δ, rˆ and νˆc0, zˆc0follows from Proposition 3.1.3 with w = m2, x = g0, y = γ0, z = (ν0, z0), and with r1(g0) = cg30,r2(g0) = CDg0.By the fundamental theorem of calculus, for any 0 < a < γ0,νˆc0(m2, g0, γ0) = νˆc0(m2, g0, a) +∫ γ0a∂νˆc0∂γ0(m2, g0, t) dt. (5.3.9)905.3. Critical parametersTaking the limit a ↓ 0 and using (5.2.3), we obtain|νˆc0(m2, g0, γ0)| ≤ O(g0) + γ0 supt∈(0,γ0]∣∣∣∣∂νˆc0∂γ0 (m2, g0, t)∣∣∣∣ . (5.3.10)The supremum above is bounded by a constant and so the first estimate of (5.3.4) follows fromthe fact that |γ0| ≤ rˆ(g0) (since rˆ(g0) can be taken as small as desired).Proof of Theorem 2.8.1. By Proposition 5.1.4, and by taking rˆ smaller if necessary, K0 = K+0satisfies the estimate required by Theorem 5.2.2 whenever (m2, g0, γ0) ∈ D(δ, rˆ). The existenceof the sequence (2.8.2) then follows from Theorem 5.2.2 and Proposition 5.3.1.91Chapter 6ConclusionWe end with a discussion of some open problems that may be accessible by extensions to therenormalization group method discussed in this thesis. We will try to point out some of themain obstacles that must be overcome.6.1 Other modelsIn order to apply the renormalization group to the models we have considered, we had toexpress them as perturbations of a Gaussian measure whose covariance admits an appropriatefinite-range decomposition. Here we discuss other models that can be written in this way.6.1.1 Long-range modelsIn [117], Wilson and Fisher suggested studying models in dc− dimensions with > 0 small anddc = 4 the upper-critical dimension. Building on this, approximate values for 3-dimensionalcritical exponents were computed in [45,64,85]. One approach to the rigorous implementation ofthis idea involves studying models in dimension d (an integer) whose upper-critical dimensionis dc + . This is not as problematic as considering models in fractional dimensions, as theupper-critical dimension dc need not be the actual dimension of some ambient space.Given a massless covariance C ′, the upper-critical dimension is simply a number dc suchthat some class of models scales like a Gaussian model with covariance C ′ if and only if d > dc.For instance, suppose we choose C ′ to decay likeC ′0x |x|−(d−α) (6.1.1)with α = 12(d+ ). Such a choice is given byC ′ = (−∆)−α/2 (6.1.2)for α ∈ (0, 2). Then recalling Remark 1.6.3, we might expect that dc = 2α. In particular, ifα = 12(d+ ) with d ≤ 3, then d = dc − .This approach has been used to implement the renormalization group below the upper-critical dimensions in [1, 19, 27, 94]. Recently, Slade [107] has extended the approach discussedin this thesis to compute anomalous (non-Gaussian) critical exponents for long-range versionsof the weakly self-avoiding walk and the |ϕ|4 model. In particular, he showed that, as ν ↓ νc926.1. Other modelsfor these models, the susceptibility χ satisfiesχ (ν − νc)−(1+n+2n+8α+O(2)). (6.1.3)By extensions of [107] to use observable fields, we think it should be possible to identify thescaling behaviour of the two-point function and possibly other correlation functions for theselong-range models. In particular, this would make it possible to confirm the intriguing predictionof [51], which states thatη = 2− α (6.1.4)if d = dc − for small . In other words, unlike the susceptibility, deviations from mean-fieldbehaviour of the two-point function cannot be detected to any order in .Remark 6.1.1. Models at and above the upper-critical dimension exhibit asymptotic freedom.In our context, this means that ‖Kj‖Wj → 0, νj , zj → 0, and gj → 0 in the massless regimem2 = 0. Below dc, we do not have asymptotic freedom, as reflected by the lack of exactasymptotics in (6.1.3). In some ways, this is advantageous (see [107]), but in others it createsadditional difficulties that must be overcome.6.1.2 The O(n) model and self-avoiding walkRecall that the Hamiltonian of the O(n) model was defined in (1.4.15). On Λ, it takes the formHJ(σ) = −12σ · Jσ, σ ∈ Sn−1. (6.1.5)This was derived from the |ϕ|4 model by taking a suitable g → ∞ limit. The restriction tosmall coupling g is deeply embedded into the method we use, but the Kac-Siegert transformation(see [17]) offers an alternative approach to the study of this model.Namely, let Ω = (Sn−1)Λ and let dσ denote the product measure on Ω, where Sn−1 isequipped with the uniform measure. The partition function of the O(n) model is given byZJ =∫Ωe−HJ (σ) dσ. (6.1.6)When J is a positive-definite symmetric matrix, the Gaussian measure dµJ(ϕ) with covarianceJ is well-defined and satisfies the elementary identitye−HJ (σ) = e12σ·Jσ =∫(Rn)Λeσ·ϕ dµJ(ϕ). (6.1.7)Interchanging the order of integration, we can writeZJ =∫(Rn)Λe−∑x∈Λ L(ϕx) dµJ(ϕ), (6.1.8)936.2. Other observable quantitieswhereL(t) = − log∫Sn−1eσ0·t dσ0, t ∈ Rn (6.1.9)is the negative logarithm of the Laplace transform of the sphere. Since L is a rotation- andreflection-invariant analytic function and L(0) = 0, it has the formL(t) = ν|t|2 + g|t|4 +∞∑k=3c2k|t|2k. (6.1.10)Letting J = (−∆ + γ2)−1, we havedµJ(ϕ) ∝ e− 12 (γ2|ϕ|2+ϕ·(−∆ϕ)). (6.1.11)Thus, we can express the partition function as a perturbed |ϕ|4 model.By a procedure as in Section 2.2, the analysis of this model can be reformulated in terms ofthe evolution of an effective interaction Zj with initial condition Z0 = (I0 ◦K0)(Λ). Once again,the initial error coordinate K0 will be coupled to I0, but we expect that the critical parametersνc0, zc0 can be identified by an implicit function theorem argument as in Section 5.3.However, estimates on K0 (which are straightforward to obtain by a more careful computa-tion of (6.1.10)) indicate that K0 is not of order g30, which is required to invoke Theorem 5.2.2.In fact, K0 = O(g3/20 ).One approach to possibly overcoming this issue is the following: First, note that the irrel-evant error coordinate should shrink by a factor of O(L−1) after each renormalization groupstep. Thus, after the first jg = blogL g−3/2c steps, we should be left with an error coordinateKj of size O(g3). A careful analysis of the renormalization group flow is required during thesefirst jg steps. However, the flow of coupling constants in this regime need only be computed tofirst order; indeed, any second-order terms would in any case be of higher order than the errorterm, which is of order g3/2.Remark 6.1.2. Similarly, it is possible to re-cast the strictly self-avoiding walk as a pertur-bation of weakly self-avoiding walk using a supersymmetric integral representation obtainedin [26]. The covariance of the form (−∆ + γ2)−1 in this case corresponds to a model of spread-out self-avoiding walk with exponentially decaying jump probabilities. Once again, the initialerror coordinate is not of order O(g30).6.2 Other observable quantitiesHere we discuss some open problems concerning the models studied in this thesis.946.2. Other observable quantities6.2.1 The correlation lengthOur results concerning the finite-order correlation lengths ξp are insufficient for recovering thepredicted behaviour of the true correlation length ξ. The estimate (3.2.2) shows that the errorsin the approximation (3.2.1) of the two-point function decay at any desired polynomial rate, butthis is not sufficient for studying ξ, which would need exponentially decaying errors. The currentestimates follow from the covariance bounds (2.3.4) on the decomposition of [5]. Although itmay not be possible to improve the bounds for this particular decomposition, this should bepossible for the decomposition of [22] (see [22, p. 445]).However, even if this were possible, exponentially decaying errors would require that theweights `j decay like e−cLj above the mass scale. This, in turns, would cause the weights `σ,jdefined by (2.4.21) to grow so quickly that the third bound in (4.3.1) would fail in a major way.Thus, it seems new ideas would be needed to study the correlation length (note, however, thatthe correlation length for the 1-component |ϕ|4 model was successfully studied by a block-spinrenormalization group method in [69]).6.2.2 Inversion of the Laplace transformOne of the main motivations for studying the susceptibility and finite-order correlation lengthfor walks is the possibility of recovering information about the growth of the partition functioncT and the mean-squared distance 〈|X(T )|2〉 as T →∞. In particular, recalling the discussionin Section 1.1, one may try to derive logarithmic corrections to the predicted scaling relations(1.6.22)–(1.6.23) as a consequence of Theorem 1.7.1(ii)–(iii).This approach was successfully used in [24], where the mean-squared displacement of a hier-archical model of weakly self-avoiding walk is recovered by inversion of the Laplace transform.This requires control over the two-point function in a sector of the complex plane larger thanwhat has been achieved here on the Euclidean lattice.6.2.3 The broken symmetry phaseThe authors of [61] studied weakly self-avoiding walk on a four-dimensional hierarchical latticein the phase ν < νc. They employed a renormalization group method similar to the oneused here in order to show that the walks exhibit a broken supersymmetry in this phase. 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We let Xn = Pn(X) be the projection of Xand note that Xn is a simple random walk on Zdn.We call h = (hx)x∈Zd a field of path functionals if hx : (Zd)[0,∞) → R is a function oncontinuous-time paths for each x ∈ Zd; a simple example is given by the local time functional.We assume that the random field h(X) = (hx(X))x∈Zd has finite support almost surely, i.e.,with probability 1, hx(X) = 0 for all but finitely many x. Denote by h(Xn) the correspondingrandom field for Xn, i.e., for x ∈ Zdn,hx(Xn) =∑y∈Zdhx+ny(X). (A.1.1)Given a positive integer k, we define Qk ⊂ Zd by Qk = {y ∈ Zd : 0 ≤ yi < k, i = 1, . . . , d}.Then, for integers n, k ≥ 1,∑y∈Qkhx+ny(Xkn) =∑y∈Qk∑z∈Zdhx+ny+knz(X) =∑y∈Zdhx+ny(X) = hx(Xn), (A.1.2)and it follows by summation over x ∈ Zdn that∑x∈Zdknhx(Xkn) =∑x∈Zdnhx(Xn). (A.1.3)Lemma A.1.1. Let n, k ≥ 1 and let f and g be nonnegative fields of path functionals withfinite support almost surely. Then∑x∈Zdknfx(Xkn)gx(Xkn) ≤∑x∈Zdnfx(Xn)gx(Xn). (A.1.4)104A.2. Convergence of the finite-volume approximationProof. By (A.1.3) and (A.1.2),∑x∈Zdknfx(Xkn)gx(Xkn) =∑x∈Zdn∑y∈Qkfx+ny(Xkn)gx+ny(Xkn). (A.1.5)By nonnegativity and two more applications of (A.1.2),∑x∈Zdn∑y∈Qkfx+ny(Xkn)gx+ny(Xkn) ≤∑x∈Zdn∑y∈Qkfx+ny(Xkn)∑y∈Qkgx+ny(Xkn)=∑x∈Zdnfx(Xn)gx(Xn). (A.1.6)A.2 Convergence of the finite-volume approximationFor L ≥ 2 and N ≥ 1 note that ΛN is the torus Zdn with n = LN . Thus, XLNis the simplerandom walk on ΛN . For FT = FT (X) any one of the functions LxT , IT , CT of X defined in(1.6.5)–(1.6.7), we write FN,T = FT (XLN ). For instance, with n = LN ,LxN,T =∫ T01Xnt = xdt, IN,T =∑x∈ΛN(LxN,T )2. (A.2.1)We apply Lemma A.1.1 with k = L and n = LN for three choices of f , g:IN+1,T ≤ IN,T (fx = gx = LxT ), (A.2.2)CN+1,T ≤ CN,T (fx =∑e∈U Lx+eT , gx = LxT ), (A.2.3)∑x∈ΛN+1|∇eLxN+1,T |2 ≤∑x∈ΛN|∇eLxN,T |2 (fx = gx = |∇eLxT |). (A.2.4)Summation of (A.2.4) over unit vectors e ∈ Zd also gives∑x∈ΛN+1|∇LxN+1,T |2 ≤∑x∈ΛN|∇LxN,T |2. (A.2.5)The following is a re-statement of Proposition 1.8.4.Proposition A.2.1. Let d > 0, g > 0 and γ < g. For all ν ∈ R,limN→∞Gx(g, γ, ν) = Gx(g, γ, ν) (A.2.6)andlimN→∞χN (g, γ, ν) = χ(g, γ, ν). (A.2.7)105A.2. Convergence of the finite-volume approximationIn fact, χN and χ are analytic in Reν > νc and χN → χ uniformly on compact subsets of thisdomain.Proof. It suffices to prove (A.2.6) and (A.2.7). Analyticity is a property of the Laplace trans-form and uniform convergence on compact sets follows from Montel’s theorem. For pointwiseconvergence, we will only prove the case γ ≥ 0. The proof for γ < 0 can be found in [13].Fix x ∈ Zd, and consider N sufficiently large that x can be identified with points in ΛN .By (1.6.20), (A.2.2) and (A.2.5)cN,T (x) ≤ cN+1,T (x). (A.2.8)Thus, (A.2.6) follows by monotone convergence, once we show thatlimN→∞cN,T (x) = cT (x). (A.2.9)To show this, recall that we are identifying the vertices of ΛN with nested subsets of Zd.We can thus define ∂ΛN to be the inner vertex boundary of ΛN . We setc∗N,T (x) = EΛN0(e−Ug,γ,T1X(T )=b1{X([0,T ])∩∂ΛN 6=∅})(A.2.10)c∗T (x) = E0(e−Ug,γ,T1X(T )=b1{X([0,T ])∩∂ΛN 6=∅}). (A.2.11)Since walks which do not reach ∂ΛN make equal contributions to both cT (x) and cN,T (x), wehavecT (x)− c∗T (x) = cN,T (x)− c∗N,T (x). (A.2.12)Thus,|cT (x)− cN,T (x)| = |c∗T (x)− c∗N,T (x)| ≤ c∗T (x) + c∗N,T (x). (A.2.13)Let PΛN0 and P0 be the measures associated with EΛN0 and E0, respectively. With Yt a rate-2dPoisson process with measure P,c∗T (x) + c∗N,T (x) ≤ P0(X([0, T ]) ∩ ∂ΛN 6= ∅) + PΛN0 (X([0, T ]) ∩ ∂ΛN 6= ∅)≤ 2P(YT ≥ diam(ΛN ))→ 0 (A.2.14)as N →∞. This completes the proof of (A.2.6).Finally, by monotone convergence of GN to G, for ν ∈ R,limN→∞χN (g, γ, ν) =∑x∈ZdlimN→∞Gx,N (g, γ, ν)1x∈ΛN = χ(g, γ, ν), (A.2.15)which proves (A.2.7).106Appendix BMoments of the free Green functionIn this appendix we prove Proposition 3.3.1.B.1 Main resultThe following is a re-statement of Proposition 3.3.1. Since we are only dealing with the freeGreen function, we setGx(m2) = Gx(0, 0,m2). (B.1.1)Proposition B.1.1. Let cp be the constant defined by (1.7.4). For all dimensions d > 2 andall p > 0, as m2 ↓ 0, ∑x∈Zd|x|pGx(0,m2) = cppm−(p+2)(1 +O(m)). (B.1.2)In particular, ξp(0, ε) = cpε−1/2(1 +O(ε1/2)) as ε ↓ 0.The last sentence in the the proposition follows immediately from (B.1.2) and the fact thatχ(0,m2) = m−2 (recall (1.5.12)), so it suffices to prove (B.1.2).The case p = 2 of (B.1.2) can be obtained easily from the identity∑x∈Zd|x|2Gx(m2) = −∆RdGˆ(0), (B.1.3)where Gˆ is the Fourier transform of G. Higher even moments could in principle be computedby further differentiating Gˆ. We adopt a different approach for general p > 0, based on thefinite range decomposition of (−∆Zd + m2)−1 given in [5, 22]. This finite range decompositionalso provides the basis for the renormalization group method.B.2 Riemann sum approximationWe will make use of the following elementary result.Lemma B.2.1. Let f : Rd → R be a Lipschitz function with support in a box of side t centredat the origin. Then there is a constant C such that for any > 0,∣∣∣∣∣∣∫Rdf(x) dx− d∑x∈Zdf(x)∣∣∣∣∣∣ ≤ C(t)d. (B.2.1)107B.3. Covariance decompositionProof. For any x ∈ Zd, let Sx() denote the square of side centred at x ∈ Rd. Then∫Rdf(x) dx =∑x∈Zd∫Sx()f(y) dy. (B.2.2)By the mean value theorem, there exists yx = yx() ∈ Sx() such that∫Sx()f(y) dy = df(yx). (B.2.3)Thus, ∣∣∣∣∣∣∫Rdf(x) dx− d∑x∈Zdf(x)∣∣∣∣∣∣ ≤ d∑x∈Zd|f(yx)− f(x)|. (B.2.4)By the Lipschitz condition on f , each summand on the right-hand side is O(). By the supportassumption on f , there are at most O(td/) such summands and the result follows.B.3 Covariance decompositionThe finite-range decomposition of the finite-volume covariance discussed in Section 2.3 is derivedfrom a decomposition of the infinite-volume covariance (whose construction is the main resultof [5]) of the formGx(m2) =∞∑j=1Cj;x(m2). (B.3.1)Recall that the finite-range property refers to the fact that Cj;x(m2) = 0 if |x| ≥ 12Lj , whereL > 1 is fixed arbitrarily. We review some properties of this decomposition, from [5,10], beforeproving Proposition B.1.1. The positive-definiteness of the finite range decomposition is notneeded here, and L need not be large.The terms Cj;x(m2) are defined in [10, Section 6.1] byCj;x(m2) =∫ 12L0φ∗t (x;m2)dtt(j = 1)∫ 12Lj12Lj−1φ∗t (x;m2)dtt(j ≥ 2)(B.3.2)(in [10], the notation Cj;0,x and φ∗t (0, x;m2) was used instead). Here, φ∗t is a function of x ∈ Rdand m2 > 0 given in [5, Example 1.1]. It satisfies the finite range property that φ∗t (x;m2) = 0for |x| > t. It was also shown in [5] that there exists a function φt satisfying the same finiterange property but giving a decomposition of the continuum Green function:(−∆Rd +m2)−10x =∫ ∞0φt(x;m2)dtt. (B.3.3)108B.4. Proof of main resultMoreover, by [5, (1.37)], for |x| ≤ t,φ∗t (x;m2) = φt(x;m2) +O(t−(d−1)(1 +m2t2)−k). (B.3.4)This allows us to approximate the discrete Green function by the continuum one, for which themoments are easily computed. We have set the constant c present in [5] equal to 1, which wecan do by rescaling φ∗t .As t approaches 0, the error bound in (B.3.4) degenerates. However, to estimate (B.1.1), itsuffices to restrict to x 6= 0. Then, since x ∈ Zd, the finite range property permits replacementof the lower bound in the range of integration for j = 1 in (B.3.2) by 12 , and the contributiondue to j = 1 can be estimated in the same way as the terms j ≥ 2.Also, by [5, (1.34)], for any k there is a constant Ck such that|Dxφt(x;m2)| ≤ Ckt−(d−1)(1 +m2t2)−k. (B.3.5)We fix a choice of k which obeys k > 12(p + 1) and use only this choice. By [5, (1.38)], thereexists a function φ¯ such thatφt(x;m2) = t−(d−2)φ¯(xt;m2t2). (B.3.6)B.4 Proof of main resultProof of Proposition 3.3.1. We begin by writing∑x∈Zd|x|pGx(m2) =∑x∈Zd|x|p∞∑j=1Cj;x(m2) = M(m2) + E(m2), (B.4.1)where the main and error terms are respectivelyM(m2) =∑x∈Zd|x|p∞∑j=1∫ 12Lj12Lj−1φt(x;m2)dtt, (B.4.2)E(m2) =∑x∈Zd|x|p∞∑j=1(Cj;x −∫ 12Lj12Lj−1φt(x;m2)dtt). (B.4.3)We first compute the main term M . By (B.3.6),φt(x;m2) = md−2φmt(mx; 1). (B.4.4)109B.4. Proof of main resultTherefore, by Riemann sum approximation,∑x∈Zd|x|p∫ 12Lj12Lj−1φt(x;m2)dtt(B.4.5)= m−(p+2)md∑x∈Zd|mx|p∫ 12Lj12Lj−1φmt(mx; 1)dtt(B.4.6)= m−(p+2)∫Rd|x|p∫ 12Lj12Lj−1φmt(x; 1)dtt+O(L(p+1)jL−2k(j−jm)+),where the error estimate follows from (B.3.5) and (2.4.18). Summation over j givesM(m2) = cppm−(p+2) +O(m−(p+1)), (B.4.7)where we used (B.3.3) for the first term, and we used 2k > p + 1 and Lemma 3.3.2 for thesecond term.For the error term, it follows from (B.3.2), (B.3.4), and the observation that the lower boundin the range of integration for the j = 1 term in (B.3.2) can be changed to 12 thatCj;x =∫ 12Lj12Lj−1φt(x;m2)dtt+O(L−j(d−1)(1 +m2L2j)−k)1|x|≤Lj . (B.4.8)Therefore, again using (2.4.18), we haveE(m2) =∞∑j=1∑|x|≤Lj|x|pO(L−j(d−1)L−2k(j−jm)+) (B.4.9)=∞∑j=1O(L(p+1)jL−2k(j−jm)+). (B.4.10)With 2k > p+ 1 and Lemma 3.3.2, this gives E(m2) = O(m−(p+1)), and the proof is complete.110Appendix CAn implicit function theoremIn this appendix, we prove Proposition 3.1.3.C.1 Implicit function theorem with a parameterWe make use of [89, Chapter 4, Theorem 9.3], which is a version of the implicit functiontheorem that allows for a continuous, rather than differentiable, parameter. While the precisestatement of [89, Chapter 4, Theorem 9.3] takes this parameter from an open subset of a Banachspace, by [89, Chapter 4, Theorem 9.2], the parameter can in fact be taken from an arbitrarymetric space. With this minor change, we restate [89, Chapter 4, Theorem 9.3] as the followingproposition.Proposition C.1.1. Let A be a metric space, let W,X be Banach spaces, and let B ⊂ W bean open subset. Let F : A × B → X be continuous, and suppose that F is C1 in its secondargument. Let (α, β) ∈ A× B be a point such that F (α, β) = 0 and D2F (α, β)−1 exists. Thenthere are open balls M 3 α and N 3 β and a unique continuous mapping f : M → N such thatF (ξ, f(ξ)) = 0 for all ξ ∈M .We also use the following lemma, which is a small modification of [89, Chapter 3, Theo-rem 11.1]. In particular, it considers functions that may only be left- or right-differentiable.Lemma C.1.2. Let F be a mapping as in the previous proposition with A ⊂ Rm1 × Rm2. Inaddition, suppose that F is left-differentiable (respectively, right-differentiable) in α2 at (α, β),with α = (α1, α2). If f is a continuous mapping defined in a neighbourhood of α, such thatF (ξ, f(ξ)) = 0, then f is left-differentiable (respectively, right-differentiable) in α2 at α.C.2 Main resultThe above results lead to the following proposition, which is a re-statement of Proposition 3.1.3.Recall that D(δ, r) is defined in (2.8.1) byD(δ, r) = {(w, x, y) ∈ [0, δ]3 : y ≤ r(x)}. (C.2.1)111C.2. Main resultProposition C.2.1. Let δ > 0, and let r1, r2 be continuous positive-definite functions on [0, δ].SetD(δ, r1, r2) = {(w, x, y, z) ∈ D(δ, r1)× Rn : |z| ≤ r2(x)}, (C.2.2)and let F be a continuous function on D(δ, r1, r2) that is C1 in (x, z). Suppose that for all(w¯, x¯) ∈ [0, δ]2 there exists z¯ such that both F (w¯, x¯, 0, z¯) = 0 and DY F (w¯, x¯, 0, z¯) is invert-ible. Then there is a continuous positive-definite function r on [0, δ] and a continuous mapf : D(δ, r) → Rn that is C1 in x and such that F (w, x, y, f(w, x, y)) = 0 for all (w, x, y) ∈D(δ, r). Moreover, if F is left-differentiable (respectively, right-differentiable) in y at somepoint (w, x, y, z), then f is left-differentiable (respectively, right-differentiable) at (w, x, y).Proof. Take any (w¯, x¯) ∈ [0, δ] × (0, δ] and let R(w¯, x¯) be the maximal radius s such that forall (w, x, y) ∈ B(w¯, x¯, 0; s) there exists z such that both F (w, x, y, z) = 0 and DZF (w, x, y, z)is invertible. By continuity of (DZF (w, x, y, z))−1 near (w¯, x¯, 0, z¯), and by Proposition C.1.1(applied to the restriction of F to A × B, for some A 3 (w¯, x¯, 0) and an open set B 3 z¯), wehave R(w¯, x¯) > 0 and there is a continuous functionfw¯,x¯ : B(w¯, x¯, 0;R(w¯, x¯))→ Rn (C.2.3)such that F (w, x, y, fw¯,x¯(w, x, y)) = 0 for all (w, x, y) ∈ B(w¯, x¯, 0;R(w¯, x¯)). Moreover, theunique solution to F (w, x, y, z) = 0 is given by z = fw¯,x¯(w, x, y) for all such (w, x, y). Byan application of Lemma C.1.2 (with α1 = (w, x), α2 = y), we see that fw¯,x¯ is left- orright-differentiable in y wherever F is. By another application of Lemma C.1.2 (with α1 =(w, y), α2 = x), we see that fw¯,x¯ is C1 in x.Set R(w¯, 0) = 0 for all w¯ ∈ [0, δ], and letDf =⋃(w¯,x¯)∈[0,δ]2B(w¯, x¯, 0;R(w¯, x¯)). (C.2.4)We define f(w, 0, 0) = 0 and, for x > 0,f(w, x, y) = fw¯,x¯(w, x, y) for (w, x, y) ∈ B(w¯, x¯, 0;R(w¯, x¯)). (C.2.5)By uniqueness, this function is well-defined. Continuity of f at (w, 0, 0) follows from the factthat |f(w, x, y)| ≤ r2(x). The remaining desired regularity properties of f follow from those ofthe fw¯,x¯. It remains to show that D(δ, r) ⊂ Df for some continuous positive-definite function ron [0, δ].First, let us show that R is continuous on [0, δ]2. Let x¯ > 0 and fix 0 < < R(w¯, x¯). Thenfor any (w¯′, x¯′) ∈ [0, δ]×(0, δ] such that |(w¯, x¯)−(w¯′, x¯′)| < , we have B(w¯′, x¯′, 0;R(w¯, x¯)−) ⊂B(w¯, x¯, 0;R(w¯, x¯)) by maximality of R. It follows that R(w¯′, x¯′) ≥ R(w¯, x¯) − . By a similarargument, R(w¯′, x¯′) ≤ R(w¯, x¯) + , so |R(w¯, x¯) − R(w¯′, x¯′)| ≤ . Thus, R is continuous on[0, δ]× (0, δ]. Continuity at x¯ = 0 follows from the fact that R(w¯, x¯) ≤ r1(x¯) uniformly in w¯.112C.2. Main resultFor x¯ ∈ [0, δ], letr(x¯) = inf(R(w¯, x¯) : w¯ ∈ [0, δ]). (C.2.6)Since R(·, x¯) is continuous, r(x¯) > 0 for x¯ > 0. Moreover, 0 ≤ r(0) ≤ r1(0) = 0, so r ispositive-definite. Continuity of r follows from joint continuity of R. For any (w, x, y) ∈ D(δ, r)(with this choice of r),|(w, x, y)− (w, x, 0)| = |y| < r(x) ≤ R(w, x), (C.2.7)so (w, x, y) ∈ B(w, x, 0;R(w, x)). We conclude that D(δ, r) ⊂ Df .113
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Renormalization group analysis of self-interacting walks and spin systems Wallace, Benjamin 2017
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Title | Renormalization group analysis of self-interacting walks and spin systems |
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Wallace, Benjamin |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | The central concern of this thesis is the study of critical behaviour in models of statistical physics in the upper-critical dimension. We study a generalized n-component lattice |φ|⁴ model and a model of weakly self-avoiding walk with nearest-neighbour contact self-attraction on the Euclidean lattice ℤd. By utilizing a supersymmetric integral representation involving boson and fermion fields, the two models are studied in a unified manner. Our main result, which is contingent on a small coupling hypothesis, identifies the precise leading-order asymptotics of the two-point function, susceptibility, and finite-order correlation length of both models in d = 4. In particular, we show that the critical two-point function satisfies mean-field scaling whereas the near-critical susceptibility and finite-order correlation length exhibit logarithmic corrections to mean-field behaviour. The proof employs a renormalization group method of Bauerschmidt, Brydges, and Slade based on a finite-range covariance decomposition and requires two extensions to this method. The first extension, which is required for the computation of the finite-order correlation length (even for the ordinary weakly self-avoiding walk and |φ|⁴ model), is an improvement of the norms used to control the evolution of the renormalization group. This allows us to obtain improved error estimates in the massive regime of the renormalization group flow. The second extension involves the identification of critical parameters for models initialized with a non-zero error coordinate coupled to a marginal/relevant coordinate. This allows us, for example, to realize the two-point function and susceptibility for the walk with self-attraction as a small perturbation of the corresponding quantities without self-attraction, whose asymptotic behaviour was determined by Bauerschmidt, Brydges, and Slade. This establishes a form of universality. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-07-04 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0348692 |
URI | http://hdl.handle.net/2429/62135 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2017-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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