C o n v e r g e n c e o f L a t t i c e T rees to S u p e r - B r o w n i a n M o t i o n above the C r i t i c a l D i m e n s i o n by Mark Holmes MSc (Math), University of British Columbia, 2001 MSc[Dist] (Stat), University of.Auckland, 1999 BCom/BSc[Hons] (Finance/Stat), University of Auckland, 1998 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D o c t o r o f P h i l o s o p h y in. T H E F A C U L T Y O F G R A D U A T E STUDIES (Mathematics) T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a July 2005 © Mark Holmes, 2005 Abst rac t A lattice tree is a finite connected set of lattice bonds containing no cycles. Lattice trees are interesting combinatorial objects and an important model for branched polymers in polymer chemistry and physics. In addition they provide an interesting example of critical phenomena in statistical physics with similar properties to models such as self-avoiding walks and percolation. We use the lace expansion to prove asymptotic formulae for the Fourier trans-form of the r-point functions (quantities which count critically weighted trees con-taining r fixed points) for a spread-out model of lattice" trees in Zd for d > 8. Our results therefore provide additional evidence in support of the critical dimen-sion dc = 8. The spread out model allows bonds between vertices x,y 6 Zd with \\% — y\\oo < L, providing a small parameter L~* needed for convergence of the lace expansion. We extend the inductive approach (to the lace expansion on an interval) of van der Hofstad and Slade [19] to prove convergence of the Fourier transform of the 2-point function (r = 2). We then proceed by induction on r , equipped with the lace expansion on a tree [21]. The asymptotic formulae for the r-point functions imply convergence of cer-tain expectations of the spread out lattice trees model formulated as a measure valued process, to those of the canonical measure of super-Brownian motion. Ap-pealing to the hypothesis of universality, we expect that the results also hold for the nearest neighbour model. Our results together with the convergence of the survival probability would imply convergence of the finite-dimensional distributions of our process to those of the canonical measure of super-Brownian motion. Convergence of the survival probability remains an open problem. Contents Abstract ii Contents iii List of Figures vi Acknowledgements viii 1 Introduction 1 1.1 Background and motivation 1 1.1.1 Combinatorics and statistical physics 2 1.1.2 Probability and measure-valued processes 4 1.2 The model 6 1.3 A measure-valued process 10 1.4 The r-point functions 13 2 The lace expansion 17 2.1 Graphs and Laces 17 2.1.1 Classification of laces 23 2.2 The Expansion 25 3 The 2-point function 28 3.1 Organisation 28 3.2 Recursion relation for the 2-point function 29 3.3 Assumptions of the induction method . 35 3.4 Verifying assumptions 37 3.5 Proof of Theorem 1.4.3 44 4 The r-point functions 48 4.1 Preliminaries 48 4.2 Application of the Lace Expansion . 54 ii i 4.3 Decomposition of 57 4.4 Bounds on the £ . 60 4.5 Proof of Theorem 4.1.8 62 4.6 Proof of Theorem 1.4.5 65 5 Diagrams for the 2-point function 70 5.1 Definitions and Notation 70 5.2 Proof of Proposition 5.1.1 74 5.2.1 Diagrams with an extra vertex 81 5.3 General Diagrams 82 5.4 Diagram pieces 90 5.4.1 Proof of Lemma 5.4.1 91 5.4.2 Proof of Lemma 5.4.2 93 6 Diagrams for the r-point functions 95 6.1 Proof of Proposition 4.3.2 95 6.2 Proof of Lemma 6.1.1 97 6.3 Proof of Lemma 6.1.2 100 6.4 Proof of Lemma 6.1.3 105 6.4.1 Acyclic laces with 2 bonds covering the branch point 105 6.4.2 Acyclic laces with 3 bonds covering the branchpoint 109 6.5 Proof of Lemma 4.3.3 122 6.6 Proof of Lemma 4.2.1 124 6.6.1 Proof of the first bound of Lemma 4.2.1 125 6.6.2 Proof of the third bound of Lemma 4.2.1 128 6.6.3 Proof of the second bound of Lemma 4.2.1 . 129 7 Convergence to CSBM. 134 7.1 The canonical measure of super-Brownian Motion . 134 7.1.1 Branching Random Walk 134 7.1.2 B R W converges to C S B M 136 7.2 Lattice trees 137 7.2.1 ISE 139 7.3 Finite dimensional distributions 141 7.4 Proof of Theorem 1.3.1 152 7.5 A note on convergence of finite dimensional distributions 157 iv Appendix A 159 A . l Motivation 159 A.2 Assumptions on the Recursion Relation 160 A.2.1 Assumptions S,D,Ee,Ge 161 A.3 Induction hypotheses 163 A.3.1 Initialisation of the induction 165 A.3.2 Consequences of induction hypotheses 165 A.4 The induction advanced 169 Bibliography 170 v List of Figures 1.1 Example of a lattice tree 1 1.2 Example of a lattice tree 7 1.3 Particle picture of a lattice tree . 8 1.4 Examples of shapes . . . 15 2.1 A graph on a network Af and its associated subnetwork A 18 2.2 A graph containing a bond in 1Z 19 2.3 Graphs and laces on star-shaped networks 20 2.4 Construction of a lace from a connected graph 23 2.5 Minimal and non-minimal laces 23 2.6 Cyclic and acyclic laces 24 3.1 A lattice tree as a backbone and mutually avoiding branches . . . . 29 4.1 A lattice tree and its corresponding network shape 51 4.2 r-point error terms 56 4.3 The relation £ - = 1 rj = r + 3 64 4.4 Examples of degenerate shapes 66 5.1 Diagram pieces Jl4!' and Amijm2 72 5.2 A typical opened diagram , 73 5.3 Diagram for N = 1 85 5.4 A typical Feynman diagram . . 86 6.1 The single bond lace on [ - M 2 , Mi] and the corresponding [0, M i + M2]. 98 6.2 A lace on [ - M 2 , Mi] and the corresponding [0, M i + M 2 ] 99 6.3 Some cyclic laces containing only 3 bonds 101 6.4 Diagrammatic bound for a basic cyclic lace '. 101 6.5 A degenerate cyclic lace and its diagrams 104 6.6 Diagrammatic bound for a basic acyclic lace 106 6.7 Diagrams for acyclic laces with 2 bonds covering the branch point . 108 vi 6.8 Another acyclic lace with 2 bonds covering the branch point 110 6.9 Illustration of a 4-star Lemma 110 6.10 Illustration of another 4-star Lemma . I l l 6.11 Basic acyclic laces with only 3 bonds covering the branch point, and their decomposition into opened subdiagrams 112 6.12 Application of Lemma 6.4.2 . . 114 6.13 Application of Lemma 6.4.3 . . . 114 6.14 Another application of Lemma 6.4.3 115 6.15 Decomposition of a diagram from a non-minimal lace 116 6.16 Construction of a lace from a graph that covers multiple branch points 130 vii Acknowledgements This work was supported by the following awards from the University of British Columbia: Ki l l am Predoctoral Fellowship, Josephine T. Berthier Memorial Fellow-ship, John R. Grace Fellowship. The author would like to thank those associated with the foundation and the administration of these awards. The author would also like to thank, in alphabetical order, Remco van der Hofstad, Edwin Perkins and Gordon Slade for their ideas and encouragement. M A R K H O L M E S The University of British Columbia July 2005 vii i Chapter 1 Introduction This chapter serves as an introduction to the terminology, ideas and context of this thesis. In Section 1.1 we discuss some of the motivation for this thesis and give a brief outline of some of the relevant existing results. In Sections 1.2 and 1.3 we introduce the model that we study and state the main result. We conclude this chapter by defining some quantities that are the main focus of this thesis, and briefly discuss how they are connected to the main result. 1.1 B a c k g r o u n d a n d m o t i v a t i o n A lattice tree in Zd is a finite connected set of lattice bonds containing no cycles (see Figure 1.1). Lattice trees are an important model for branched polymers. They are combinatorial objects, so are of interest in combinatorics and graph theory. As we shall discuss shortly, our model for lattice trees is relevant to statistical physicists as a lattice model that exhibits a phase transition, with the behaviour at criticality being of particular interest. We can also describe our model as an example of a non-Markovian measure-valued process which converges (in dimensions d > 8) to a Figure 1.1: A nearest neighbour lattice tree in 2 dimensions. 1 well known measure-valued Markov process in the scaling limit. Thus our results are also appealing to probabilists and researchers interested in stochastic processes. 1.1.1 Combinatorics and statistical physics Lattice trees provide an interesting example of critical phenomena in statistical physics with similar properties to models such as self-avoiding walks (a model for linear polymers) and percolation. Let ln be the number of n-bond (nearest neigh-bour) lattice trees that contain the origin. A n elementary question in combinatorics or graph theory would be "what is /„?". Even in two dimensions the answer is not known for large values of n. However it is known [24] that > ^ ^J- in all dimensions, and a standard subadditivity argument then shows that Ifi —)• A > 0 as n —>• oo. The bounds C i n - C 2 l o g n A n < l n < C 3 „ ^ A n j ( 1 ! ) were proved in [23] and [26] respectively. Using the notation f(x) ~ g(x) to mean l inx^oo = 1, it is widely believed that ln~C\nn1-6, (1.2) where 6 is called a critical exponent for the model. Critical exponents convey infor-mation about the macroscopic or asymptotic properties of the model. The exponent 0 is believed to depend on d, but not on the type of lattice or the type of bonds allowed (provided modest regularity conditions such as symmetry and finite range hold). A n important example for our purposes is the unrestricted 2-point function, pp(x) = 2~lreT(x) P^T w n e r e T(x) is the set of lattice'trees containing the origin and x and # T is the number of bonds in T. The function pv(x) is a power series with i the coefficient of pN being the number of lattice trees containing 0 and x consisting of AT bonds. This power series has nontrivial radius of convergence pc = j, at which it is believed that p(x) changes from having exponential decay in \x\ for p < pc to power law decay C PPc(x) ~ l < p | d - 2 + , , ' a S M -> °°> (L3) I I where « represents some asymptotic behaviour that we do not state precisely at present. This kind of fundamental change in the properties of the model at p — pc is sometimes referred to as a phase transition. The critical exponent r\ in (1.3) is also thought to depend on d, but not on the type of lattice or bonds. This lack of dependence on the details of the model is called universality, and models with the same critical exponents are said to be in the same universality class. It should be pointed out that universality is a widely 2 believed hypothesis in statistical physics rather than a rigorous mathematical theory. However there are many rigorous examples which give evidence in support of the hypothesis. Different critical exponents of a model are not independent of each other, and may obey a scaling or hyper-scaling relation (if the relation includes the dimension d) or inequality. A good source of information on critical exponents for lattice trees (self-avoiding branched polymers) is [9]. Lattice trees are self-avoiding objects by definition (since they contain no cycles). It is plausible that the self-avoidance constraint imposed by the model becomes less important as the dimension increases in the following sense. We might expect a randomly chosen branching lattice object in d dimensions and containing N. bonds to be more likely to be self-avoiding as d increases. In fact there is considerable evidence that for dimensions d > 8 the self-avoidance constraint is negligible in terms of the macroscopic view of the model. It is believed that for d > 8 the critical exponents cease to depend on the dimension and correspond to those of a simpler model, that does not have the constraint. The simpler model is called the mean-field model, and the dimension dc above which the constrained model has the same macroscopic properties as the mean-field model is called the critical dimension. Lubensky and Isaacson [25] proposed dc = 8 as the critical dimension for lattice trees and animals. There are few rigorous results for lattice trees for d < 8. The scaling limits of many models in statistical physics in 2 dimensions are believed to be described by a class of processes called Stochastic Loewner Evolution (SLE), [30]. The S L E processes are candidates for the scaling limit of a model where the scaling limit is believed to have a property called conformal invariance. The scaling limit of lattice trees in 2 dimensions is not expected to have this property. Brydges and Imbrie [4] used a dimensional reduction approach to obtain strong results for a continuum (i.e. not lattice based) model for d = 2,3. Appealing to universality, we would expect lattice trees to have the same critical exponents as the Brydges and Imbrie model. More is known in high dimensions, where the asymptotic behaviour should correspond to the mean-field model for lattice trees, branching random walk. Tasaki and Hara [29] showed in the context of lattice animals that the finiteness of the square diagram 2~2X y z Ppc(x)ppc(y ~ x)Ppc(z ~ v)Ppc{z) implies mean-field critical behaviour for the susceptibility xip) = YxPp(x)- The s a m e methods and results apply to lattice trees. Hara and Slade [12], [13] proved the finiteness of the square diagram for sufficiently spread-out lattice trees (and animals) for d,> 8, and for the nearest neighbour model for d >^ 8, as well as the mean-field critical behaviour of various quantities. Hara, van der Hofstad, and Slade [11] proved for a sufficiently spread out model that for d > 8, (1.3) holds with r\ = 0. This is the same exponent 3 as for branching random walk. In [11] and this thesis the major tool of analysis is a technique known as the lace expansion, (introduced by Brydges and Spencer [5]). This technique is highly combinatorial in nature. 1.1.2 Probability and measure-valued processes Most of the discussion in the following three paragraphs can be found in standard graduate level probability texts, for example [3]. Fix a probability space (O, T, P) and suppose that Xi are independent iden-tically distributed real valued random variables with mean 0 and finite variance a2, and let Sn = 2~27=i ^i- A fundamental result in probability theory, called the cen-tral limit theorem states that converges weakly to a standard Gaussian random variable, Z. More precisely, defining probability measures //„(•) = P{~^ G •), and /*(.) = P ( Z G •) = £-±=e-£dz, (1.4) then fj,n =4- /i. Convergence takes place in a metric space of probability measures on R equipped with the weak topology (for example the Prohorov metric), M i ( R ) , so that /j,n ==» fj, if and only if for every bounded continuous / : R —)• R, f / dfj,n —> J f du. We use the notation E(1[f(X)] = J f d/j, where it is understood that X is a random variable with distribution fi. Therefore we can also write /i n =^4» H E^JfiXn)] -> Ep[f(X)], for every bounded continuous / : R -»• R To prove weak convergence in R (convergence in the space of probability measures on R with the weak topology) it is enough to show convergence of the Fourier transforms Ef,n[eikXn] -> ^ [ e i f c X ] (or more traditionally E[eik^] -> E[eikz] = e-V) to that of the Gaussian. In other words the functions {fk(x) = elkx : k G [ - T T , TT]} constitute a convergence determining class. These results can easily be generalised to Revalued random variables. Note that the constraint that the Xi be independent and identically distributed may be relaxed (for example Xi stationary, ergodic with E[Xn+\\Tn] — 0) and the central limit theorem may still hold. Setting So = 0, the collection {5„} n >o is a random walk on R, and writing X f = t > 0, defines a real-valued stochastic process {X^}t>o that is right continuous with left limits for each n, i.e. {X™}t>o G D(R). Define probability measures /j,n on the Borel sets of D(R) by ^„(«) = P({X^}t>o € •)• Another fun-damental result in probability states that nn ==>• W, where W is Wiener measure. It is perhaps more commonly said that {X"}t>o converges weakly to a continuous, real valued stochastic process E>t called Brownian motion, or that random walk con-verges to Brownian motion in the scaling limit. To prove convergence in the space of probability measures on D(R) (weak topology) it is enough to prove convergence 4 of the finite dimensional distributions and tightness. The {tin} are tight if for every e > 0 there exists a compact K C D(R) such that sup„/j, n(K c) < e. Convergence of the finite dimensional distributions by definition means that for every m G N, t G [0, oo) m , and every bounded continuous / : K m -> K , E,n [f{Xl,..., XI)) -+ Ew[f(BtlBtm)]. (1.5) To verify (1.5), it is enough to show convergence of the corresponding Fourier trans-forms E^n e t Due to the independence of the Xi, the process {X^}t>o is a Markov process. That is, the future of the process is independent of the past given the present. Since the increments (consider the Xj) of the process also have mean 0, {X^}t>o is a martingale (ElX^^T^} = X™ where = cr({X"} s< t)). Brownian motion is also a Markovian martingale. Brownian motion paths almost surely have Hausdorff dimension dA2 and are almost surely self-avoiding in 4 or more dimensions. As such, Brownian motion is a sensible candidate for the scaling limit of self-avoiding walk (neither Markovian nor a martingale) for d > 4. A result of Hara and Slade shows that for d > 4, self-avoiding walk converges to Brownian motion in the scaling limit. In this case tightness follows from a negative correlation property of the model. The following brief introduction to some important measure-valued processes is described in more mathematical detail in Chapter 7. Let Y be a non-negative integer-valued random variable with mean 1 and positive variance. Critical branch-ing random walk in d dimensions is a process that starts with a single particle at time 0 located at the origin, and at each time n 6 N, each particle a alive at time n independently gives birth to Ya = Y particles at independently and ran-domly chosen neighbouring vertices and then dies instantly. It can be described by a measure-valued process Xn where for each fixed time n, Xn is a finite measure (Xn e My(Md)) on the Borel sets of R d with Xn(B) being the number of particles a alive at time n whose spatial location is some x € B. In this way, a realisation of a measure-valued process describes the evolution in time of the distribution of mass. The mean offspring number of 1 is critical. It can be shown that this process dies out almost surely, but that the expected time when this happens is infinite. The process is Markovian due to the independence conditions and with the critical birth rate the total mass process is also a martingale. In a similar way to what was done for the simple random walk case, we can define the branching random walk process for all t > 0, so that it is right continuous with left limits. Wi th appropriate scaling of space, time, and mass, critical branching random walk converges weakly (i.e. convergence in the space of measures on D(Mjp(KF L !))) to a measure-valued process Xt called super-Brownian 5 motion (SBM). This is of course a statement that fi'n ==>• No for some measures fx'n € MF(£)(M J p(M d )) and some other measure No called the canonical measure of super-Brownian motion (CSBM). Tightness of the measures u-'n can be verified using martingale methods. Now the support process {At}t>o, where At is the support of the measure Yt = J 0 Xsds of a S B M Xs has Hausdorff dimension 4 A d and has no self intersections in dimensions d > 8 (No almost everywhere). This is the appropriate way to say that S B M is self-avoiding for d > 8. Intuitively, by comparison with the self-avoiding walk results we might expect that our critical lattice trees model (described as a measure-valued process with appropriate scaling) converges weakly to C S B M in the same sense as branching random walk, for d > 8. Studying a different but related limit conjectured by Aldous [2], it was shown in [7] that sufficiently spread out lattice trees in dimensions d > 8 converge to integrated super-Brownian excursion (ISE) as the total size of the tree goes to infinity. ISE is a probability measure on probability measures on R d , i.e. 1 e Mi (Mi (M d ) ) which describes the distribution of the total mass of C S B M (conditioned to be 1). ISE contains no information about time evolution, however some results concerning ancestry were also proved in [7]. • In this thesis, we prove convergence of the finite dimensional distributions of an appropriately defined lattice trees process to those of C S B M , for d > 8. This convergence is obtained by proving convergence of the Fourier transforms of relevant quantities and using the existence of a certain exponential moment of C S B M . The main tool used in the proof is the lace expansion, in the form of both (an extension of) the inductive approach of [19] and the lace expansion on a tree of [21]. Tightness remains an open problem. The processes in question are neither martingales nor Markovian, so many of the standard methods for proving tightness do not immediately apply. 1.2 T h e m o d e l We now present the basic definitions of the quantities of interest. We restrict our-selves to the integer lattice Z d . Definition 1.2.1. 1. A bond is an unordered pair of distinct vertices in the lattice. 2. A cycle is a set of distinct bonds {v\V2,V2Vz, • • • ,vi-ivuviVi}j for some I > 3. 3. A lattice tree is a finite set of vertices and lattice bonds connecting those vertices, that contains no cycles. This includes the single vertex lattice tree that contains no bonds. 6 y y Figure 1.2: A nearest neighbour lattice tree in 2 dimensions. The backbone from x to y of length n = 17 is highlighted in the second figure. 4- Let r > 2 and let Xi, i G { 1 , . . . , r} be vertices in T. Since T contains no cycles then there exists a minimal connected subtree containing all the Xi, called the skeleton connecting the X{. If r = 2 we often refer to the skeleton connecting x\ to X2 as the backbone. R e m a r k 1.2.2. The nearest-neighbour model consists of nearest neighbour bonds {x\,X2} with xi,X2 G Z d and \xi — X2\ = 1. Figure 1.2 shows an example of a nearest-neighbour lattice tree in Z 2 . We use Z + to denote the nonnegative integers {0 ,1 ,2 , . . . } . Def in i t ion 1.2.3. 1. For x E Zd let Tx = {T : x <E T}. Note that this set always includes the single vertex lattice tree, T = {x} that contains no bonds. We also let Ty(x) = {T G Ty : x G T}, and often write T(x) for To(x), the set of lattice trees containing the vertices 0 and x. 2. For T £ To we let T{ be the set of vertices x inT such that the backbone from 0 to x consists of i bonds. In particular for T G % we have To — {0}. A tree T G 7o is said to survive until time n ifTn ^ 0. 7 — r <P 1 ^ 3 1 1 Figure 1.3: A nearest neighbour lattice tree T in 2 dimensions with the set Tj for i = 10. 3. For x = (xi,.. .,xr-i) G Z d ( r _ 1 ) and fi G Z ^ _ 1 we we write x £ T „ i / i ; 6 T n i /or eac/i i and define 7fi(x) = {T G 7o : x G T„} . If we think of T G To as representing a migrating population in discrete time, then Tj can be thought of as the set of locations of particles alive at time i. Figure 1.3 identifies the set Tio for a fixed T. Similarly 7fi(x) can be thought of as the set of trees for which there is a particle at Xi alive at time n, for each i. In order to provide a small parameter needed for convergence of the lace expansion, we consider trees taking "steps" of size < L for some large parameter L. The steps are weighted according to a function D which is supported on [—L, L]d and which has total mass 1. Thus D represents a kind of step probability function. We define this formally in the following subsection. The methods and results in this paper rely heavily on the main results of [11] and [19]. Since the assumptions on the model are stronger in [11], we adopt the finite range L, D spread out model of [11]. The following definition and the subsequent remark are taken, almost verbatim from [11]. Def in i t ion 1.2.4. Leth be a non-negative bounded function o n R d which is piecewise continuous, symmetric under the Zd-symmetries of reflection in coordinate hyper-planes and rotation by | , supported in [— 1, l ] d , and normalised 1,dh(x)ddx = 8 1). Then for large L we define h(x/L) 22x&.ih{x/L) Remark 1.2.5. Since Ylxezd h(x/L) ~ Ld using a Riemann sum approximation to ^d h(x)ddx, the assumption that L is large ensures that the denominator of (1-6) is non-zero. Since h is bounded, ~Y^x^%dh,{xjV) ~ Ld also implies that \\D\\oo<§-d. ' (1.7) We define a2 = 2~2X \x\2D(x). The sum Y^,x \x\rD(x) can be regarded as a Riemann sum and is asymptotic to a multiple of U for r > 0. In particular a and L are comparable. A basic example obeying the conditions of Definition 1.2.4 is given by the function h(x) — 2~ dIr_ 1 1]d(a:) for which D(x) = (2L + l)~d I[-L,L]dnZd(x)-Definition 1.2.6 (L, D spread out lattice trees). Let fi/j = {x € ZD : D(x) > 0}. We define an L,D spread out lattice tree to be a lattice tree consisting of bonds {x,y} such that y — x £ VLD-The results of this thesis are for L, D spread out lattice trees in dimensions d > 8 . Appealing to the hypothesis of universality, we expect that the results also hold for nearest-neighbour lattice trees. However from this point on, unless otherwise stated, "lattice trees" and related terminology refers to L, D spread out lattice trees. Definition 1.2.7 (Weight of a tree.). Given a finite set of bonds B and a non-negative parameter p, we define the weight of B to be . WP,D(B)= PD(y-x), (1.8) {x,y}eB with W P ) D(0) = 1. If T is a lattice tree we define WP,D{T) = WP,D(BT), (1.9) where BT is the set of bonds ofT. Definition 1.2.8 (p{x)). Let Pp(x)= YI wpMn (i.io) T£T(x) 9 Clearly we have p p(0) > 1 for all L,p since the single vertex lattice tree contains no bonds and therefore has weight 1. A standard subadditivity argument [24] shows that there is a finite, positive pc at which 2~2xPp(x) converges for p < pc and diverges for p > pc. Hara, van der Hofstad and Slade [11] proved the following Theorem, in which 0(y) denotes a quantity that is bounded in absolute value by a constant multiple of y. T h e o r e m 1.2.9. Let d > 8 and fix v > 0. There exists a constant A (depending on d and L) and an LQ (depending on d and u) such that for L > LQ, A \ ( £ ( r f - 8 ) A 2 \ / L 2 \ PPAX) = a2(|a.| v !)d-2 1 + ° \ ^ \ x \ V l)((d-8)A2)^ J + ° \{\X\ V 1)2-* J (1.11) Constants in the error terms are uniform in both x and L, and A is bounded above uniformly in L. We henceforth take our trees at criticality and write W(-) = WPctD(-), and p(x) = pPe(x). (1.12) Hara, van der Hofstad and Slade [11] also proved that pcp{0) < 1 + O (L~2+U) and ^"^^V'cTv.rj1 • ( 1 ' 1 3 ) where the constants in the above statements depend on v and d, but not L. 1.3 A m e a s u r e - v a l u e d p rocess Let M p ( R d ) denote the space of finite measures on R D with the weak topology. For each i, n € N and each lattice tree T, we define a finite measure XT e M F ( R D ) by Xn,T = ^l £ g ( 1 1 4 ) where 6X(B) = IxeB for all B e #(M d). The constants C\,Ci depend on L and d and wil l be stated explicitly later. Figure 1.3 shows a fixed tree T and the set Tj for i = 10. For this T, the measure X1^ assigns measure ^ to each vertex in the set Tialy/Cin = {x : yJC^nx € Tin.}. We extend this definition to all t € R t by ' Xt'T = Xlnl • (1-15) 10 Thus for fixed n, T and t > 0, we have { X t n , T } € D(MF(Rd)). For a Polish space (complete, separable metric space) E we let D(E) = D([0, oo),E) denote the space of right continuous paths with left limits taking values in E. Then D(E) equipped with the Skorokhod topology is also Polish ([8], Theorem 5.6). Let Mp(E) denote the space of finite measures on a Polish space E. Then Mp{E) equipped with the weak topology is also Polish ([6], statement 3.1.1.). The above discussion says that D{MF{^)) (with the appropriate topologies) is a Polish space. Next we must decide what we mean by a "random tree". We define a prob-ability measure P on the countable set 7o by P({T}) = , so that p(£)=Er£*T(T), . j g c T ( L l g ) P(0) Lastly we define the measures /x n € Mp(D(MF(Rd))) by Hn(H) = C 3 n P ({T : {X?'T}tm+ € H}) , H€ B(D(MF(Rd))), (1.17) where B(E) denotes the Borel cr-algebra on E and C3 is another constant that wil l be stated explicitly later. We expect that /j,n ==» No, where No is the canonical measure of super-Brownian motion. This convergence, which we wil l sometimes call convergence as a stochastic process follows from convergence of the finite-dimensional distributions and tightness (see for example [3] Theorems 8.1 and 15.1). The precise definition of this convergence is technical, and thus we postpone its formalisation until Chapter 7. In particular, No(-X"e / 0M) (where 0 M denotes the zero measure) is finite but becomes infinite as e \ 0. Therefore it is natural to consider No on the set where extinction occurs after time e. We note in Chapter 7 that to prove the statement of convergence of the finite-dimensional distributions we would require the asymptotics of the survival proba-bility P ( T n > 0). Without the survival asymptotics we prove Theorem 1.3.1, which is the main result of this thesis for probabilists (statistical physicists may be more interested in Theorems 1.4.3 and 1.4.5), in which {Y^1} denotes a process chosen according to the finite measure u,n and {^t} denotes super-Brownian excursion, i.e. a measure-valued path chosen according to the a-finite measure No. We also use Vp to denote the set of discontinuities of a function F. A function Q : M / r ( R r f ) m —> M is called a, multinomial if Q(X) is a real multinomial in {Xi(l),..., Xm(l)}. A function F : M p ( R r f ) m -> C is said to be bounded by a multinomial if there exists a multinomial Q such that | F | < Q. 11 T h e o r e m 1.3.1. There exists LQ S> 1 such that for every L > LQ, with /x n defined by (1.17) the following holds: For every s,X > 0, m G N , t G [0, oo) m and every F : M F ( M d ) m C bounded by a multinomial and such that N Q ( X ( - G Vp) = 0, (.1) [F{Y?)Y?{1)] £ N o [F{Yr)Ys{\) and (2) i j ^ F ( y P ) / { r n ( 1 ) > A } ^ E n o F{Yr)I{Ys{i)>x} (1.18) (1.19) The factors in Theorem 1.3.1 involving the total mass at time s, are essen-tially two ways of ensuring that our convergence statements are about finite mea-sures. In particular these factors ensure that there is no contribution from processes with arbitrarily small lifetime. As we have already noted in Section 1.1, it is often sufficient to prove results such as (1.18-1.19) for a suitable class of test functions. For any measure fi on R D and (j) : M R F —> C we define /j,(<j>) = fRd 4>dp. In particular, c (1.20) For k G [ - 7 r , 7 r ] r f , let (j>k{x) : Z D -> C be defined by <f>k(x) = eik'x. We indirectly prove the following Lemma in Chapter 7. L e m m a 1.3.2. Suppose that for every r > 2, every k G R ( R - 1 ) D , and every t G E, fJ-n EN0 r-1 3=1 (1.21) Then the conclusions of Theorem 1.3.1 hold. 12 Note that for t € (0, oo) r 1 we have, j=i • CaCr-1 r - l j=i r - l p(0)n p(0)n T€T0 j=l xj-.VnChxjeT[ntjl E (n«M*i)) E W(T) I r e r L n I J ( 5 t ) (1.22) r - 2 /nC2 CzC[ r - l r - 2 E E V ^ 2 E W(T). p(0)n This suggests that it might prove useful to examine the quantities 2^TerL £ j (x) ^ C H -1.4 T h e r -po in t f u n c t i o n s Definition 1.4.1 (2-point function). For ( > 0, n 6 N , and x £ Rd we define, tn(x;0 = C E W(T). (1.23) TeTn(x) We also define tn(x) = tn(x; 1). Definition 1.4.2 (Fourier Transform). Given an absolutely summable function f : l) —> K, we /ei /(A;) = 2~ x^ elfc'xf(x) (k € [—7r, 7T]'J denote the Fourier transform off-In [19] the authors show that if a recursion relation of the form n+l /n+i(fc;2) = E 9m(k;z)fn+i-m(k;z) + en+i(k;z) (1.24) m=l holds, and certain assumptions 5, D , i?, and G on the functions / . , g, and e. hold then there exists a critical value zc of z such that fn(k,zc) (appropriately scaled) converges (up to a constant'factor) to the Fourier transform of the Gaussian density as n —> oo . In Appendix A we extend this result (based on the ideas of [18]) by generalizing assumptions E and G according to a parameter p > 1, where the p = 1 13 case is that which is proved in [19]. In Section 3.2 we show that tn(k;() obeys the recursion relation n+l tn+i(k;() = Y ^m-i{k-,OCPcD(k)tn+i-m{k;C) + T?n+i{k; (), (1.25) m=l where Trm(x;() is a function that is defined in Section 3.2. After massaging this relation somewhat, the important ingredients in verifying assumptions E and G for our lattice trees model are bounds on nm using information about p(x) and t[(k;C,) for I < m. The quantities 7? m _i (&;() are defined using a technique known as the lace expansion. The lace expansion is discussed in Chapter 2 and it enables us to express 7 r m _ i in terms of Feynman diagrams, that can be bounded using (1.13) and bounds on ti(k;() for I < m. As in previous work already discussed, the critical dimension dc = 8 appears in this analysis as the dimension above which the square diagram PW(o) = Y, p(*)p(y - *)P(Z - y)p(z) (i-26) x,y,z converges. Ultimately we verify assumptions Ep and Gp for our lattice trees model with p = 2 and thus the results of Appendix A are valid. The parameter ( appears in (1.4.1) as an additional weight on bonds in the backbone of trees T € Tn(x). Those trees are already critically weighted by pc (a weight present on every bond in the tree) as described by Definition 1.2.7 and (1.12) and exhibit mean-field behaviour in the form of Theorem 1.2.9. One might therefore expect a Gaussian limit for tn with C = 1. The following theorem follows from the induction approach of Appendix A , together with a short argument showing that the critical value of ( obtained from the induction is (c = 1. T h e o r e m 1.4.3. Fix d > 8, t > 0, 7 e ( 0 , l A ^ ) and 5 G (0, ( l A ^ - 7 ) . There exists a positive LQ = Lo(d) such that: For every L > Lo there exist positive A,v depending on d and L such that hbz) - + 0 ( ? ) + 0 ( ^ ) + 0 f c v W ) • with the error estimate uniform in jfc € E d : k2 < c l °g (L w t J ) | ? where C = C(^) and the constants in the second and third error terms may depend on L. Based on Theorem 1.4.3 and (1.22), we choose C2 = va2 in (1.14). 14 < Figure 1.4: The unique shape a(r) for r = 2,3 and the 3 shapes for r = 4. Def in i t i on 1.4.4 (r-point funct ion). For r> 3, he N ^ " 1 ) and x e R ^ " - 1 ) we To state a version of Theorem 1.4.3 for r-point functions for r > 3 we need the notion of shapes. A shape is an abstract set of vertices and edges connecting those vertices. The degree of a vertex v is the number of edges incident to v. Vertices of degree 1 are called leaves. Vertices of degree > 3 are called branch points. We are primarily concerned with shapes that have a binary tree topology as follows. There is a unique shape for r = 2 consisting of 2 vertices (labelled 0, 1) connected by a single edge. The vertex labelled 0 is called the root. For r > 3 we have 115=3(2.7 — 5) r-shapes obtained by adding a vertex to any of the 2(r - 1) - 3 edges of each (r — l)-shape, and a new edge to that vertex. The leaf of this new edge is labelled r - l . Each r-shape has 2r — 3 edges, labelled in a fixed but arbitrary manner as 1,. . . , 2r — 3. This is illustrated in figure 1.4 which shows the shapes for r = 2,3,4. Let E r denote the set of r-shapes. We make the edges in a 6 S r directed by directing them away from the root. By construction each r-shape has r — 2 branch points, each of degree 3. Thus the unique shape for r = 3 (Figure 1.4) has 3 leaves and 1 branch point. Given a shape a € S r and k 6 R ( r - 1 ) d we define «(a ) € R ( 2 r _ 3 ) d as follows. For each leaf j in a (other than 0) we let Ej be the set of edges in a of the unique path in a from 0 to j. For / = 1,... , 2r — 3, we define define W(T). (1.28) r e r f i ( x ) r - l (1.29) 3=1 Next, given a and s € M; .(2r-3) we define c;(a) € R; ,(r-i) by 4- 4-(1.30) 15 Finally we define = {3 :Z{a) = i}. (1.31) This is an r - 2-dimensional subset of M+ r ~ 3 ^. For r = 3we simply have Ri(a) = {(s,ti-slt2-a):se[0,ti/\t2]}. (1.32) It is known [1] that for r > 2, 0 < <i < ti • • • < t r _ i and (j>k(x) = elk'x, E N 0 r - l 3=1 aezrjRi(a) 1=1 2r-3 o e 2d ds. (1.33) For v = 3 this reduces to * l A ' 2 (ki+k2)2s fc?(t!-.) fc|(t2-a) 2d C?5. (1.34) T h e o r e m 1.4.5. F i z d > 8 , 7 e (0,1A ^ ) and r5 € (0, (1A ^ ) - 7 ) . T/iere exists Lo = -ko(d) ^ 1 such that: for each L > LQ there exists V — V(d,L) > 0 such that for every t 6 (0, c o ) ^ - 1 ) , r > 3, R > 0, and \\k\loo < R, L"tj I ^ va2n = n r - 2 y r - 2 ^ 2 r - 3 - 2r-c e - ^ ^ r - ' d s - r - O ( 4 n5 (1.35) where the constant in the error term depends on t, R and L. Based on Theorem 1.4.5 we choose C\ = V~XA~2 and C 3 = VAp(0) in (1.14) and (1.17). Theorem 1.4.5 is proved in Chapter 4 using the lace expansion on a tree of [21]. The proof proceeds by induction on r, with Theorem 1.4.3 as the initializing case. Lattice trees T e 7fi(x) can be classified according to their skeleton (recall Definition 1.2.1). Such trees typically have a skeleton with the topology of some a G S r and the lace expansion and induction hypothesis combine to give the main contribution to (1.35). The relatively few trees that do not have the topology of any a € S r are considered separately and are shown to contribute only to the error term of (1.35). Theorems 1.4.3 and 1.4.5, combined with the observations (1.22) and (1.33) verify the conditions of Lemma 1.3.2. Thus assuming Lemma 1.3.2, Theorems 1.4.3 and 1.4.5 are sufficient to prove the main result, Theorem 1.3.1. Lemma 1.3.2 and Theorem 1.3.1 are proved in Chapter 7. 16 Chapter 2 The lace expansion The lace expansion on an interval was introduced in [5] for weakly self-avoiding walk, and was applied to lattice trees in [12, 13, 7, 11]. It has also been applied to various other models such as strictly self-avoiding walk, oriented and unoriented percolation and the contact process. The lace expansion on a tree was introduced in [21] and was applied to networks of mutually avoiding, SAW joined with the topology of a tree. Our analysis requires some modifications to the definitions of connected graph and lace given in [21]. In this chapter we follow [21] with some small modifications and define the notion of a lace on a star-shaped network. In Section 2.1 we introduce our terminology and define and construct laces on star shaped networks of degree 1 or 3. In Section 2.2 we analyse products of the form OsteA^l + Ust] a n a " perform the lace expansion in a general setting. Such products will appear in formulas for the r-point functions in Chapters 3 and 4. 2.1 G r a p h s a n d L a c e s Given a shape a € S r , and n G N 2 * " - 3 we define Af = Af(a,n) to be the skeleton network formed by inserting rij — 1 vertices into edge i of a, i = 1,. . . , 2r — 3. Thus edge i in a becomes a path consisting of rij edges in Af. A subnetwork AA C Af is a subset of the vertices and edges of Af such that if uv is an edge in AA then u and v are vertices in Ad. F ix a connected subnetwork AA C Af. The degree of a vertex v in AA is the number of edges in AA incident to v. A vertex of A4 is a leaf (resp. branch point) of AA if it is of degree 1 (resp. 3) in AA. A path in AA is any connected subnetwork A4i C A4 such that Mi has no branch points. A branch of AA is a path of A4 containing at least two vertices, whose two endvertices are either leaves or branch points oi AA, and whose interior vertices (if they exist) are not leaves or branch points of AA. Note that if b' € M\ C AA is a branch point of AA\ then it is also a branch point of Ad but the reverse implication 17 Figure 2.1: A shape a € S r for r = 4 with fixed branch labellings, followed by a graph T on J\f(a, (2,4,3,1,1)), and the subnetwork Ab(T). does not hold in general. Similarly if v € Mi is a leaf of M then it is also a leaf of Mi but the reverse implication does not hold in general. Two vertices s, t are neighbours in M if there exists some branch in M of which s, t are the two endvertices (this forces s and t to be of degree 1 or 3). Two vertices s,t oi M are said to be adjacent if there is an edge in M that is incident to both s and t. For r > 3, let 6 denote the unique branch point of M neighbouring the root. If r = 2, let b be one of the leaves of J\f. Without loss of generality we assume that the edge in a (and hence the branch in Af) containing the root is labelled 1 and we assume that the other two branches incident to b are labelled 2,3. Vertices in M may be relabelled according to branch and distance along the branch, with branches oriented away from the root. For example the vertices on branch 1 from the root 0 to the branch point (or leaf if r = 2) b neighbouring the root would be labelled 0 = ( l , 0 ) , ( l , l ) , . . . , ( l , n 1 ) = 6. Examples illustrating some of the following definitions appear in Figures 2.1-2.2. Def in i t ion 2.1.1. 1. A bond is a pair {s, t} of vertices in M with the vertex labelling inherited from j\f. Let E x denote the set of bonds of M. The set of edges and vertices of the unique minimal path in M joining (and including) s and t is denoted by [s,t]. The bond {s,t} is said to cover [s,t]. We often abuse the notation and write st for {s, t}. 2. A graph on M is a set of bonds. Let QM denote the set of graphs on M. The graph containing no bonds will be denoted by 0. 3. Let Tl — UM denote the set of bonds which cover more than one branch point of M. If r < 3 then Tl = 0 since in this case M Q M cannot' have more than one branch point. Let QjJ^ = {T G GM '• T D TZM = »- e- the set of graphs on M containing no bonds in Tl. 18 Figure 2.2: A graph T £ Q{Af) that contains a bond in Tl. The bond in ft appears darker. For simplicity, only the leaves and branch points of are explicit. 4- A graph V £ QM is a connected graph on AA if, as sets of edges, U stgr[s,*] = AA (i.e. if every edge of M is covered by some st £ T). Let QMn denote the set of connected graphs on Ai, and Q ^ , c o n = QJ^ I"1 GJA'• 5. A connected graph F £ QMn is said to be minimal or minimally connected if the removal of any of its bonds results in a graph that is not connected (i.e. for any st£T,T\st^ Qffl). 6. Given V £ QM and a subnetwork A C M we define = {st £ V : s, t £ A}. 7. Given a vertex v £ AA and T £ QM we let AV(T) be the largest connected subnetwork A of M containing v such that V\_A is a connected graph on A. Note that A could be a single vertex. In particular Av{$) — v. 8. Let £tf be the set of graphs T £ Qjj1 such that Ab(T) contains a vertex adjacent to some branch point b' ^ b of Af. Note that this set is empty if r < 3, since then Af contains at most one branch point. Note also that if b is adjacent to another branch point of Af, then even 0 £ since ,4.6(0) = b. The existence of A,(r) is clear since if A\ and A2 are connected subnetworks of AA containing v such that r|^ . is a connected graph on Ai,. then A = A\ U A2 also has this property. For A £ {0,1,2,3}, n £ N A let <SA(n) denote the network consisting of A paths meeting at a common vertex v, where path i is of length rij > 0 (contains rii edges). This is called a star-shaped network of degree A . By definition of our networks Af(a,n), with n £ N 2 7 - - 3 , for any T £ Qj/1 \ E^, Ab(T) contains at most one branch point and is therefore a star-shaped subnetwork of degree 3 (if it contains a branch point), 2, 1, or 0 (if Ab{T) is a single vertex). Since it contains no branch point, a star shaped network <S1(n) of degree 1 may be identified with the interval [0, n], and we can write S[0, n] for Sl(n). Similarly a star-shaped network S2(n\, n 2 ) 19 Figure 2.3: Two graphs on each of S1(8) and <S3(4,4,7). The first graph for each star is connected. The second is disconnected. The connected graph on <S3(4,4, 7) is a lace while the connected graph on Sl(8) is not a lace. of degree 2 may be identified with the interval [—ri2, ni] and we can write <S[—TJ.2, ni] ' for (S 2(ni, 712). Our main interest will be connected graphs on star-shaped networks. Figure 2.3 shows graphs on each of <S1(8) and <S3(4,4,7). The first graph in each case is connected, while the second is disconnected. Def in i t i on 2.1.2. Fix a connected subnetwork A4 C j\f'. Let V € QM^',con be given and let v be a branch point of M. If M. contains no branch points then we let v be one of the leaves of M.. Let C T be the set of bonds Sit% in T which cover the vertex v and which have an endpoint (without loss of generality U) strictly on branch A4e (i.e. ti is a vertex of branch A4e and ti ^ v). By definition of connected graph, Tg will be nonempty. From Tve we select the set Yg,max for which the network distance from ti to v is maximal. We choose the bond associated to branch A4e at v as follows: 1. If there exists a unique element ofFe'max whose network distance from S j to v is maximal, then this S j t j is the bond associated to branch M.e at v. 2. If not then the bond associated to branch Me at v is chosen (from the elements Y<v,max wflose n e i w o r k distances from Si to v are maximal) to be the bond Siti with Si on the branch of highest label. Def in i t i on 2.1.3 (Lace) . A lace on a star shape S = <SA(n), with n G N A , A G {1,3} is a connected graph L € such that: • If st € L covers a branch point v of S then st is the bond in L associated to some branch Se at v. • If st € L does not cover such a branch point then L\st is not connected. 20 We write C(S) for the set of laces on S, and £ (<S) for the set of laces on S consisting of exactly N bonds. Note also that the definition of a lace can be extended to star-shapes of higher degree (e.g. see [21]) and even to more complex networks (for example networks with general tree topology). However we do not require such generality for our analysis. See Figure 2.3 for some examples of connected graphs and laces. We now describe a method of constructing a lace L r on a star-shaped network <S of degree 1,2 or 3. Note that the only (connected) graph on a star-shape of degree 0 (i.e. a single vertex) is the graph F = 0 containing no bonds, and we define 1/0 = 0. Def in i t ion 2.1.4 (Lace const ruct ion) . Let S be a star-shaped network of degree I, 2, or 3. In the latter case, b is the branch point, otherwiseformer b denotes one of the leaves of S. Fix F G ^ ^ ' C O N . Let F be the set of branch labels for branches incident to b. For each e in F, • Let s f i f be the bond in F associated to branch Se at b, and let be be the other endvertex of Se. • Suppose we have chosen {s\tl,... , s f i f } and that u ' = 1 [sf i f] does not cover be. Then we define t\+l = max{£ G Se : 3 s € Se, s tf such that st G F}, \ ) sf+1 = min{s G Se : stf+1 G F}, where max (min) refers to choosing t (s) of maximum (minimum) network distance from b. Similarly s <f,t if the network distance from t to b is greater than the network distance of s from b. • We terminate this procedure as soon as be is covered by Li\=l[s^tf], and set Lr(e) = {sf «?,...,*?<?}. Next we define -L r = UeeFLr(e), and given a lace L € C(S) we define C(L) = {steEs\L:LLlJst = L} (2.3) to be the set of bonds compatible with L. In particular if L G C(S) and if there is a bond s't' G L (with s't' ^ st) which covers both s and t, then st is compatible with L. The following results are proved for star-shaped networks in [21] for the different notion of connectivity. The proofs presented here are very similar. (2.2) 21 P r o p o s i t i o n 2.1.5. Given a star shaped network S = SA(n), A G {1,3}, and a connected graph F G Qcon(S), the graph Lr is a lace on S. Proof. By construction, every branch of S is covered by L r so L r is a connected graph on S. Now suppose st G Lr covers the branch point (or leaf if A = 1) b of <S, with s G Se, t G Sei (where e' = e if 5 = b or t = b). Then st was chosen as the bond in F associated to Se or Se>, so in particular it is the bond in Lr associated to Se or Sei. Now if st G Lr does not cover b then s and t are on the same branch Se for some e and so st = s\t\ for some i. Now observe that if L r \ st is a connected graph on S then we would not have chosen s\t\ = st in the construction of L r - • P r o p o s i t i o n 2.1.6. Let F G g~n>con. Then L r = L if and only if L CF is a lace andF\L CC(L). Proof. If L r = L, then L is a lace by Proposition 2.1.5. By definition any bond st G T \ L that covers b is compatible with L since L r contains the bond s't' in T associated to each branch Se at b, and s't' is therefore also the bond in L U st associated to Se at b. Similarly if st G F \ L does not cover b then there are bonds in L r chosen from all bonds F to satisfy the optimal covering criteria (2.1). Therefore these same bonds satisfy those criteria when choosing from bonds in L r U st, so that LLUS* = L and st is compatible with L. For the reverse direction, let L C F be a lace and F\L C C(L). Assume that Lr + L. Then (a) there exists st G L r D (F \ L) or (b) there exists st G L n (T \ L r ) . For (a), if s i G L r D (T \ L) covers the branch point then by definition of L r it is the bond in F associated to some branch <Se. Therefore for any lace V C F, st is the bond in L' U st associated to <Se so st is not compatible with any lace L' C F. Since st G F \ L we have a contradiction. If st G L r D (F \ L) does not cover the branch point then st = s\t\ for some e,i. Then for this fixed e there is a smallest i such that s f t ? G L r fl (T \ L). Then this bond is not compatible with L and we again have a contradiction. For (b), since F \ L C C(L), we must have that every bond in F associated to a branch Se is in L. Since L is a lace, these are the only bonds in L which cover b and they are also in L r by definition. Therefore the st G L D (F \ Lr) must satisfy s,t G <Se, s,t ^ b. Since L is a lace, L \ st is not connected, and therefore since L r is a connected graph and st £ Lr there must exist s't' in L r H (T \ L) and by case (a) we have the result. • 22 Figure 2.4: A n illustration of the construction of a lace from a connected graph. The first fi gure shows a connected graph T on a star S^n^ n^ n3y The intermediate figures show each of the Lr{e) for e G F},, while the last figure shows the lace LY-Figure 2.5: Basic examples of a minimal and a non-minimal lace for A = 3. For the non-minimal lace, a removable edge is highlighted. See Figure 2.4 for an example of a connected graph T on a star-shaped network of degree 3, and its corresponding lace Lr-2.1.1 Classification of laces Def in i t i on 2.1.7 ( M i n i m a l ) . A lace on S is said to be minimal if the removal of any bond from the lace results in a disconnected graph on S. A lace L on a star shape <S of degree 1 or 2 is necessarily minimal by Defini-tions 2.1.3 and 2.1.1. For a lace on a star shape of degree 3 this need not be true. See Figure 2.5 for an example of a minimal and a non-minimal lace for A = 3. There is a more general version of the following Lemma for laces on star-shaped networks of higher degree, but we present only the results needed for our analysis. L e m m a 2.1.8. (a) For a star shaped network S of degree A G {1,2,3}, any min-imally connected graph F G Qcon(S) is a lace. 23 Figure 2.6: Basic examples of a cyclic and an acyclic lace. (b) For any non-minimal lace L G C{S3), there exists a bond st G L (that covers the branch point) such that L\st G £(S) and L\st is minimal. Proof. For (a), let Y G Gcon{S), and let b be as in Definition 2.1.4. Let st G T cover b and suppose s G <Sei and t G Se2 where Sei are branches of <S (we may have e\ = e2). If st is not the bond in Y associated to <Sei then T\st covers Ser Therefore if st is not the bond associated to either iS e i or Se2 then Y \ st covers <S so that Y is not minimal. By Definition 2.1.3 this is enough to prove (a). For (b), let L G C(S3) be non-minimal. Then there exists st G L such that L\st is connected. By Definition 2.1.3, st must be the edge in L associated to some branch e, and in particular it covers the branch point. Since «S3 is a star shape of degree 3 this means that L contains exactly 3 bonds covering the branch point. Now observe that L\st satisfies the definition of a lace, and contains exactly 2 bonds covering the branch point. It follows immediately that L \ st is minimal since a graph T with only 1 bond covering the branch point of S3 cannot be a connected graph on .5 3 . • As in part (b) of Lemma 2.1.8, a non-minimal lace contains a bond st that is "removable" in the sense that L \ st is still a lace. In general such a bond is not unique. One can easily construct a lace on a star shaped network of degree 3 for which each of the bonds sit\,... ,S3^3 covering the branch point satisfy L \ Sj i j G £(S). Def in i t i on 2.1.9 ( C y c l i c ) . A lace on a star shaped network <S3 is cyclic if the edges covering the branch point can be ordered as {sktk : fc = 1,.. . , 3}, with tk and sk+i on the same branch for each k (with S4 identified with S\). A lace that is not cyclic is called acyclic. See Figure 2.6 for an example of this classification. 24 2.2 T h e E x p a n s i o n Here we examine products of the form I l s t e E . ^ + Ust]- Following the method of [22] we can express such a product as ii[i+ust}= n [ i + ^ j - f n [i+ust])(i-n^+^V(2-4) Define K(M) = Ylst£EM\R.[l + Ust]- Expanding such a product we obtain, for each possible subset of \ a product of Ust for st in that subset. The subsets of E x \ 1Z are precisely the graphs on AA which contain no elements of 7Z, hence K(M)= Y HU't> (2-5) r e c x * s t e T where the empty product Ylst€$ Ust = 1 by convention. Similarly we define AM)= Y nu*t (2-e) TeQj}n'con s t ^ If M is a single vertex then J{M) = 1. If S is a star-shaped network of degree 1 or 3 then E E n^ = E E n LeC(S) V^gcson: st€T LeC{S)steL VeQ%on: s't'er\L Ly — £ -^r = ^ O O = E E n u « E n Le£(5)s*6L r'CC(L)s't '6r ' N=l LeCN(S) steL s't'eC(L) (2.7) where the second to last equality holds since for fixed L, {r G Qgin : Lr = L} = { L U T ' : T' C C(L)} by Proposition 2.1.6. The last equality holds as in the discussion preceding (2.5) since expanding Y[s't'eC(L)[H~Us't'] we obtain for each possible subset of C(L), a product of Ust for st in that subset. Recursion type expression for K(Af) Recall that Af = Af{a, n) where a G E r and n G N 2 7 - - 3 , for some r > 2. If r = 2 then let 6 be the root of Af. Otherwise let b be the branch point neighbouring the root of Af. In each case let S^- be the largest connected subnetwork of Af containing b and no vertices that are adjacent to any other branch points of Af (5 -^ could be empty or a single vertex). Observe, that for any graph T G Qjf^\S^f.i the subnetwork Ab(F) contains no branch point of A/" other than b (if r > 3) and hence is a star shape of degree 0, 1 or 3. 25 Def in i t i on 2.2.1. If M. is a connected subnetwork of M then we define j\f\M. to be the set of vertices of M that are not in M together with the edges of M connecting them. In general (M \ M) U M contains fewer edges than j\f, and Af\M. need not be connected. However if M c 5^- then N\A4 has at most 3 connected components (at most 1 ifr — 2) and we write (Af\M)i, i = 1,2,3 for these components, where we allow (A/- \ M)i = 0. Definition 2.2.1 allows us to write K{M)= E Uu«+ E veg^\£b^^er resetter 3 E E n^n E n ^ + E I I « ACS^-. r e e f e r i = i r . e g - n ^ sHieri reefrster be A (2.8) S t ) where the sum over A is a sum over connected subnetworks of j\f containing b and no vertices adjacent to any other branch points of j\f. Some of the (N \ A)i may be a single vertex or empty and we define Er^ee^ FL^er, ^s»t» = 1- Defining (AO = E r e ^ Uster U,u we have K(M) E J(A)]lK((M\A)i) + E^(M). be A Depending on A/", the first term of (2.9) may be zero since «S -^ may be empty. The fact that for any A contributing to this first term, the subtrees (Af\A)i are of degree ri < r is what allows for an inductive proof of Theorem 1.4.5. If r = 2 then ftf contains no branch point. In this case we may identify the star-shaped network 5 1 (m) with the interval [0, m] and (2.8)-(2.9) reduce to K([0,n}) = E J([0,m})K([m + l,n}), (2.10) which is the usual relation for the expansion of K(•) on an interval for this notion of connectivity (see for example [11]). Otherwise b is a branch point of M and we let K($) = 1, and Ii = I{(Af) be the indicator function that the branch i is incident to b and another branch point 6j. Therefore for a fixed network M,ni — 2fy = rij — 2I2 (AO is equal to either 712 — 2 (if branch 2 is incident to b and another branch point bi) or Then (2.8)-(2.9) give *W= E E J(SA(m))]lK((Af\SA(m))i) + E^(Af), (2.11) mi<ni m2<n2-2l2 i = 1 "13 < 713 - 2/3 26 where <SA(m) is a star-shaped network satisfying ( {b} , if rh = 0 ) 5 3 (m) , if mi ^ 0 for all i 1 <S[0, mj] , if rrii ^ 0, and rrij = 0 for j ^ i S[—TUj, mi] , if j > i, rrij / 0, rrii ^ 0, and = 0 for k / (2.12) In the case where there is another branch point be that is adjacent to b in A/" (so that r i2 or 713 is 1), the sum over at least one of m 2 , m 3 in (2.11). However note that this case contributes to the term E^(J\f), as required. The combinatorial analysis of • E^(Af) and • the contribution to (2.4) from graphs containing a bond in TZ is difficult and we postpone it until Chapter 6. Neither term appears in our analysis of the 2-point function in Chapter 3. 27 Chapter 3 The 2-point function 3.1 O r g a n i s a t i o n In this chapter we prove Theorem 1.4.3 using an extension of the inductive approach to the lace expansion of [19]. The extension of the induction approach is described and proved in a general setting in Appendix A . Broadly speaking there are two main ingredients involved in applying the results of Appendix A . Firstly we must obtain a recursion relation for the quantity of interest, the Fourier transform of the 2-point function, and massage this relation so that it takes the form n+l fn+i(k;z) = ]P gm(k;z)fn+i-m(k;z) + en+l(k;z), with m = i t 3 - 1 ) / o ( M = l , h(k;z)=zD(k), e i ( * ; z ) = 0 . Secondly we must verify the hypotheses that certain bounds on the quantities fm for 1 < m < n appearing in (3.1) imply further bounds on the quantities gm,em, for 2 < m < n + 1. This second ingredient consists of reducing the bounds required to diagrammatic estimates, and then estimating the relevant diagrams. In Section 3.2 we prove a recursion relation of the form (3.1) for a quantity closely related to the Fourier transform of the 2-point function. In Section 3.3 we state the assumptions of the inductive approach for a specific choice of parameters corresponding to our particular model. In Section 3.4 we reduce the verification of these assumptions to proving a single result, Proposition 3.4.1. Assuming Propo-sition 3.4.1, the induction approach then yields Theorem 3.4.3, which we show in Section 3.5 implies Theorem 1.4.3. The diagrammatic estimates involved in proving Proposition 3.4.1 provide the most model dependent aspect of the analysis and these are postponed until Chapter 5. 28 Figure 3.1: The first figure is of a lattice tree T € 7^(0, x) for n = 17. The second figure shows the backbone which is also a (self-avoiding) walk OJ, while the third shows the branches emanating from the backbone, which are also mutually avoiding lattice trees RQ, ..., RN. 3.2 R e c u r s i o n r e l a t i o n for t he 2-po in t f u n c t i o n Recall Definitions 1.2.4, 1.2.6, and 1.2.8. Also recall from Definition 1.4.1 that the two point function is defined as tn{x) = C £ W(T). (3.2) TeTn{x) Every tree T € Tn(x) consists of a unique backbone (which is a self-avoiding walk) OJ connecting 0 = OJ(0) to x = oj(n) that contains n bonds, together with branches emanating from each vertex in the backbone. The branches emanating from the backbone vertices are themselves lattice trees Ro,... ,Rn, and by the definition of lattice tree (applied to T) they must be mutually avoiding. Since each Ri contains the vertex oj(i), the mutual avoidance of the Ri incorporates the self-avoidance of the backbone OJ. See Figure 3.1 for a pictorial view of this discussion. Let Ust = U{Rs,Rt) = {-]; ******** (3.3) I 0, otherwise. Then r io<s<t<n[ l + Ust] is the indicator function that all the Ri avoid each other. Summarising the above discussion and using the fact that the weight W(T) of a tree factorises into (bond) disjoint components (see Definition 1.2.7) we can write, tn(x-o = c E ! (3.4) w : 0 • £ W(R0) £ WiRt)--- YI II i^ + Ustl tfoerw(o) Ri€TuW Rn£Tu,n) 0<s<t<n 29 where the first sum is over simple random walks of length n from 0 to x. To simplify this expression, we abuse notation and replace (3.4) with tn(x;() = C Y w^)f[ E ' W W II V + Ust]. (3-5) w : 0 - > x , i=0Ri€Tu(i) 0<s<t<n \ui\ — n Recall Definition 2.1.1 and the discussion following it. The set of vertices [0,n] corresponds to the set of vertices of Af(a,n), where a is the unique shape in £ 2 -Since this J\f contains no branch points, we have TZ = 0 and therefore from Section 2.2 we have Uo<s<t<n i1 + u*t] = K{M) = K([0,n]). Hence n tn{xyO = C Y W(")H E W(Ri)K([0,n]). (3.6) \u\ = n Definit ion 3.2.1. For m > 0 we define m *m{x;Q = C Y ^ M j J Y W(Ri)J([0,m\). (3.7) w : 0 ->• x i = 0 fii£7L,(i) \us\ = m Note that form = 0 this is simply J2Ro€7-0 W(Ri) = p(0) ifx = 0 and zero otherwise. Definit ion 3.2.2. Let f,g. We define the convolution of absolutely summable func-tions f and g to be the function (f*g)(x) = Yf(y^-y)- (3-8) Clearly, by the substitution u = x — y we have (f * g) = (g * /). Moreover since T,y,z&z^\f(y)9(z-y)h(x-z)\ < co by Fubini, (f*{g*h)){x) = ((f*g)*h)(x), and we can do pairwise convolutions in any order. The following recursion relation is the starting point for obtaining a relation of the form (3.1). Proposit ion 3.2.3. n tn+i(x;Q = ^ ( 7 T m * C p c D * i n _ m ) ( a ; ; C ) + 7 r n + i ( x ; C ) + p(0) (CPcD*i n ) (a ; ;C) . (3.9) 30 Proof. By definition n+l tn+l(x;0 = C + 1 E W(")H E W(Ri)K([0,n + l]). (3.10) \u\ = n + l . Equation (2.10) gives n K([0, n + 1]) = K([l,n + 1]) + E J([°> m])K([m + 1,n + 1]) + J([0,n + l]). (3.11) m=l Putting this expression into equation (3.10) gives rise to three terms which we consider separately. 1. The contribution from graphs for which 0 is not covered by any bond: For this term we break the backbone from 0 to a; (a walk of length n + l ) into a single step walk and the remaining n-step walk as follows. n+l C N + 1 E ^ M l l E ^ ( ^ [ l . n + l] w:0->x, «=0 Bi£T(u(i)) \u\=n + l = E w(Ro) E E W w i ) * (3.12) |«i| = i n+l E CWi^H E W(Ri)k[l,n + l), w2:y->x, i=l RiGT(w 2(i-l)) JOJ2| = n where K[l, n + l] depends on Rx,..., Rn+i but not RQ. Therefore using the substitutions R'j = Rj+x this is equal to p(°) E E CW(OJX)X ye^D wi:0-*y, M = i E C n^(^)fl E W{Xj)K[0,n] |W2| = n • = P(0) E Pc(D(y)tn(x-y;0 y€&D = p{0)pc((D*tn)(x). 31 2. The contribution from graphs which are connected on [0,n + 1]: n+l C + l Y W(u,)j[ Y W(Ri)J([0,n + l])=-Kn+1(x;O (3.14) |w| = ra + 1 3. The contribution from graphs which are connected on [0, m] for some ra 6 {1 , . ' . . , n}: For this term we break the backbone from 0 to x (a walk of length n + 1) up into three walks, of lengths ra, 1, n — m respectively C " + 1 E ^ M l l E W(Ri)J2J[0,m]K[m + l,n + l] co-.O^x, ' i=0 RieTuii) m=l |w|=n + l n / m \ = E E E E r ^ i ) n E J M X m=l « v c i : o - » u , V^O^ieTL^i) / |cui j = m E C ^ ( ^ 2 ) X |w2| = l . £ C N _ M W > 3 ) ( fi E ^ ) W [ r a + l , n + l ] . = n — m (3.15) Now [0,ra] and [m + l , n + 1] are disjoint, so J([0, ra]) and K([ra + l , n + 1]) contain information about disjoint subsets of {B4 : i G '{0, . . . , n + 1}}. Using 32 the substitutions i?' = Rj+m+i this is equal to: E E E E CWCi) Ml E W{R4)]J[0,m]x t m=l u v wi : 0->u, V = 0 / i i G r " l ( i ) / / n—m \ pcC.D(<; - u) E C " " m W ( w 3 ) II E W ^ K ^ n ~ m l 1^ 31 = ra — m n = E EE 7 r m(u ;OVcC,D[v - u)tn-m{x -v\Q m=l u v n = E ^m * pc<^D * in-m^ (x> ^ • 771=1 (3.16) • Dividing both sides of Equation (3.9) by p(0) and taking Fourier transforms we get P(0) ^m(fc;C) „ , m A „ f^n^n-mjk-X) , ^ W l ^ C ) , , n x . r v i / ' " ^ 0 ^(orp(0)c^(")~7(oT~+~7^ (3.17) We now massage (3.17) into the form (3.1) required for the analysis of Ap-pendix A . Def in i t i on 3.2.4. For fixed ( > 0, define 1) z = p(0)(pc. %) fo(k",z) = 1, fi(k;z) = g\{k\z) = zD(k), and ei{k;z) — 0. 3) For n>2, fn(k;z) = p(0) ' p(0) en(k;z) = gn_i(k;z) h(k;Q P(0) zD(k) + ^n(fc;C) p(0) ' (3.18) We note from (3.17) with n = 0 that since £()(£) = p(0)/ x =o,we have to(k) = p(0) and p(0) P(0) (3.19) 33 Therefore for n > 2 For n > 3 this is tu \ tu \ ? [ ! ( * i O , nn{k;() en(k;z)=gn„1(k;z)-^ + - M - . en(k;z) = ^-2^°zD(k)^P- + M k ' X ) (3.20) (3.21) P(o) • v ' p(0) p(0) Lemma 3.2.5. T/ie choices of fm, gm, em above satisfy Equation (3.1). Proof. The case n = 0 is trivially true by definition of /o, / i , g\ and e i . We use (3.19-3.20) for the case n — 1 so that, n+l E 9m{k;z)fn+i-m{k;z) +en+i{k;z) m=l = gi(k;z)fi(k;z) + g2{k;z)f0{k;z) + e2(fc;z) *i(fc;C) = zD(k)zD(k) + ni^PzD(k) + zD{k) p{0) p(0) - 3l>(fc) + P(0) (3.22) - p(0) (^*> + *^*> p(o) + P (0) = *2(fc;C) P(0) ' by (3.17) for n = 1. For n > 2, 71+1 E 9m(fc;2)/n+l-m(fc;2) + e n + i (fc;z) = gi(k;z)fn(k-z) + gn{k;z)fi(k;z) + gn+i{k; z)f0{k; z) + en+i(k;z)+ n - l E ^)/i+i-m(fc; m=2 = *D(k)^£P- + ^^-zD(k)zD(k) + ^P-zD(k)+ (3.23) P(0) 7r n _i(A;;C) p(0) zD(k) P(0) *i(fc;C) P(0) zD(fc) P(0) ^n+i(fc;C) P(0) n - l E m=2 7rm-^i(A;; C) tn+1_m(k; C) P(0) P(0) 34 The second term cancels with the second part of the fourth term. The last term added to the third term and the first part of the fourth term gives m=l which appears on the right side of (3.17). The remaining terms here are the remain-ing terms on the right side of (3.17), hence by (3.17) the entire quantity is equal to = fn+1(k;z) as required. ' • 3.3 A s s u m p t i o n s o f t he i n d u c t i o n m e t h o d The induction approach to the lace expansion of [19] is extended in Appendix A with the introduction of two parameters 6 and p* and a set B C In this chapter we apply the extension with the choices 6 — ^ p , p* — 2, B = {2} and we define — d p = L = L 2 . The induction method is discussed thoroughly in Appendix A, and so we simply restate the assumptions in this section, and verify them in the next section. We have already shown in Section 3.2 that for our choices of fm,9m,em as given in Definition 3.2.4, n+l fn+i(k;z) = Y 9m(k;z)fn+1-m(k;z) +en+i(k;zy (n > 0), (3.25) m=l with fo{k;z) = 1. Assumption S. For every n € N and z > 0, the mapping k i-> fn(k; z) is symmetric under replacement of any component ki of k by — /CJ, and under permutations of the components of k. The same holds for en(-;z) and gn(-;z). In addition, for each n, \fn(k; z)\ is bounded uniformly in k e [—TT, ir]d and z in a neighbourhood of 1 (which may depend on n). Assumption D . We assume that fi(k;z) = zD{k), ei(k;z)=0. (3.26) In particular, this implies that g\(k\z) = zD(k). Define a(k) = 1 — D(k). As part of Assumption D, we also assume: (i) D is normalised so that D(0) = 1, and has 2 + 2e moments for some e > 0, i.e., \x\2+2eD(x) < oo. (3.27) 35 (ii) There is a constant C such that, for all L > 1, Halloo < CL~\ a2 = a2L<CL2, (3.28) (iii) There exist constants 77,ci,C2 > 0 such that ClL2k2 < a(k) < c2L2k2 (\\k\lao < L" 1 ) , (3-29) a ( f c ) > » / (Hfclloo > L'1), (3.30) a(k)<2-n {ke[-TT,n]d). (3.31) For h : [—n, ir]d —> C, we define (3.32) The relevant bounds on / m , which a priori may or may not be satisfied, are that \\D2fm(-;z)\\2 < - ^ r , \fm(0;z)\<K, \V2 fm(0; z)\ < Ka2m, (3.33) L a m 4 for some positive constant K. We define B = L~L (3.34) The bounds in (3.33) are identical to the ones in (A. 13), with our choices if p* — 2, B = {2}, and 9 = 4=±. Assumption E. There is an Lo, an interval / C [1 — a, 1 + a] with a G (0,1), and a function K i - > Ce(K), such that if (3.33) holds for some K > 1, L > Lo, z £ I and for all 1 < m < n, then for that L and z, and for all k G [—7r, ir]d and 2 < m < n +1, the following bounds hold: |em(*;z)| < CeiK)?™,-^, \em(k; z) - em(0;z)| < C e ^ o W K ^ . (3-35) Assumption G. There is an Lo, an interval J C [1 — a, 1 + a] with a € (0,1), and a function K Cg(K), such that if (3.33) holds for some K > 1, L > Lo, z £ i and for all 1 < m < n, then for that L and z, and for all A; G [—7r, 7r]d and 2 < m < n +1, the following bounds hold: \9m(k;z)\ < Cg(K)Bm-d-^, |V 2 5 m (0;^)| < Cg(K)a20m^, (3.36) \dzgm(0;z)\<Cg(K)8m-d-^, (3.37) \gm(k;z)-gm(0;z)-a(k)a-2V2gm(0;z)\ < Cg{K)8a{k)1+*'m'^', (3.38) with the last bound valid for any e' G [0,1 A (^)) . 36 3.4 V e r i f y i n g a s s u m p t i o n s Assumption S: The quantities fn(k; z), n = 0,1,. . . are (up to constants), Fourier transforms of tn(x,Q, which are symmetric by symmetry of D. Hence the /„ have all required symmetries. Similarly 7rm(x,£) are symmetric by symmetry of D, so that the quantities gn,en also have the required symmetries. Now /o = 1 is trivially uniformly bounded in k and z < 2. Furthermore for n > 1, using the bound F i l l + Ust] < 1 in (3.5) we obtain E x 0 < ((pc)np(0)n+1 ExD^(x) = (CPc)np(fj)n + 1, where denotes the n-fold convolution of D(»). Therefore for n>l, \ fn(k,z)\ < ^°ffiC) < (CPcp(0))n = zn so that /„ is bounded uniformly in k G [—7r, ir]d and z in a neighbourhood of 1 and therefore satisfies the weak bound of Assumption S. Assumption D: By Definition 3.2.4 we have fi(k,z) = zD(k) and e\ = 0. Addi-tionally, all moments of D are finite, so choosing e = 1 ensures that (3.27) and 3.28) hold trivially (see Remark 1.2.5). The remaining conditions (iii) are verified by van der Hofstad and Slade in [19]. We therefore turn our attention to verifying assumptions E and G. Recall from Definition 3.2.4 and (3.20) that for n > 2, gn and en could be expressed in terms of the quantities 7 r m for m < n. In Chapter 5 we will prove the following proposition. Proposition 3.4.1 ( 7 r m bounds). Suppose the bounds (3.33) hold for some z* G (0,2), K > 1, L > Lo and every m < n. Then for that K, L, and for all z G [0, z*], m < n + 1 and q G {0,1,2}, where ( = p ^ p , the constant C = C(K,d) does not depend on L, m and z, and v > 0 is the constant appearing in Theorem 1.2.9. We choose v < 1 in (1.13) so that 2 - ^ > 1 and therefore / 3 2 - ^ < L~i. The proof of Proposition 3.4.1 involves reformulating irm in terms of laces and estimating Feynmann diagrams corresponding to those laces. For now we concentrate our efforts on verifying assumptions E and G assuming Proposition 3.4.1. Assumption E: Suppose there is some z* G (0,2), K > 1, L > LQ such that (3.33) holds for all m < n. Let z G [0, z*}. Recall that e\(k\ z) = 0 and observe from (3.20) (3.39) 37 t h a t \e2(k;z)\ = P(0) P(0) < z 7fi(fc;0 P(0) + vr2(fc;C) P(0) < (3.40) P(0) + P(0)2^ < 2— w h e r e w e h a v e a p p l i e d P r o p o s i t i o n 3.4.1 w i t h |7rm(A;;()l < E x l^mO^OI) a n d h a v e a l s o u s e d p(0) > 1. S i m i l a r l y f o r 3 < m < n + 1, \em(k', z)\ — 7ri(fc;C) , 7Tm(fc;C) P(0)2 + P(0) < p v u r P W i / 9 (0 ) 2 (m-2) d f i •zC(K)B'-T + p{0)m 2 d-4 m 2 T h u s w e h a v e o b t a i n e d t h e f i r s t b o u n d o f A s s u m p t i o n E . I t f o l l o w s i m m e d i a t e l y t h a t C'{K)d2-^ \em{k;z) - e m ( 0 ; z ) | < (\em{k;z)\ + \em(0;z)\) < m 2 (3.42) f o r a l l m > 2. B y (3.30) t h i s s a t i s f i e s t h e s e c o n d b o u n d o f A s s u m p t i o n E f o r Halloo > L~l. T h u s i t r e m a i n s t o e s t a b l i s h t h e s e c o n d b o u n d o f A s s u m p t i o n E f o r Halloo < f o r w h i c h w e u s e t h e m e t h o d o f [21]. L e t h : Zd —> E b e a b s o l u t e l y s u m m a b l e , a n d s y m m e t r i c i n e a c h c o o r d i n a t e a n d u n d e r p e r m u t a t i o n s o f c o o r d i n a t e s . N o w h(k) - h{0) < h(k)-h(0)-^2h(0) 2d + ^ V ^ ( O ) 2d Y,[ccB(k-x)-l-^Yix2i)h{x) 2d + 2d V2 / i ( 0 ) (3.43) B y s y m m e t r y w e h a v e t h a t 1 1 - £ \x\2h(x) = - £ £ x i h ( x ) = E X ) H ( X ), (3.44) i—\ x w h i c h i m p l i e s — ^ j -V 2 / i ( 0 ) = Ylx 2^?i=i(kjX j)2h(x). O n t h e o t h e r h a n d i f i ^ j t h e n . E s X{Xjh(x) = 0, s o t h a t E x ( ^ ' X)2HX) a l s o e q u a l s Y,x E f = i ( ^ * a ; * ) 2 M : c ) - T h u s w e c a n r e w r i t e (3.43) a s h(k) - h(0) < £ ^cos(fc • x) - 1 + i(fc • x)2^ h(x) + 2d V2h(0) (3.45) 38 We claim that there exists a constant c, such that for all 77 € [0,1], | cos(i) -1 + \t2\ < c i 2 + 2 j ? . To see this note that f o r > 1 the left hand side is bounded above by 2 + \t2 < \t2 < §rj 2 + 2 j?. For \t\ < 1 the left hand side is bounded above by 0 0 t2n t2(n-l-n) k ^ k <2"»! k ^ (3.46) where the constant is independent of 77. This verifies the claim. Putting this result into (3.45) we get h(k)-h(0) <C^2\(k-x)2+2nh(x)\ + Jfcp 2d V2h{0) (3.47) In particular if we choose 77 = 0 then (3.47) becomes d h(k)-h(o) < C ^ ^ ( ^ ) 2 I M X ) I + %v2m x j=l <C\k\2Y,x)\Hx)\. (3.48) Now em(fc; z) — em(0; z) is equal to (gn-i{k\z) -gn-i{0;z)) ^ +gn-i(0\z) + p(0) p(0) P«>) By (3.47) with 77 = 0, and Proposition 3.4.1 with q = 1 we have that , o - 2 / 3 2 - ^ |5fm(fc;C) -S?m(0;C)l < C(i^)fc2 V , • m 2 Therefore |em(fc;z) — em(0;z)| is bounded above by \gm-i(k;z) -gm-i(0;z C{K)B2-^ + |^ m _i(0;z) |C( i f ) fc / p(0) p(0) + C(.rY)fc: < p (Q) fc2a2 |gm-i (fc; z) - gm-i (0; z) I + \gm-\(0; z) |fc2cr2 + d - 6 ra 2 (3.49) (3.50) o282-f p(0)m 2 (3.51) Thus recalling that <7i(fc;z) = zD(k) we have 92 |e2(fc;z) -e 2 (0;z) | < C(K)B*-f , - 2 fcV P(0) za(fc) + zfc cr +. d _ 6 2 ~ (3.52) 39 For m > 3, recall that gm-i{k\z) = ^m~(20()fc;C)zD(k) which gives \gm-\{k;z) - o m _ i ( 0 ; 2 ; ) | < |S?m-2(*; 0 - 7Tm-2(0; C)|JD(0) + a(fc)|7r m _ 2 (0; C)| < p(0) L C(K)k2a2B2-ir C(K)a{k)p2--d . d-6 ( m - 2 ) ~ + ( m - 2 ) T ~ ' (3.53) Therefore for m > 3, |em(A:;z)-em(0;z)| < C{K)R2~ + a{k)0 zk2o2(32-^ k2a + 2„2 , • d - 6 1 d - 4 1 , . d - 4 1 d - 6 \(m-2) 2 (m - 2) 2 (m — 2) 2 m 2 (3.54) Both (3.52) for m = 2 and (3.54) for m > 3 are bounded above by C ' ( K "]? ( 6 f c ^ for m 2 PHoo < £ - 1 by (3.29) and the^act that a2 ~ L 2 (see Remark 1.2.5). Assumption G: Suppose there is some z* e (0,2), K > 1, L > Lo such that (3.33) holds for all m <n. Let 2 € [0,2*]. As for Assumption E , we may apply Proposition 3.4.1 to obtain for 2 < m < n + 1 bm(fc;z)| = zD(k) 7Tm-l(fc;C) p ( o ) < zC(K)p2-ir • C'(K)/32-d p(0)(m - 1) V < d - 4 m 2 (3.55) which gives the first bound of Assumption G . For the second bound we note that by symmetry the first derivatives of nm and D vanish at 0. Hence for m > 2 fc=o \V2gm(0;z)\ = zD(k)-P(0) P(0) | V 2 7 r m _ i ( 0 ) + 7 r m _ x ( 0 ) V 2 D ( 0 ) | , z (c{K)B2-^a2 , C{K)P2-&-it A C'{K)B2^o2 p ( o ) \ d - 6 m 2 d - 4 m 2 d - 6 m 2 This verifies the second bound of Assumption G. Next for m > 2, we have that gm(k;z) = 7 f m - i ( f c ; C ) Z l f f l = ^ m p(0) " V z m _ 1 £ m _ i ( f c ; C A 0(fc) P(0) (3.56) (3.57) where ^ " ^ - T ^ does not depend on z (or (). Therefore . ! /5r m _i ( f c ;C)\ 5(fc) \dzgm(k;z)\ Hi:: yim—l D(k) P(0) P(0) d - 6 m 2 < (3.58) 40 which proves the third part of assumption G. Now for ||fe||oo > L1-, (3.30) applies and we have that for m > 2, \gm(k; z) - gm(0; z) - a(k)a~2V2gm{0; z)\ S • *=* + i + a { K ) *=* (3.59) m 2 m 4 m 2 61/ < a(fc) d-6 ' m 2 since a(k) > r), and where the constant depends on r/. This satisfies the final part of assumption G for \\k\loo > For ||fc||oo < £ - 1 > we again use the method of [21]. By the triangle inequality we bound \gm{k;z) - gm{0;z) - a(k)a'2V2gm{0;z)\ by \k\2 gm(k;z) -gm{0;z) - -^rrV2gm(0; z) 2d + (l-D{k))o-2-^ 2d |V25m(0;«)|. (3.60) Recall that for m > 2, gm(k; z) = ^yC^m * D)(k). On the first term we apply the analysis of the first term of (3.43), to the symmetric function irm * D. Choosing 77 = e' we see that the first term of (3.60) is bounded by zO\k\2+2<'Y\X\2+2e'\(*rn-l*D)(x)l (3.61) x with the constant independent of e'. We claim that (3.62) If e' = 1 then the bound (3.62) holds trivially. If e' < 1 then (3.62) is Holder's inequality with f(x) = \x\2^'\(nm^*D)(x)\1¥, g(x) = \(7rm^D)(x)\1^, I ± ^ + I ^ = 1. (3.63) Applying Proposition 3.4.1 with q = 0 gives X>m-1 * D)(x)\ < £ km-l(v)| E D(X -V)Z C { K ) ^ d • (3-64) m 2 We now apply Proposition 3.4.1 with q = 0,2 together with the inequality (a + b)4 < 8 (a 4 + b4) (obtained by squaring the inequality (a + b)2 < 2(a2 + b2) and 41 applying the same inequality again) to get E N 4 l (^ - i . * D)(*)\ <s(EM4K-i(y)lEp^ - v)+ * \ y x Ekm-i(y)iEk-yl4^ -y)) 2/ a; / < C ( £ lvlVm-lfo)l + E l^-l(2/)k4) o - 4 C ( K ) ^ - 1 f oAC(K)B l~^ a4C(K)B2-^ < d-8 "F d^4 m 2 m 2 < d - 8 m 2 (3.65) Note that we have used Remark 1.2.5 to obtain 2~2X \x\rD(x) < Car with the con-stant independent of L (it may depend on r). Putting (3.64) and (3.65) back into (3.62) we get Eki2+2£'i^-iwi<fcw2_6; 2 / ^ntis\ni-d-4 \ m 2 o^C(K)B l—d d - 8 m 2 < a 2 ( l + , ' ) C ( K ) / ? 2 - f (3.66) m 2 Combining (3.66) with (3.61) gives \k\2 gm(k; z) - gm{0; z) - ^ " V 2 5 m ( 0 ; z) 2 | i . | 2 U + e ' < < C(K)02-ir(a2\k\2) d - 6 r , m 2 C(K)32-^a(k)i+£' d - 6 ; m 2 (3.67) when ||fc|| < L - 1 . This satisfies the required final bound of Assumption G. It remains to verify this bound for the term inside the second absolute value in expression (3.60). For this term we write 2d a2 x and proceed as for the first term to obtain (3.68) l-D(k) _ \kf a* ~~2d < c\k\2+2£' E \x\2+2e'\D(x)\ < c\k\2+2e'L2^l+e'\ (3.69) 42 Together with Proposition 3.4.1 with q = 1 this gives (1-D(k))a-2-2d |V2<U0;z)| < C{K)d2-^a2 (\k\2L2)1+e' d-6 m 2 (3.70) which satisfies the required final bound of Assumption G for < L 1 . R e m a r k 3.4.2. We have actually verified slightly stronger statements than those of Assumptions E and G. For the purposes of proving Theorem 3.4-3 we were only required to verify the bounds of Assumptions E and G for z = z*, however, we proved that if the bounds (3.33) hold for some z* then the bounds of Assumptions E and G hold uniformly in z G [0, z*\. We have now verified that Assumptions S,D,E,G all hold provided Propo-sition 3.4.1 holds. Thus subject to proving Proposition 3.4.1, we may apply the induction method of Appendix A and obtain Theorem A.2.1 which for our model is the following. T h e o r e m 3.4.3. Fix d > 8, 7 G (0,1A and S G (0, (1A - 7 ) . There exists a positive LQ = Lo(d) such that: For every L > Lo there exist A',v,zc depending on d and L such that the following statements hold: (a) T ( k Zc 1 P (0) Al - — = Ae 2d 1 + 0 k' n" O 1 d-8 n 2 (3.71) with the error estimate uniform in {k G K D : 1 — D(k/Vva2n) < 1 l o g n } . (b) va2n l + O r d <r (c) for every p > 1, (d) The constants zc, A' and v obey < C d d . d-i L?n2p 2 (3.72) (3.73) 1 = £ 9m(0;zc), m=l 1 + E m = i e m ( 0 ; z c ) A' = v — — E m = l m f f ™ ( ° ^ c ) E ^ = 1 V 2 5 m ( 0 ; z c ) °~2 E m = l m ? m ( ° ; ^ ) ' (3.74) 43 The constants A',v satisfy A' = 1 + O [L 2 j , v = 1 + O [L 2 J . Also zc = To reiterate the induction method shows that (3.33) holds for all m, provided Proposition 3.4.1 holds. 3.5 P r o o f o f T h e o r e m 1.4.3 In this section we show that Theorem 1.4.3 follows from Theorem 3.4.3(a). Com-paring the two Theorems and setting A = A'p(0) (recall that Cc = p(o)Pc)i ^ *s dear that to prove Theorem 1.4.3 it is sufficient to prove the following two Lemmas Lemma 3.5.1. For d, 7, 8 and LQ as in Theorem 3.4-3, there exists a constant Co = Co(d, 7) such that = Ae~i' + 0 ( £ ) + C ( ^ ) + 0 ( _ J ^ ) , , 3 , 5 , with the error estimates uniform in {k G R d : \k\2 < Cblog(|_nrJ V I ) } . Lemma 3.5.2. The critical value C,c = pfyPc in Theorem 3.4-3 is 1. The significance of Lemma 3.5.1 is to incorporate the continuous time variable t into the asymptotic formula (3.71) and to present a more palatable region of E d on which the error estimates in are uniform. Proof of Lemma 3.5.L The statement is trivial for \nt\ = 0, so we assume that [nt\ > 1. Incorporating a time variable b y n H [nt\, k >->• k\J^- into (3.71), and using A = A'p(0) we have that r^ n ij (y^hf^! Ccj is equal to r-2T-n'Cc\ =Ae 2dn 1 y/vaz[nt\ ' « J \[nt\-^ (3.76) where the error estimate is uniform in k e Md : 1 - D ( V j L ^ j < 7 H " 1 logLntJ }> - (3.77) We claim that there exists a constant CQ such that {k : \k\2 < Co log([nrJ)} C H n j t . 44 Define GnMk:\\kU<^K I (3. 7 8) By (3.29) and using the fact that a ~ L, there exists C\ > 0 such that for k € G n >t, , - f l | - ^ ] < ^ . (3.79) 1 v ^ p j I - [nt\ v Now since WkW2^ < \k\2 and using the fact that a ~ L, there exists a constant Co < ^ such that {A; : |A ; | 2 <Colog(LnrJ )}cG n , t . (3.80) Then for k2 < C 0 log([niJ) we have 1 D I k ^ \ < Clk" < g i g o l Q g ( N J ) y/va2[nt\ I ~ [nt\ ~ [nt\ 7log(Ln*J) (3.81) < [nt\ Thus verifies the claim, and thus (3.76) holds with the error estimate is uniform in {k : \k\2 < Co log([nrJ)}. Since \nt\ < nt in the first error term of (3.76), and fc2[nt] k2t e 2dn — e 2<i we have proved Lemma 3.5.1. • The significance of Lemma 3.5.2 was discussed immediately before the state-ment of Theorem 1.4.3 in Section 1.4. Essentially, £ was a weight introduced so that we could apply the induction method of Appendix A . That £c should be 1 is intuitive since the lattice trees are already critically weighted (by pc) and this idea is the basis of the following proof. Proof of Lemma 3.5.2. The susceptibility, x(z) is defined as x(z) = YU(0;z) = J2 * n (Q ; C ) P(0) n HK ' x TeTN(0,x) (3.83) where £ = z P(0)Pc ' 45 Let z'c denote the radius of convergence of x(z)- By Theorem 3.4.3 there exists a zc > 0 (resp. ( c) such that c ^ E E wm^A, (3.84) x TeTn(x) so that v 7 x T e r „ ( i ) Thus the radius of convergence of x{0 (resp. x(z)) is Cc > 0 (resp. zc). Write J2x p(x) = limM->oo £ | x | < M p(x) a n d observe that £ N < M (|B|vi)«i-» Ma. It follows from Theorem 1.2.9 that J2XP(X) = °°- T h u s (3.85) K W n x T6T„(0 , x ) r v ' (3.86) Recall from (1.16) that P{T G Tn(0,x)) = Er6Tnp((0^) W ( T ) which implies that £ c < 1. 3.4.3 states that for every fc, X Setting = 0 we have . Then Theorem (3.87) A £ P ( T G T n ( 0 , * ) ) -+! , (3.88) and dividing (3.87) by (3.88) gives P ( T e T n ( 0 , x ) ) E u P ( T G T n ( 0 , W ) ) -» e 2 d . Let Z„ be Z d-valued random variables defined by P{Zn = x) = ^p^Jn^x)) Then (3.89) is the statement that E (3.89) . , (<>,«) E t t p(r€r„(o ,«) ) -E e ik-Z (3.90) for every fc, where Z ~ j\f(0,I,i). This is equivalent to Z , and thus for every R > 0 we have G S(0,i?) 1 —> P (Z G 5(0, J2)). (3.91) 46 where B(0, R) denotes the ball with centre 0 and radius R in (R , | • |). Choose Ro such that P(Z G B(0,R0)) > | . Then there exists an i V 0 = N0(Ro) such that for every n > No, Zn V 1 P l — ^ e B ^ R o ) ) ^ - . (3,92) , V<7 vn Therefore for every n> NQ, \x\<RoV(T2vn Applying (3.88) to the denominator, we find that there exists Ni > No such that for every n> Ni, ^ E P(TeTn(0,x))>-, i.e. YI P(T&Tn(0,x))>-. \x\<RoV(r2vn \x\<Ra\/a2vn (3.94) Bounding £r e T„(o,x) W(T) by p(z) = £ m E re r . f o , , ) ^ follows that In c We also have from (1.13) that, £ E - ( l / v ( i ) ^ - c ( L ' f i o ) n - ( 3 - 9 6 ) \x\<RoV<r2vn \x\<RoV(r2vn Thus from (3.96) and (3.94), £ < Cn for every n > no- This requires that (c > 1 and we have the result. • Assuming that Proposition 3.4.1 holds, we have now verified Lemmas 3.5.1 and 3.5.2, and hence we have proved Theorem 1.4.3. We postpone the proof of Proposition 3.4.1 to Chapter 5. 47 Chapter 4 The r-point functions We have shown Gaussian behaviour (Theorem 1.4.3) of the 2-point function with appropriate scaling in Chapter 3. We now wish to prove the analogous result for r-point functions, Theorem 1.4.5. The proof is by induction on r, with Chapter 2 already having verified the initializing case r = 2. We use the technology of the lace expansion on a tree of van der Hofstad and Slade [21] as expressed in Chapter 2, and prove the result, assuming certain diagrammatic bounds. The diagrammatic estimates are again postponed until Chapter 6. 4.1 P r e l i m i n a r i e s Recall from Definitions 1.2.3 and 1.4.4 that for fixed r > 2, fi £ Z ^ - 1 and 5c G K R - 1 , we have Tfi(x) = {reTo :x i eT n | , t = l , . . . , r - l } (4.1) and $(x)= J2 W{?)- (4-2) T€T f i (x ) For T G T(x), let T^X be the backbone in T from 0 to a;. Def in i t i on 4.1.1. A lattice tree B is said to be an (ri, x) bare tree if 1) B G 7fi(5) and 2) ^Z\B^Xi =B. We let B ( n , x ) denote the set of (ri, x) bare trees. If B G B ( n , x ) then we write TB = {T G 7a(x) : T^XI = B ^ X I , i 6 l , . . . , r — 1} for the set of lattice trees containing B as a subtree. 48 Since every T G Tj(x) has a unique minimal connected subtree (U^TU^) connecting 0 to the xi, i = 1,. . . , r — 1, we have = E E W(T). (4.3) B€B(n,x) TeTB Definition 4.1.2 (Branch point). Let B G B(n,x). A vertex x G B is a branch point of B if there exist i,j G {1, . . . , r — 1}, i ^ j such that Xi and Xj are distinct leaves (vertices of degree 1) of B and B^Xi C\B^Xj — B^x. The degree of a branch point x G B is the number of bonds {a, b} G B such that either a = x orb — x. As they are defined in terms of the leaves of B G B(h,x), branch points of B depend on B but not the set B(n, x) of which B is a member. In particular if B is also in B(n',x') then our definition gives rise to the same set of branch points. By definition, a branch point that is not the origin must have degree > 3. Definition 4.1.3 (Degenerate bare tree). For fixed r, a bare tree B G B(n,x) is said to be non-degenerate if B contains exactly r — 2 distinct branch points, each of degree 3, none of which is the origin. Otherwise B is said to be degenerate. We write Brj(n,x) for the set of degenerate trees in B(fi, x) and set B£,(h,x) = B(n,x)\BD(n,x). Clearly from (4.3) we have <rn(*) = E E W(T) + E E W{T). (4.4) B£BcD(n,x)T€TB B6B D (n ,x ) T€TB Definition 4.1.4. Let B G B(h,x). Two distinct vertices y, y* in B are said to be net-neighbours in B if the unique path in B from y to y* contains no other branch points of B other than y, y*. A net-path in B is a path in B connecting the origin or a branch point in B to a net-neighbouring branch point or leaf in B. Lemma 4.1.5. Fix r>2,he W"1, x G Z ^ " 1 ) . 1. If B G B£,(n, x) then B consists of 2r — 3 net-paths joined together with the topology of a for some a G S r. 2. If B G B£>(h, x) then B contains fewer than 2r — 3 nonempty netpaths and fewer than r — 2 branch points that are not the origin. Proof. Induction on r . For r = 2, there are no degenerate bare trees and the result is trivial. Suppose the result holds for all r' < r. 49 1. Let B G B ^ r i , x) . Then B contains r - 2 branch points, each of which is of degree 3, none of which is the origin. Let x ^ 0, xr-\ be the unique branch point in B net-neighbouring xr-\. Removing the netpath B^Xr_1 \ B^x, we have that x is a vertex of degree 2 in B* = B \ (B^Xr_1 \ B^x) and therefore B E B ( ( n i , . . . , n r _ 2 ) , {xi,...,£7—2)) contains r — 3 branchpoints, each of degree 3, none of which is the origin. Thus B* is nondegenerate. By definition of a netpath and the fact that x is not a branch point of B*, we see that B* contains two fewer netpaths than B. The induction hypothesis gives that B* consists of 2(r — 1) — 3 net paths joined together with the topology of a* for some a* G E r - i - Therefore B contained 2r — 3 netpaths joined together with the topology of a € XV-i, where a is the shape obtained by adding a vertex to the edge of a* corresponding to the unique net-path in B* containing x = and adding an edge to that vertex. 2. Suppose now that B G Bf)(n,x). If B contains no branch point other than perhaps 0, then trivially for r > 3, B contains fewer than 2r — 3 net paths. Otherwise we use the same decomposition as for part 1, and let the ^degree of the branch point x ^ 0 be I. If / = 3 then B* above is a degenerate bare tree and the result hold by induction. If I > 3 then B* contains one fewer netpath and the same number of branch points as B. By induction B* G B ( ( n i , . . . ,nr-2), (xi,... ,xr-2)) contains at most 2(r — 1) — 3 netpaths and ( r - l ) - 2 = r - 3 branch points that are not the origin. Therefore B contained at most 2r - 4 netpaths and r — 3 branch points that are not the origin. Def in i t i on 4.1.6. For a fixed shape a G S r and n G N 2 ^ - 3 we let J\f(a,n) be the abstract network shape obtained by inserting rij — 1 vertices onto edge j of a, j = 1,. . . , 2r — 3. Each edge j of a has two vertices in a incident to it. We define branch JVJ of Af to be the smallest connected subnetwork of Af that contains the vertices j\,J2-Let B G B£,(ii, x) . We say that B has network shape Af(a,n) if B and Af(a, n) are graph isomorphic and for each i the graph isomorphism maps leaf i of Af(a,n) to Xi. Fory = (yi,...,y2r-3)€Z'd(2r~3\ we define TM\a,n)(y) t o be the set of lattice trees T G 75 such that there exists x G Z d ( r _ 1 ) , i i G _ 1 and B G B ^ ( i i , x ) such that 1. Te T B , 2. B has network shape Af(a,n), and 50 / i x 3 I I Figure 4.1: A shape a G £ 4 with labelled edges, and a nearest neighbour lattice tree T G TM{afi)&) for n = (3,5,7,7,2), y = ((2, -1 ) , (-2, -3 ) , (2,3), (3,4), (2,0)). Also T G 7fi(x) where n = (17,12,8) and x = ((7,6), (6,2), (0, -4)) . Note for example that y i + y 3 + y 4 = X]_. 3. if the endvertices of netpath Bj are Uj,Vj G M.d, where B^Uj C B^Vj then Vj — Uj = yj, for each j = 1,... , 2r — 3. Suppose T G Ttf(aji)(y), with corresponding x, ri, B as in Definition 4.1.6. Since B has shape j\f(a,n), we may label the netpaths {B\,... , Z ? 2 r - 3 } of B ac-cording to the edge labels { 1 , . . . , 2r ^ 3} of a. Let Ei = {j : Bj C £ L * X i } , and note that Ei is equal to the set of edges in the unique path in a from the root to leaf i, defined in Section 1.4. By definition we have 2~2jeEi Vj = x» a n d T^jeEi ni = n i -See Figure 4.1 for an illustration of this. Lemma 4.1.5 implies that if T G 7B for some non-degenerate B G B^(n,x), then T G TM{a,n){y) for some a G S r , n G P P - 3 , y G Z d ( 2 r ~ 3 ) satisfying Y,jeE{ ni ~ n*> YljeEi Vj = x * > * e i 1 ' • • • > r ~ 0 n t n e o t h e r h a n d suppose T G TjV{a,H)(y)-Let rcj be the vertex in T corresponding to leaf i of a, i = 1,... , r — 1, and let ri; = | T U X J . Then T G Ta(x) by definition. Choosing P> = U ^ T i T ^ ^ , it is easy to see that B G B(n,x) and T G TB- Finally since j\f(a,n) contains r — 2 distinct branch points, each of degree 3 (of which none are the origin), B must also have this property and thus B G B£>(n, x). For fixed a G S r , ri G P f _ 1 and x G Z d ( r _ 1 ) we write Y]s" - to mean the 51 sum over {n £ N 2 7 " 3 : YljeEi nj ~ ni> * = 1> • • • j»" - !}> and t o m e a n t n e sum over {y £ Z d ( 2 r -" 3 ) : £ V e J S . W = x i ' * = 1, •••,>" - 1}- Then £ £ ^ c n = £ £ £ E ^cn- (4-5) See Figure 4.1 for a concrete example of this idea. Def in i t i on 4.1.7. For fixed r > 2, a £ E r , network shape Af — Af{a,n), and netpath displacements y — ( y i , . . . yir-z) € Z d ( 2 r _ 3 ) we define tM{a,n){y)= E ^(r)- (4-6) Recall the definition of K from (1.29). We are now able to state the main result of this chapter, Theorem 4.1.8. T h e o r e m 4.1.8. Fix d > 8, 7 £ (0,1A ^ ) and 6 £ (0, ( 1 A - 7 ) . There exists LQ = Lo(d) S> 1 such that: for each L > Lo there exists V = V(d,L) > 0 such that for every r > 2, a £ E r , n £ N 2 7 " - 3 , R > 0, and « £ [ - i i , #]( 2 r - 3 ) d , F — (53=) - ™ ~ n ^ w - ( E - ( E ^ (4.7) where A and v are the constants appearing in Theorem 1.4-3 and.the constants in the error terms may depend on r and R. The constant V is defined in Definition 4.3.1 and reflects the presence of non-trivial interaction near branch points of our binary tree networks Af where three trees must meet at a single point but are otherwise mutually avoiding. In view of (4.4) and (4.5) we have that ^ ) = E ^ E E E * % ^ + E ^ E E^cn _ £ e * k - x ^ ( a , n ) ( y ) + & ( k ) . We wil l show that </>£(•) gives rise to an error term. , Recall the definition of the set of edges Ej of the unique path in a from 0 to leaf j. Then Xj = £ 2 = I 3 a n d in (4.8) we use r - l r - l 2r -3 2r-3 r - l 2r -3 E K J ' x i = E kJ ' E J"J{'eB,-} = E y'' E kJ'7{'eB,-} = £ y r K l = n • & ( 4- 9) j=i j=i ;=i ;=i j=i 1=1 52 where KI was defined in (1.29). Thus the first term on the right of (4.8) is equal to £ E E E & * £ * " < ° . * ) ( f l = E E E e * * w < « , * ) 0 ) = E E ^ ( M ) ( « ) -This is more clear if we consider the case r = 3, for which there is a unique shape a (which we suppress in the notation for A/"), and a single branch point. If we denote the spatial location of the branch point by y then (ni A112)—1 tfnun2)(kiM) = E E e ^ " 1 ^ 2 " 2 E ^ ( n , n 1 - n , n 2 - n ) ( y , X l - y, X 2 - y) x i , x 2 n= l y + 0l(k), (4.H) where informally one may think of <jP as consisting of the n = 0 and n = n i A 112 terms of the sum. The first term on the right of (4.11) is equal to ( n ! A n 2 ) - l E E E E ' ( M X \ ~ W ) + M * ~ W ^ ^ n = l x i , x 2 y ( n i A n 2 ) - l - E AA/"(n ,ni-n ,n 2-n)(«l)«2,«3)-n=l (4.12) Recall from (3.4)-(3.5), and the fact that Qc = 1 that we were able to express the critical 2-point function as 71 i n ( z ) = E w ( u ) j i E n [ i + ^ i . ( 4 - 1 3 ) u : 0 ->• x, i = 0 Ri£%(i) 0<s<t<n = n using the notation T\i=o £^e"7L ( i ) W W rio<s<t<n f1 + U*t] to represent E W(R0)--- E ^ ( ^ ) IT + (4.14) floer^o) «ne7L(„) o<«<*<n The product nt^ +^ t] incorporats the mutual avoidance of the branches Ri emanat-ing from the backbone u (which is a random walk), and we analysed this product using the lace expansion. For higher-point functions, the backbone structure in question may be interpreted as a branching random walk, with the temporal (resp. spatial) location and ancestry of the branching given by Af(n, a) (resp. y). 53 Definition 4.1.9. Fix N(n,a). We say that to is an embedding of Af into Zd if CJ is a map from the vertex set of Af into Z r f that maps the root to 0 and adjacent vertices in Af to D(-) neighbours in Z d . Let 0*Af(y) be the set of embeddings LV of Af into Zd such that the embedding oji of branch i has displacement y^. We now express the r-point function (4.6) in a form similar to that previously obtained for the two point function (3.5). For a collection of sets of vertices {Rs}s€j^f, define as in (3.3), Ust = U(Rs,Rt) = { Q j o t h e r w i g e (4.15) Recall from Definition 2.1.1 that E// = {st : s,t £ Af,s t}. Also note that a vertex s £ Af is uniquely described by a pair (i,m.j), where i is an edge in a and mj < nj. We write F h e A f X ^ e r ( } a s shorthand notation for E E E •" E • «•«) K o £ 7 ^ ( 0 ) ^(1,1)67^,(1,1) J?( l ,2)G7L(i ,2) • R ( 2 r _ 3 , n 2 r _ 3 ) 6 7 L ( 2 r - 3 , n 2 r _ 3 ) Then, tN(a,n)(y)= E ^)II E W(RS) ]J[1 + U,}, (4.17) venial) serf RS€TU(S) beE^-where 1. the sum over a; is a sum over all embeddings of the network shape i.e. over all bare trees with the required network shape and displacements, 2. the sums over Rs are sums over all branches at vertices s of the embedding CJ, 3. the factor fJb [1 + U],] ensures that the branches are mutually avoiding so that only combinations of branches that form lattice trees are counted. Equation (4.17) follows from (4.6) since any combination (to £ Cl/s(y), {Rs}secj) such that the Rs are all mutually avoiding lattice trees, uniquely defines a lattice tree T £ Tjv'ta.n) a n d vice versa. 4.2 A p p l i c a t i o n o f t h e L a c e E x p a n s i o n We now apply the expansion described in Section 2.2. Let ifim = E ww IT E WM (IT i 1 +) (1 ben / (4.18) 54 Then by expressions (2.4) and (4.17) we can write tM{a,n){y)= £ £ W{RS)K{N)-4%{y), (4.19) where K(JV) = ILeT^ \1 ^ e w ^ s e e shortly that <^(K) is an error term. Another such error term comes from Mv)= £ W{UJ)1\ £ £ (4.20) ue&M(y) seA/" f l s e r (w(s) ) r e n t e r where 6 is the branch point neighbouring the origin and £jy is defined in part 8 of Definition 2.1.1. Recall the definition of a branch from the second paragraph of 2.1. Let nb = (ni,n2,nz) be the vector of branch lengths for branches incident to b and let G = G(JV) C {2,3} be the set of branch labels for branches incident to b and another branch point of X. Define UnB{X) C Z\ and UnB{N) C l\ by 7b' ' nnb = {m:0<m,i<Y, * = 1,2,3} n {nl : mi < n{ - 2, i G G} %nb = ({rh : 0 < m{ < nu i = 1,2,3} D {m : < n; - 2, i G G}) \ U^. Note from (2.11) that U Hfib = {m : mi < n i , m 2 < n 2 - 2/2, ra3 < ^ 3 - 2 J 3 } and that this is empty if = 1 for some i G G. Equations (2.8)—(2.11) give an expansion for K(j\f) which yields 3 tM(y)= £ W(OJ)H YI W ^ £ ^ A (m))n^((A / ' \ l S A (m)) i ) wen (^jT) se/\f R3eT(w(s)) men^ 1=1 + rAy) + <f>Ar(y)-4>}v(y}, (4.22) where 3 ^(y)= £ £ ^ ( ^ ) £ J(<SA(m))n^((^\5A(m)) i). uenM{y) S<EAT RseT{u(s)) meHnb i = l (4.23) See Figure 4.2 for an illustration of these definitions. In accordance with Definition 2.1.1, the first term on the right side of (4.22) does not contribute in cases where b is adjacent to another branch point of j\f (which implies that r > 4 and n2 A n 3 = 1). For r = 3 there is only one branch point, 6, hence <f>tf(y) = 4>j\f(y) = 0-Lemma 4.2.1 states that in fact for large T i — o o = infi<j<2r—3 Uj, all the terms 0^-, <jfyf and <f>Jf are error terms, so the main term in (4.22) is QhT{afi){y) = tAT(a,n)(y) ~ ^//(v) ~ <l>j\f(y) + ^( j7)> ( 4 - 2 4 ) 55 Figure 4.2: A n example of graphs on A/*(a, ft) with a € E 5 a shape with edge labels shown at the bottom and ft = (3,4,4,3,6,4,3). The first graph contains an edge in 1Z so contributes to (fP1. The second graph does not contain such an edge but branch 2 is covered so this graph contributes to 4>F. In the third graph, branches 2 and 3 are not covered, but r i 2 - 2 > 772,2 = 2 > ^ = | and this graph contributes to which is the first term on the right of (4.22). Taking Fourier transforms of (4.22) or (4.24) we obtain where the constants implied by the O notation depend on r. The proof of Lemma 4.2.1 involves estimating diagrams and is postponed until Chapter 6. ,7T (4.25) Lemma 4.2.1. The error terms defined in (4-18)-(4-23) satisfy (4.26) 56 4.3 Decomposition of In this section we show that QM can be expressed as a convolution of a function and functions t^, for j — 1,2,3, and ultimately that QM can be expressed as a Gaussian term plus some error terms. The Mj are network shapes with aj G E r j . and rj < r. This permits analysis by induction on r. We first define the quantity ^^{u) and then the constant V appearing in Theorem 1.4.5 in terms of this function. We then state some bounds on the function 7 r M ( « ) in Proposition 4.3.2 and Lemma 4.3.3 that are the main ingredient for the proof of Theorem 1.4.5. The proofs of Proposition 4.3.2 and Lemma 4.3.3 are postponed until Chapter 5. The convolution expression for QM(V) involving TT^ appears in Lemma 4.3.4, and for the Fourier transform in (4.42). Finally we express QM as a Gaussian term plus some error terms in (4.43). These error terms are bounded in Section 4.4. Definition 4 . 3 . 1 . Suppose is a star-shaped network of degree A G {1,2,3}. defined by branch lengths M as in (2.12). Let u G Z3d. We define = E II E Wi^JiS^). (4.27) M M Note that if Mj = 0 then figA/^ is empty unless Uj = 0. In particular ifS^ = {0} is a single vertex (star-shaped network of degree 0) then we define TTQ(U) — p(0)/^-_gj. Now by (2.7) we can write oo J ( ^ ) = E E n i i + u ^ N=l LeCN(SA)beL b'eC(L) (4.28) = E ( - I ) W E I IH*) n N=I LecN(s^)beL b'eC(L) so that for TT^U) = £ £ = 1 ( - 1 )* n^{u) where E E ^ II E W{Ri)H(-U>) Y[[1 + UV]. LecN(sA)^enA() its* Ri€T(u(i)) \>&L \>'ec(L) M M (4.29) Note that n^lu) > 0 since —U\, > 0. We also define V^ E E E^^n^^-^^cEE^w- (4-3°) M e z ^ u e z M i 7 e z M *=i M V 57 The following Proposition is proved in Chapter 6 and is the main ingredient for the proof of Theorem 4.1.8. In order to state the proposition in a tidy manner, we introduce the notation [M] = My 1. (4.31) Proposition 4.3.2. There exists a constant C independent of L such that for N > 1 and q G {0,1}, E K | 2 % £ ( £ ) <NHN2a2\\M\UqBN(M), uez3d (4.32) where u = (ui,U2,us), and N BN(M) = (ce2-^' n , d - 6 + E r . , i f e f i E E d - 4 =1 [Mi]— i = 1 [ M i ] T - ^ m . < M . [Mj - mj] 2 [Mk + mj] 2 (4.33) Lemma 4.3.3. Let BN(M) be defined by (4-33) there is a constant C independent of L such that E ^ 3 E B N ( M ) < ^ ^ , j = 1,2,3, and ^=1 M:Mj>nj ^ E ^ E n^iioo^iv(M) < \ ^=1 M<n \ 1 0 - d s |n||oo2 , (4.34) if d^ 10 log||n||oo, if d= 10. Given M G TLftb we define A/^ = (jV \ <5^)j, where the notation (A/- \ M)i was defined in Definition 2.2.1. Note that the dependence of j\f~ on M is suppressed in the notation. Let vectors y G lPT-^d and v G Z 3 d and 5 4 cJV with M G be given. We write B^- for the set of branch labels of H that are branches in j\f~ but not and we write $ for the vector of yj such that j G Sjy—. Then we define Vvi = (Vi ~ VuVi)- (4-35) Lemma 4.3.4. Let yVi denote the vector of displacements associated to the branches of Ni (determined by v, y, and the labelling of the branches ofjV as in (4-35)). Then QjV(afi)(y) = E E ^ M ^ I I ^ E ^ ^ - ^ / V - i V v i ) - (4-36) MeuHb « 1=1 Vi 58 Proof. First from (4.22) and (4.24) we have QM{«,n){y)= £ £ E ^(^)^ A (M))f [ m i -M e « 5 t wen^(f) seAf RaeT^^s) i=i (4.37) However, as in the proof of (3.9) for the two point function, we may split up the branching random walk w G 0^/-(y) into 4 branching random walks (some of which may be empty) to obtain 3 . E W\u>) = E E Wi^WYPcDivi-Ui) E W(ui). (4.38) Trivially, 3 n E w(RA)= n E w(Ra)ii n E W ^ ^ seArRseT„(3) seS^RseTuis) i=i S i e A r - R S i e T u . ( S i ) where the products of the form s G j\f~ are products over vertices in the network shape JV~ . Since by definition, j\f~ and are vertex disjoint (i.e. have no vertex in common), equations (4.37)-(4.39) show that QAf(a,n){y) 1S equal to ( \ E E E n E W(R.)J{S%) x f3 M \ I ] E ^ ^ - ^ ) E W I I E W(RSi)K(jVr-) (4-40) 3 = E E ^ M ^ I I ^ E " 0 ^ as required. • Given /? G [—7r, 7 r ] 2 r 3 we let K 6 = (KI, K 2 , K 3 ) , and K* denote the vector of Then Ki, i = 1,2,. . . , 2r - 3 such that 2 is the label (inherited from M) of a branch of Afj 3 eiR-y _ gi^-u JJ eiitjivj-uj)eiK]-gVj _ (4.41) 59 From Lemma 4.3.4 we have rc6 Finally we write QAA(K) = 2 \[Ae-in^ + £P(K) +4(K) + ^ D ( K ) + #(*), (4.43) where the are defined by £gw= E - 1 ) ) ^ E ^ (^ )n^ (^ ), E C {1,2,3} \leE ) MeHnb ^ X E^<D . 3 4(«)=P? E (^ (^ -^ (oOn^ )^' - . J = 1 ( 4 4 4 ) Mennb \ i = l 3 i=i J £Y(K) = Vr-"2Y[Ae-i^2Vc £ ^ M ( O ) -The first term is obtained by writing D(KJ) = (l + {D(KJ) - 1)^, the second by writing % ( K 6 ) = ( % ( 0 ) + ( ^ ( K 6 ) - ^ ( 0 ) ) ) and so on. 4.4 B o u n d s o n t he £ . In this section we prove bounds on the quantities (4.44), as stated in Lemma 4.4.2. A l l of these terms will turn out to be error terms in our analysis and in general rely on estimates for TTJ^{K) such as those appearing in Proposition 4.3.2. Each term except £ m d wil l also use naive bounds of the form appearing in Lemma 4.4.1, in which #M denotes the number of branches in AA (recall the definition of a branch from the second paragraph of Section'2.1). Lemma 4.4.1. There exists a constant K, independent of L, AA. and K such that for any network AA tM(Z)<K*M. (4.45) 60 The proof of Lemma 4.4.1 is elementary, but we also postpone this proof until Chapter 6. Using Lemma 4.3.3 with rij = 1, oo E = E E E*£(*> < E E ^ w * (4.46) where the constant is independent of L. In particular since 7TQ(0) = 1, this proves that V = l + 0{32-^). L e m m a 4.4.2 (£~ bounds) . For all K, ^ ) = o ( i 2 K l (4-47) = { o d ^ l V i o g i H U ) , i /d = io, # W = O [ E 4 T V (4-49) Proof of (4-4V- F ° r l> £ E we bound Y$=itjVj(Kj) a n d Y I M ^ M ^ ^ by constants using Lemma 4.4.1 and (4 .46) . This leaves us with \£gm<c E n°M- (4-5°) E C { 1 , 2 , 3 } 3^E E^H For each nonempty E we may bound all but one of the a(ki) by 2. This gives 3 |#(*) |<C5>(«i) . ( 4 - 5 1 ) In particular since CL(KJ) < 2 this quantity is also bounded by a constant C. If | | £ j | | o o ! > L~L, then C < C | | K 6 | | 2 L 2 and we have obtained (4.47) for ||KJ||OO > L~L. By (3.29) we have for Halloo < L _ 1 that ' O ( K J ) < CL2K2 (4 .52) which proves the first bound for | | K J | | O O < L~L. • 61 Proof of (4-48)- We bound the by a constant and apply (3.48) with 3d instead of d with nb = ( K I , I , • • • , K 2 , i , • • • , « 2 , d , K 3 , i ; • • • > «3,d) to bound the difference ^ M ( ^ ) — ^M^b)- ^n doing so we obtain ^ M ( O ) - % ( K & ) < C | « 6 I 2 E \Uj\2^a{u)\. « e z 3 d (4.53) This gives us l4(*) l<C£ l^ l2 E Kl2l%(^ )l-Applying Proposition 4.3.2 and Lemma 4.3.3 we obtain I n?6|2_2||^>b||C i ef l £ t v 0) (4.54) Sl(H) <C £ | « | 2 ^ 2 | | M | | o o i V 5 ^ ( M ) < cfi—d M<nb » I | K " | c r l l n MOO if d ^ 10 K 6 | 2 a 2 l o g | | n 6 | | 0 0 , if d = 10, (4.55) as required. • Proof of (4-49)- We bound each exponential by a constant, leaving Men-,, (4.56) Next we observe that M € Wjj6 only if Mj > ^ for some j € {1,2,3}. The required bound then follows from Proposition 4.3.2 and Lemma 4.3.3. • It follows immediately from (4.47) that and from (4.48) that = o n \ I o n = < 0 <r2n k - l V l l n ' I l L ^ ^ n (4.57) if d ± 10, (4.58) 0 n * i v ^ i i * i i , \ i f d = 1 0 -4.5 P r o o f o f T h e o r e m 4.1 .8 . We prove Theorem 4.1.8 by induction on r (or equivalently on the number of branches 2r — 3 in Af). For r = 2 recall that A = A'p(0), so (as in the proof 62 of Theorem 1.4.3) we have by Theorem 3.4.3 and Lemma 3.5.2 that Vva2 n = Ae-i^+0(^)+0 n d - 8 | ' 2 1 (4.59) with the error terms uniform in {K G Rd : \ K \ 2 < C 0 logrii}. This yields the required result for r = 2. Now fix r and M = j\f(a,n) with a G S r and rt G N 2 7 * - 3 , and assume the theorem holds for all r-j < r. By (4.25) and Lemma 4.2.1, we have that Vva2 n = QAT K Vva2 n + V ' = i nt 2 J (4.60) Next from (4.43), (4.57)-(4.58) and (4.49), we have that QAT ( ^ 3 ^ ) is equal to Vr-2U2Li3Ae-i% plus cind K V va'n + o l g ^ ) + 0 ( i " i ^ : i i r ) + 0 ( E 4 , j—1 n • (4.61) plus the error term (4.58) Since 5 < A 1 in the statement of Theorem 4.1.8 we have V 0 < 1 — S and these error terms satisfy the error bounds of the Theorem. It remains to show that £ l n d (-7=%= ) is an error term of the required type. From (4.44) we have ind V va'n =pi E %(°V nv Vva2 n 2 r - 3 - T / r - 3 JJ A e - ^ Meunb By the induction hypothesis applied to rj < r, we have 1=1 K* K3 Vva2n *2 * = ^ - 2 ^ - 3 TJ e + 0 E ,<J=S + 0 ( E \^\2nf-s) n (4.62) (4.63) where the sums and products are over branch labels of branches in JVJ . If Hfib — $ then £ l n d = 0. For M G Hnb, for every j G {1,2,3} we have ^ < njt < n/. This 63 Figure 4.3: A n illustration of the relation 2~2i=i ri — ?" + 3 resulting from the decom-position of a network jSf into A/ i , when M 6 Tinb- The 3 extra vertices generated by this decomposition are indicated. enables us to replace riji by n; if necessary in the error terms of (4.63). Additionally since M € TL^ we have r = Yli=i(ri ~ l ) i o r equivalently YA=I ri = r + 3 (see Figure 4.3) and II v . 3 = 1 Kj ' \ Vva2n = y r - 3 A 2 r - 3 JJ e ^ JJ e ~ ' i=4 j = l ( 2r -3 n \ / 2 r - 3 E4r +° E 2 r - 3 ' n 2 Thus, nj=1 (^) - Vr~3 UT=i3 Ae~^ is equal to 2r -3 2 y r - 3 j 4 2 r - 3 JJ /=4 3 .2 e 2dn — I I e 2d" • 2 r - 3 | - . | 2 „ ( l - * ) \ 3 „2„ ' 2 r - 3 / t r n ./=i n, n (4.64) (4.65) Next using a telescoping sum and the inequality e ° - e 6 < C(6 - a) for 6 > a > 0 64 we see that nil- f ( n j - M j ) 2dn -n is equal to 1 = 1 \j<l KJ ( T I J — Mi) Kfnl g ~ 2dn — g ~ 2dn n »f tn , -M, - ) 3 ~ 2 3 K 2 M ; 2dn ' (4.66) 2dn l=i 1=1 Collecting terms and applying Proposition 4.3.2 and Lemma 4.3.3 we have £" ind n I / o \yvaz n a-JE^bE E 2dn '=1 " M 6 K S , f 2 r - 3 + ° E T 3 T + ° E 2 ' - 3 'iSpni1-"' i=i n. z=i n =° E 3 ISI2 K \ n '2r-Z . 1=1 n + 0 E- ir +° E 2 r ~ 3 1^12^(1-5) urn i=i n, i=i n (4.67) Since 1 — 6 > V 0 these are all error terms, and the proof is complete. • 4.6 P r o o f o f T h e o r e m 1.4.5. In this section we prove Theorem 1.4.5. By (4.8) there are two terms to consider. From (4.9), the first term of (4.8) involves a quantity that is treated in Theorem 4.1.8, summed over shapes and temporal locations of the branch points. We shall see in the proof of Theorem 1.4.5 that with the appropriate scaling this first term approximates the sum over shapes of the integral in Theorem 1.4.5. The second term of (4.8) is the contribution from degenerate trees and Lemma 4.6.2 shows that this is an error term. In proving this Lemma we wil l make use of an expression of the form (4.5) for degenerate trees. As such we introduce the notion of a degenerate shape. Definition 4.6.1 (Degenerate Shape). For r > 3, let E r be the set of rooted trees d such that 1. a. contains fewer than 2r — 3 edges, and fewer than r — 2 branch points (vertices of degree > 3) that are not the root, and 65 0 ! 2 02 1 0 12 012 Figure 4.4: The seven possible degenerate shapes for r = 3. The second (resp. third) shape is only a possible candidate for the shape of B G Brj(f i ,x) if ri2 > ni (resp. ri2 < ni). 2. for each j G {0 , . . . , r - 1} there exists a vertex in a with label i (each vertex may have more than one label), and each leaf (vertex of degree I) of a has at least one label. We call a € S r a degenerate shape. Clearly there are only finitely many degenerate shapes for each fixed r. See Figure 4-4 for the set E 3 . By Definition (4.1.3) and Lemma 4.1.5, if B G Brj(n, x) for some ii G M " - 1 , x G Z d ( r _ 1 ) then B has the topology of some a G S r . For a G S r consisting of / < 2r — 3 edges and ft G N ( we define V(a, ft) to be the abstract network shape obtained by inserting r i j — 1 vertices onto edge j of a, j = 1,...,I. Furthermore for y G 1>dl we define Tt>{a,n) (v) *° be the set of lattice trees T G To with network shape T>(a,rt) such that for each edge i in a with endvertices i\,i2 (i\ is closer to the root), the corresponding vertices u, v in T satisfy v — u = j/j. Furthermore we define tv(a,n)(y)= £ W(T). (4.68) Then as in the nondegenerate case (4.5), E E E ^ c n < E E E E E = 2^1^ ^(a,n)(0). Note that for any given ri G P T - 1 we may have many a for which the set {rt: ft n} is empty. We are now able to prove the following Lemma. Lemma 4 . 6 . 2 . For all k G [—TT, T r ] ^ - 1 ^ , | & ( k ) | < C r | | n | | £ 3 . (4.70) 66 Proof. Let I = 1(a) be the number of edges in a. Applying Lemma 4.4.1 to V we obtain tv{a,n)$)<Kl. (4.71) Therefore, (4.69) implies that |&(k)| < E E ^ E NI* 8 **" 4 < Crllfill^ 3 . (4-72) S€S r fJ^ri a e E r The second inequality holds since X ^ ^ n * s a s u m o v e r a * m o s * r ~ 3 temporal locations of branch points which are not the origin, each of which must be smaller than II njjoo by definition. D We are now ready to prove Theorem 1.4.5 which we restate below. T h e o r e m (1.4.5). Fix d >. 8, 7 € (0,1 A ^ ) and S e (0, (1 A 4=3) - 7 ) . There exists Lo = Lo(d) ~> 1 such that: for each L > LQ, t € (0, co)(r_1), r > 3, R > 0, and ||/c|oo < -R> >*J I V w 2 nr-2yr-2A2r-Z n (4.73) where the constant in the error term depends on t, R and L, and where V is the constant of Theorem 4.1.8. Proof. From (4,8) and Lemma 4.6.2 we have ~ E E - E E ^)(vfc)+"'"20 | r - 3 n (4.74) where K = « (a , k) as described in (1.29). Theorem 4.1.8 may be applied to the first term, giving $r)_ ( k = E E 2r -3 - K 2 , „ A F r - 2 A 2 r - 3 JJ ^ ( i r ) + ' 2 r - 3 ° E ^ F + 0 E 2 r - 3 i « | 2 n ] " 5 7=1 n + nr-20l M l o 3 n (4.75) 67 Considering the first error term, note that ^ y d - 8 — XI d - 8 2 s E E n -2» [ntj : ^ J 2 d - 8 2 ' 1 c E d - 8 1 d - 8 ^ I I L ^ J I l o o n^[ni\: flj 2 n 2 n A |nt | : n,- = m (4.76) < E E i + c d - 8 m < I I L " t J | | o o m 2 |n*J : n 2 i r - 2 loo — 2 < C I I L n t J I l l o ^ ^ H L n t J I I - 3 + - ^ U LntJ | n 2 i r -2 oo ) where in the last step we used the fact that since rij is fixed, the sum over n [ntJ : rij: = m is a sum over the locations of r — 3 branch points. Note that if d = 10 / 1 0 - d V Q % we interpret the quantity || |ntj ||oo 2 as log(|ntJ). Thus, since | E r | is a finite quantity depending only on r, the first error term in (4.75) is nr-20 n" (4.77) where the constant in the error term depends on r and t (and goes to 0 as t \ 0). The second error term in (4.75) is n (4.78) where we have used (1.29) with n) < (r - 1) Y%Z\ (kjIleEj) • The third error term is already of the form nr~20 (^y) where the constant depends on t. Thus it remains to show that 2r -3 - « ? / „ , > . r 2r -3 E n^w-»r-2/ ,n-H A I n l l r j = l J f i i ( t t ' i = l e 2d ds fi -2* lnt\ : J n £ N 2 r _ 3 n" = O l-j , (4.79) for each a G E r , where the constant depends on t, r and R. We rewrite the left 68 hand side as n r-2 n 2 r - 3 - « ? / „ , x r - 2 r - 3 K , a ) 2 s . L_ E n ^ - j f i i ' - ^ * n e N 2 r ~ 3 (4.80) Observe that the left hand term inside the absolutely value is the Riemann sum approximation to the integral on the right, with the approximation breaking Ri(a) into cubes of side ^ , with some overcounting or undercounting at the boundary. The set i?j(a) is a convex r - 2 dimensional subset of R 2 r ~ 3 . As such there are at most C\nT~z boundary cubes in the discrete approximation, each of volume ^ r ? , where C\ is a constant depending on t and r. Since the integrand (and summand) is uniformly bounded by 1, the contribution to the left hand side of (4.80) is O (^) where the constant depends on t and r. Within each cube of side £ we have, for all s in that cube, — « j Uj e 2d n — e" 2d Sj -n = o (4.81) By a telescoping sum representation (as in (4.66)) this gives us that for all s in that cube, j=i j=i v / (4.82) Using K21 < (r - 1) YHj=i {kjIieEj)2, this verifies (4.79) and hence proves the Theo-rem. • 69 Chapter 5 Diagrams for the 2-point function Proposition 3.4.1 was needed to advance the induction argument for the 2-point func-tion in Chapter 3. In this chapter we estimate various diagrams arising from the lace expansion on an interval (star-shaped network of degree 1) and prove Proposition 3.4.1. In Section 5.1 we introduce some definitions and notation that wil l be used throughout this chapter, and state Propositions 5.1.1, 5.1.7 and 5.1.4, and Lemma 5.1.6. Proposition 3.4.1 follows immediately from Proposition 5.1.1 by summing over N. In Section 5.2 we prove Proposition 5.1.1 assuming Propositions 5.1.7 and 5.1.4 and Lemma 5.1.6. Proposition 5.1.7 and Lemma 5.1.6 are proved in Section 5.3 and Proposition 5.1.4 is proved in Section 5.4. 5.1 D e f i n i t i o n s a n d N o t a t i o n In this section we introduce some notation and results that we need to prove Propo-sition 3.4.1. Let irm{x;() be defined by (3.7), with UST given by (3.3). Then from (2.7) and writing Ust = (—1)(—Ust) we have that for m > 1, oo ^ C H C E M ) " £ E N=l LeCN([0,m]) UJ:0-+X | w | = m ( 5 1 ) m n E wmni-ust] n [I+UM-i=0RieTu(i) steL s't'eC(L) The sum over N is actually finite, since a lace on [0, m] can contain at most m 70 bonds. We define ^(X;o=cm £ £ w{U)x LeCN([0,m]) u-.O-^x M = m ( 5 2 ) m n E wni-^ n [ 1 + ^ . and from (5.1) we have for m > 1 that Trm(x;() — 2^N=I ( — l )^ 7 r m( a ; i 0 a n ( ^ h e n c e \^m(x;C)\ < X)jv ^ mO^i 0- Therefore Proposition 3.4.1 follows immediately (by summing over N) from the following Proposition. Proposition 5.1.1. Suppose the bounds (3.33) hold for some z* G (0,2), K > 1, L > LQ and every m <n. Then for that K, L , and for all z G [0, z*], m < n + 1 and q € { 0 , 1 , 2}, Y C) < V ^ / , (5.3) x m 2 9 where £ = ^ Q z ^ , i/ie constant C = C(K,d) does not depend on L , m, z, N, and where v > 0 is the constant appearing in Theorem 1.2.9. Throughout this chapter, unless otherwise specified, C denotes a constant that depends on d and K but not on L , m, z, or N. It may change from line to line without explicit comment. Define hmi (u) = hmi (u, C) by hrrn (u) — < C2p2c(D*tmi-2*D){u), if mi > 2 (pcD(u), if mi = 1 (5.4) [ /{ u = o}, if mi = 0 where t0(u) = p(0)I{u=o}-Definition 5.1.2. For qi G {0,1}, nii G Z+ we define smuqi(x) = \x\2qihmi(x). For I < 4 we define s^ ( i ) ^ {x) to be the l-fold spatial convolution of the smi>qi. Definition 5.1.3. For n € {0,1}, let 4>Ti(x) = \x\2rip(x). For I G {1,2,3,4}, let (^^(x) denote the l-fold spatial convolution of the <f>Ti, and define </)^(x) = I{x=o}-Proposition 5.1.4. Let I G {1,2,3,4}, and k G {0,1,2,3,4}. Let G Z ' + and m = Yl\=imi- If the bounds (3.33) hold for 1 < m. < n and z G [0,2] then for all m < n + 1, and z G [0,2], I'So *«> * *!8>ll~ < rnZv+X'toVX*™?^, (5.5) m 2 '71 x+y m Figure 5.1: Feynman diagrams for M$ (a,b,x,y), AmiiTn2(a,b,x,y) and Amiio{a,b,x,y). A jagged line between two vertices u and v represents a quan-tity hmi (v — u). A straight line between two vertices u and v represents the quantity p(v — u). and l 4 ^ o l l i < ^ ™ E < ^ 2 E < ? i - ' (5.6) Definition 5.1.5. Let MM(a,b,x,y) = hm{x-a)pW(x + y-b), (5.7) and A i u ^ (hmi(y-a)hm2(x-y)pW(b-x), m2 ^ 0, Amum2(a,b,x,y) = \ ( 2 ) (5-8) We recursively define ' ( a , ^ , ! / ) E ^ A m i | m 2 ( a > ) n , « ) M { ^ 1 ) i m 2 j v _ i ) ( « ) « , a ; , y ) . ( 5 9 ) The diagrammatic representation of these quantities appears in Figures 5.1 and 5.2. 72 Figure 5.2: A n example of an "opened" Feynman diagram, M^' (a,b,x,y) arising from the lace expansion. A jagged line from U i - ± to Ui represents the quantity hmi(ui — Ui-\) (derived from the backbone from a to x). A straight line between two vertices u and v represents the quantity p(v — u) (derived from intersections of branches emanating from the backbone). Lemma 5.1.6. Setting UQ = a and U2N-1 = %> for every N > @, r2N-i M^\a,b,x,y) =X1-- £ YI hmitn - fH-i) x Y P(vi ~ b)p(vN -(x+ y)) x Vl VN 1 < I < 2N — 2 : m j , m ! + 1 ^ 0 ' ( m i , . . . , m 2 N - 3 ) (a, 6, u, v)A. • m 2 j v - i , m 2 j v _ 2 (x,y,u,v). u,v (5.10) We also make use of the following notation. Let Um,N = <rn £Z 2N-1 2N-1 mi = m, m2j > 0, r r i 2 j - i > 0 i=l (5.11) For general N > 2 we let (5.12) 73 and for N = 2 we also define G2m = {m e ftTO,2 : (mi V m 3 ) < m 2 } . (5.13) Note that for N > 3, i ? ^ U = Wm^N since m i + m 2 + m 2 i v - 2 + rri2N-i < m. Similarly for J V = 2, ^ U J ^ U G ^ , = Hm,2-P r o p o s i t i o n 5.1.7. For q € {0,1,2} and N > 1, ^ | x « ( x ; C ) < £ ^ I ^ M ^ O ^ O ) . ( 5- 1 4> Observe that there are two disjoint paths in the diagram M^\a, a, x, 0) from a to x, corresponding to taking the uppermost path and the lowest path, each have displacement x — a. In the opened diagram M^\a, b, x, y), the corresponding uppermost path may be from b to x or from b to x + y depending on m. Similarly the right endpoint of the lowest path depends on m. We define z = z(m, x, b, y) and z = z(m,x,a,y) by { x - b , if # { m 2 j : m^j ^ 0} is odd x + y — b , if #{m 2 j : m 2 j- ^ 0} is even { x + y — a , if # { m 2 j : m 2 j 7^ 0} is odd a; — a , if #{m 2 j : m 2 j 7^ 0} is even (5.15) 5.2 P r o o f o f P r o p o s i t i o n 5.1.1 E s u p ^ M f ( a , M , y ) < ^ . (5.16) In this section we prove Proposition 5.1.1, assuming Propositions 5.1.4 and 5.1.7. We prove the three cases q — 0,1,2 separately. Case 1: q = 0. Our induction hypothesis is that In view of Proposition 5.1.7 with q — 0, this clearly implies Proposition 5.1.1 with q = 0. For N — 1 note that sup Y Mm (a> fe> x , y) = sup ^ hm(x - a)p ( 2 ) (x + y - 6) a,f>,2/ x a.,b,y x supYhm{x)p(-2)(x+y-b + a) (5 17) X = s u p ^ / i m ( r c ) p ^ ( a ; + 2 ) . 74 Applying (5.5) with / = 1, k = 2 and all qi = rj = 0, this is bounded by C^]dJf as m 2 required. We consider separately the contributions to (5.16) from E N , and in the case N — 2 also the contribution from G2M. Now by (5.9) we have E s u p £ > W ( a , f t , * , ! / ) = E E E s u p ^ A m i i m 2 ( a , 6 , n , v ) m i < ^ m 2 < 2 f - m i m ' e H m _ ( m i + m 2 ) , A r _ i a ' 6 «.« X S U P E M r ? - 1 ) (U> U ' S > v) y x = Y E supJ2Amum2(a,b,u,v) (5-18) u.,t/ £ sup X]Mir1)(t«,,t/>x>y) m ' e K m _ ( m i + m 2 ) , N _ 1 u ' y ' ? / x < E £ S M 0 E Amum2{a,b,u,v)- ( C / 32 - ^ ) ^ - 1 mi<^m2<¥_mi «,„ " ( m - (mi + m2))d~24 ' where we have applied the induction hypothesis in the last step. Since m i +m,2 < in the range we are summing over, the last line of (5.18) is bounded by C I < K C ^ ^} E E S U P E Amurn2(a,b,u,v), (519) m 2 m i < 2 a m 2 < 2 | i _ m i <».& u,V d-4 where the constant C = 3 2 is independent of N. Finally we split the sum over m,2 into the two cases m2 = 0, m2 > 0 to get E E SUpYAmum2(a,b,U,v) m i < f m 2 < f - m i a , b u > v = Y s u p £ / i m i ( u - a)p(v - u)p ( 2 )(6 - t;) a,b (5.20) m i < 2 m + E E s u p £ / i m i ( u - a)hm2{u - w)p ( 2 )(6 - u) m i < f 0 < m 2 < f - m i < E E • E R ^ 2 " - L < g ^ , 75 where we have applied (5.5) with all qi = rj = 0 in the penultimate step and the fact that d > 8 in the last step . Combining (5.18)-(5.20), we get that E supE^M,^) < [C^J?\ (5-21) as required. Similarly using the symmetry of (in the form of the second equality of (5.10)) and writing n\ for rri2N-\ and n2 for m,2N-2 we get E supE^W^V) = E E E supEA.i,n2(aJ,y,«',«) (5.22) ni< 2 f L " 2 < 2 f i-ni " i ' e , H m - ( n 1 + n 2),N- 1 "''^ X< V x sup EM^_1)(«>6>«.«')• a,b,v' u Using translation invariance of AnitV,2(x,y,u',v) we proceed as in (5.18)-(5.20) to get E s u p E M f ( a , M , ! / ) < ^ ^ , . (5.23) rheFNa^y x rn 2 as required. It remains to prove the bound (5.16) for the sum over rh G G 2 ^ . Note that in this case m,2 ^ 0 and so M V ( a , 6, x, y) is equal to E P ( 2 ) ( 6 - v)hmi(u - a)hm_{mi+m3)(v - u)hm3(x - v)p(-2)(x + y - u). ^2A^ . We break the sum over rh € G ^ according to which of m\ and 7713 is larger and note that m-2 = m — (mi + ms). By symmetry of M^\a,b,x,y) and translation 76 invariance we have Y snpJ2MSyMxty) meG2a'b>y x <2 Y, Y snpJ2MSHa,b,x,y) rn,S™ / a<b>y X mi<-2 1JI3 < t n i : •"' 2 mi > mi =2 Y E s u p ^ J ^ t f t - ^ V ^ - a ) m . < f m s < m i : * u>" '(5.25) 2 m 2 > m i x ^ - ( m i + m a ) ^ - u)hm(x - v)p{-2\x + y - u) = 2 Em E s u pEE^ ( 2 ) ( 6 - « ) ^ m i H m i < " 2 ~ m 3 < mi : 1 u,v ni2 > mi ^m-(n»i+m 3 )(« ~ u)hm3(x - v)p^(x + y - u), X where in the last step we have subtracted a from each vertex and correspondingly changed variables (i.e. we have used tranlations invariance). This is bounded by 2 Y Y [suPYp{2)(b~v)hm-(mi+m3)(v ~u') I [supYhm3(x ~v') j m i < y mz < m i : m 2 > m i (5.26) Applying (5.5) with all qi,rj = 0 for the term inside the first and third braces and (5.6) with I = 1 and qi = 0 for all i for the term inside the second braces, (5.26) is bounded by 2 E E ... d-4 d-4 2 m i < f m 3 < m i : (m-(mi+m 3 )) ' m " m 2 > m i < g ^ 4 E -1=1 E 1 (5-27) m i < j m l m 3 < m i : m — ( m i + m s ) > m i < c AC/til d-4 / j d-6 ' m 2 m i < f TYl^ 2 77 and we have the desired bound since d > 8. This completes the proof of Proposition 5.1.1 for q = 0. • C a s e 2 : q — 1. Our induction hypotheses are that £ SnPYM2<N\^,x,y) < a H C ^ ] \ and rneHm,N a ' b ' y x Y supXU|2M^(a,6,*,y)< m€Hm,N a ' b ' y x m 2 d - 6 m 2 (5.28) In view of Proposition 5.1.7 with q — 1, these clearly imply Proposition 5.1.1 with 9 = 1 -For N = 1, the first statement of (5.28) is sup X |x + y - 6| 2/im(a; - a)p^ (a; + y - b) < a,b,y x o-2(C62-ir)N d - 6 m 2 (5.29) Writing p( 2 ' (a: + y — 6) = 2~2U P(u ~ °)p(x + y — u) and using \x + y — b\2 < 2(\u — b\2 + \x + y - u\2), (5.29) is bounded by 2 sup ^ 4>i(u - b)p(x + y — u)hm(x — a) + 2 s u p ^ p ( u — b)(/>i(x + y — u)hm(x — a). a,b,y x a,b,y x (5.30) Applying (5.5) to each term with I = 1, k = 2, q\ = 0 and exactly one r3- = 1, (5.30) is bounded by a ^ ^ - e ^ as required. The second statement for N-= 1 is 2 / / - » / o 2 - ^ \ J V sup Y \x ~ a\2hm(x - a)pW (x + y - b) < K , (5.31) 1 m 2 a,6,y which follows immediately by applying (5.5) with I = 1, k = 2, q\ = 1 and all = 0. For the inductive step, for each statement of (5.28) we break up the sum over m 6 %m,N into sums over m G E^, m G F£, and when N = 2, also m G G 2 ^ . For the contribution from rh G E% we write | z | 2 < 2 ( | z^ | 2 + \ZM|2) where (ZA,ZM) = < (a; - u, ti - 6) , if #{m2j (x + y - u,u - b) , if #{m,2j (x - v,v - b) , if #{m 2 j (a; + y - v, u - 6) , if #{m 2 j m 2 j 7^ 0} is odd and m2 > 0 7^ 0} is even and m2 > 0 m 2 j # 0} is odd and m 2 = 0 m2j 7^ 0} is even and m 2 = 0. (5.32) 78 Thus E supEl^ M^ M^ y) m€E»>a'b>y x <2 E S U P E \zA\2Arni,m2{a,b,u,v)M{?,~l\u,v,x,y) m£EmiNa'b>yx,u,v + 2 E S U P E AmumziaAUiV^ZMfM^'^iuiViXiy). rh€E» a'b'V x,u,v As in (5.18) the first term on the right of (5.33) is equal to (CP*-1?)' (5.33) 2 E E supEM 2A7ii,m 2(a,M^)- N s i - 4 mi<^m2<^-mi a>" « . « {m-{ml+m2)) 2 / ^ . o 2 - ^ \ J V - l <2 jzi E E SUpEl^4|2Ani,m2(a,M,v). m 2 m i < 2 m TO2<2m_mi a>b u,v (5.34) We now proceed exactly as in (5.19)-(5.21) except that we use (5.5) with exactly one Tj = 1 (instead of all rj = 0 as we did in (5.20). This yields an upper bound on (5.34) of a2m(CP2-*)N d-4 m 2 (5.35) For the second term on the right of (5.33) note that by definition, ZM is either z' or z', the displacement of the upper or lower path of M^N~^(u,v,x,y). We proceed exactly as in (5.18)-(5.20) except that the induction hypotheses give a bound 2^ S U P \zM\Mm> '{u ,v ,x,y) <a -jze m'p-W , N », U'<°'>V X (M - ( m l + m 2 ) ) 2 (m - (mi + m 2 ) ) d 2 4 (5.36) which contains an extra factor of a2m compared to that appearing in (5.18). We now proceed exactly as in (5.19)—(5.21) to get a bound on the first term of (5.33) of <y m s — . (5.37) m 2 This proves that E s u P E l ^ M i J V ) ( a , 6 , , , y ) < ( 7 2 ( C ^ ) i V . (5.38)' rneE"a>b>y * m 2 79 As in the q = 0 case of Proposition 5.1.1, the bound £ s u p ^ l z p M f l a , ^ , ; ; ) ^ 2 ^ 1 " , (5.39) fheF»a>b'y x rn 2 follows by symmetry. When N = 2, the contribution to (5.28) from m G is easily bounded as in (5.25) by applying (5.5) and (5.6) with exactly one of these having one qi or Tj — 0. This gives the desired bound of A ^C AL^^ as required. This completes the m 2 proof of Proposition 5.1.1 for q = 1 • Case 3 : q = 2. Our induction hypothesis is that £ SuPY\z\2k\2M£\aAx,y) < "4(CCf)JV- (5.40) ™eHm,Na'b'y x rn 2 the induction hypothesis In view of Proposition 5.1.7 with q — 2, this clearly implies Proposition 5.1.1 with q = 2. The proof of (5.40) is very similar to the proof of (5.28) so we just present the main ideas. The N = 1 case follows from (5.5) with I = 1, k — 2, q\ = 1 and exactly one 7-j = 1. To bound ^ supX|*|2|*|2M^W,z,y), (5-41) we use the expansions \z\2 < 2(| • | 2 + | • | 2 ) and | z | 2 < 2(| • | 2 + | • | 2 ) yielding 4 terms instead of the two in (5.33). One such term is 4 ] T sup Y \zA\2\zA\2Amum2(a,b,u,v)M^^1](u,v,x,y), (5.42) fh£E% a'b'V *,u,v on which we use the q = 0 case of Proposition 5.1.1, and (5.5) with qi = 1 and exactly one of the Tj = 1. For two of the remaining three terms arising from (5.41) we use the q = 1 case of Proposition 5.1.1 and (5.5) with exactly one of q\ = 1 or some rj = 1. The remaining term arising from (5.41) is 4 S U P £ Am.^i^b^u^v^z'^^M^'^i^v^x^y), (5.43) meENa>b>yx,u,v which we bound using the induction hypothesis and (5.5) with all qi,rj — 0 and . Collecting the 4 terms we obtain the bound £ supX;|z|2U|2M (^fl>6>x>y) < a 4 ( C ^ / ) 2 - (5.44) fh£E»a^y x m 2 80 The contribution from rh € F% also obeys the bound (5.44) by symmetry, while the contribution from rh € G 2 ^ when N = 2 is handled as for the q — 1 case of Proposition 5.1.1 except that we have exactly two of the q%,rj equal to 1 when we apply (5.5) and (5.6). This completes the proof of Proposition 5.1.1 for q = 2, and hence completes the proof of Proposition 5.1.1. • Remark 5.2.1. Observe that apart from the recursive representations of the dia-grams M(N) in (5.9) and (5.10), the only information we used to bound the diagrams was Proposition 5.1.4- This will become important when we estimate more compli-cated diagrams in Chapter 6. 5.2.1 Diagrams w i t h an extra vertex. We say that a diagram has an extra vertex on some p if it is the same as a diagram corresponding to some except one p(z) in that diagram is replaced with p^2\z). We say that a diagram has an extra vertex on some hm if it is the same as a diagram corresponding to some except one hmj(z) in that diagram is replaced with hmi * hmj-m>(z). When we consider the diagrams arising from the, lace expansion on a star-shape of degree 3 we wil l encounter diagrams with an extra vertex on some p or hm. We bound the contribution from all such diagrams by repeating the inductive analysis used in the proof of Proposition 5.1.1. We do not show all the details but the main ideas are as follows. We let n denote the location along the branch point where the extra vertex is located. If n = Yll=i mi f ° r some 1 < j < 27V — 2 then the vertex is on the p emanating from the backbone at n, or a p incident to that p (of which there are at most two). If n = 0 (resp. n = m) then the vertex is on the first p (resp. last p) in the diagram, or the p incident to it. Otherwise the vertex is at position n on the backbone (i.e. on some hmi). Let M^'n(a,b,x,y) denote the corresponding diagram with an extra vertex at n. We prove by induction on N that E E - P E ^ K ^ ^ ) ^ 1 ^ ^ - M5) meHm,N n<ma>b>y x ™> 2 For N = 1 the left hand side of (5.45) is E s u P E ( ^ n * hm-n){x - a)pW(x + y - b) + 2 sup E hm(x - a)p^(x + y - b). 0<n<ma>b>y x a'b'y x (5.46) 81 Using (5.5) with I — 2, k = 2 and all qi,rj = 0, the first term in (5.46) is bounded by — o ~ ^ — o - - ( 5- 4 7) o<n<m rn 2 m 2 Similarly using (5.5) with I — 2, k = 3 and all qi, rj = 0, the second term is bounded by C^d-1 • Adding these together we get a bound of c^dJ[ which satisfies the m 2 m 2 induction hypothesis with N = 1. For general TV > 2 we bound by using (5.9), and splitting the sum over n < m into sums over n <m\+ n%2 : n ^ ni\, and n > m\ + n%2, and the final case n = ni\. In each case the extra vertex is either on Amitm2 or M^,~l\ In the former case we use the q = 0 result in the proof of 5.1.1 on the M^'1^ part and (5.5) (increasing k or I by one due to the extra vertex) on the -A™ m 2 P a r t - •"•n t n e latter case we use the induction hypothesis on the M^,~l^'n part and (5.5) on the -Aj^ m 2 part. The contributions from rh £ F£ and rh £ are dealt with as usual. Similarly we prove menm,N n<ma^y x . \ ™ 2 (5.49) Note the factor \x — a\2 in (5.49) rather than \z\ or \z\. This is to avoid the situation that could arise of having a convolution of four p's with one of them having an extra factor \u\2 on the same diagram piece. This would violate the condition ^ + Yli=iri < 4 in Proposition 5.1.4. Using \x — a\2 instead, we wil l use path along the backbone from a to x rather than the top path or bottom path, and the induction argument goes through as before. 5.3 G e n e r a l D i a g r a m s In this section we prove Proposition 5.1.7 and Lemma 5.1.6. We begin with the proof of Lemma 5.1.6. 82 Lemma (5.1.6). Setting UQ = a and U2N-I = x> for every N > 2, M^(a,b,X,y)=Y- £ M l « 2 J V - 2 2N-1 hmi(Ui - Ui_ i ) i=l £ p(vi-b)p(vN-(x + y))x Vl,...,VN n E p(wi ~ ut-i)p(vi4f- - WI)P(vL - wd ;>2:m;=0 wt IJ {P(V± ~ UtiT{l even) + P ( « i ± 3 - Ui)I{i odd^j 1 < I < 2N - 2 : : E M ( m i , I ) , m 2 N _ 3 ) (°> 6> U ' V)Am2N-i , m 2 J V - 2 (s, V, «, «)• (5.50) Proof. For the first equality of (5.50), we prove the result by induction on N and leave the.reader to verify the easiest case, N = 2 (consider the two cases m2 > 0, m 2 = 0). For N > 3, if m 2 > 0 then by separating the terms / = 1,2 from the initial and final products in the right side of (5.9) we have that M^\a, b,x, y) is equal to £ I h™i (Ul ~ a)hm2 (u2 - u i ) E p(Vl ~ h)p(v\ ~ U2) I U\,U2 \ Vl J . " 3 V-2N-2 2N-1 H h m i ( u i - Ui-i) i=3 £ p(vi±3 - ui)p{vN - [x + y)) V2,...,VN n E p(wi ~ ui-i)p(vi4? - wi)p(vi - wi) i>4:m;=0 VJI X II (P^k ~ e v e n > + p(VL? ~ Ul^{l odd}) 3 < I < 2N - 2 : = E Arnum2{aAui,U2)M\^~^m2N_i)(uL,U2,X,y), Ul,U2 (5.51) by definition of Ami>m2 and the induction hypothesis. This proves the result when m2^0 83 If m2 = 0 then by separating the / = 1,2 terms from the first product and m I = 2 term from the second product in (5.9) we have that M M !(a,b,x,y) is equal to E E ( hmAui ~ a)h0{u2 - « i ) £ p ( v i ~ b)p{w2 ~ui)p{vi ~ w2) J Ul,W2 M 2 \ Vl- . / E - E 1*3 U2N-2 " 2 J V - 1 V2, — ,VN JJ hmi(Ui - U i - l ) i=3 E p(u2±2 - w2)p(vN - (x + y)) 2 even} + p(w/±3 - Ui)I{i odd}) = £ A m i ) m 2 ( a , 6 , 7 J i , W 2 ) M ( ( ^ ) m 2 ; v _ i ) ( u 1 , u ; 2 , a ; , y ) , JJ £ P(™f - « i - i ) p ( « l ± 2 - wi)p{vi - W[) l>4:mi=0 wi 3 < I < 2N - 2 : » n j , m j + i # 0 M l ,«>2 (5.52) by definition of Amijjn2 and the induction hypothesis. This proves the result when m2 = 0, and thus completes the proof of the first equality of (5.50). The proof of the second equality is the same by symmetry of the expression for in the first equality, by considering the cases m 2 ^ - 2 > 0 and rn2N-2 = 0 and separating the terms I = 2N — 1,2N — 2. „. • We now prove Proposition 5.1.7. P r o p o s i t i o n (5 . 1 .7 ) . For q G {0,1,2} andIN > 1 X > | 2 % f e C ) < E £M 2 «MW(O,O,* ,O) . x rheHm,N x (5.53) Proof. We prove the stronger result that * £ ( * ; 0 < E MW(ot6,x,o). (5.54) Recall the definition of n^ix; r) from (5.2). For N = 1 there is only one lace L = {0m} on [0, ra] and every other bond 84 0 Figure 5.3: The Feynman diagram corresponding to the lace containing one bond. The jagged line represents the quantity hm(x), while straight line between 0 (resp. x) and v\ represents the quantity p{v\) (resp. p(x — v\)). is compatible with {Om}, so by (5.2) m 7 4 ( * ; C ) = C m Y, W ^ U E W(Ri)[-Uom}U^1 + U^ to : 0 —^ x J=0Kte7L(i) b^Oro \LO\ = m = E WiRo) E ^ ( ^ m ) h ^ 0 m ] x ( 5 5 5 ) m—1 r E II E ww> II t1 + ^ i -w : 0 -> x i = 1 b^Om |w| = m Note that everything in this expression is non-negative. Now —Uom = I{R0nRm^D} so TTm(x'-> 0 i s nonzero if and only if there exists v € Z d such that v € i?o FI i?m and therefore E w(Ro) E w(#m)[- tfom] < £ £ E RoeTo RmeTx v RoeTo(v) Rm€Tx{v) ^ ^ V If m = 1 then the last line of (5.55) is C E W(u) = (PcD(x), . (5.57) u : 0 —> x H = i as required. For m > 2, n^Omt1 + ^b] ^ I L ^ - c ^ m - J 1 + U*t\ a n d l e t t i n g I/i ( r e s P -85 Figure 5.4: A n example of the Feynman diagrams arising from the lace expansion. A jagged lines from to Ui represents the quantity hmi (ui — v,i-\) (derived from the backbone from 0 to x). A straight line between two vertices u and v represents the quantity p(v — u) (derived from intersections of branches emanating from the backbone). y 2 ) be the location of the walk UJ after 1 step (resp. m — 1 steps) we have rn— 1 = m <J2J2^D^^D(x-y2)x 2/12/2 . ' (5.58) m - 2 c m - 2 £ W(0J')H £ W W l f l l + tfk] W':J / I ->2/2 3=° RjtTw'U) b |w'| = m - 2 =/i m (a;) . Combining (5.55)-(5.58) gives the desired result for A 7 = 1. See Figure 5.3 for the diagrammatic representation of this bound. For N > 2 the reader should refer to Figure 5.4 to help understand the following derivation. Firstly L G £^ ( [0 , 7 7 1 ] ) if and only if L = { s i T J i , . . . , sjv£;v} where s\ — 0 , = m and for each i , Sj+i < rjj and Sj+i — U-\ > 0 . Hence from (5.2), TT^(X;() is equal to m AT cm E E W ^ ) U E w(Ri)l[[-u.iti] n [1 + ^ ] . { s i t l , . . . « A r t j v } w : 0 - f x . i = 0 Ri^Ta{i) i = 1 \>eC(L) £CN'([0,m]) |oj| = m (5.59) Now everything in this expression is positive, and every bond b = st such that s i < s < t < S2, or <AT_I < s < t < /JJV , or Sj+i < s < t < ti, or /jj < s < t < Si+2, is 86 compatible with L = . . . , SJV^JV}- Therefore (5.59) is bounded above by m N cm £ E w(U)H E ^ ) I I [ - M ' X { s i t i , . . . s j v * A r } w : 0 -> I i = 0 Ri£%(i) i = 1 e ^ ^ d O . m ] ) |w| = m N-l N-2 n [ i + n [i+^n n ti+^n n l>e(si ,S2) b 6 ( t j v - i , * j v ) i = 1 l>e(si+i,t») i=l be(tj,sj+2) (5.60)' where for b = s i we are using the notation b € (a, b) to mean a < s < t < b. For L = {si t i , . . . , s jvt jv}.€ £*([0 ,m]) we define rh(L) G Z 2 ^ - 1 by m i = s 2 - 0 , m 2 A T _ i = m-tN-i, m 2 i = * i - « i + i > ^ 2 i - i = « t + i ( 5 . 6 1 ) Then m2i > 0, m . 2 i _ i > 0 and X ^ i " 1 m i = mi s o m € %m,N- Similarly for any fh G %rn1N we define L ( m ) = { s i * i , . . . ,5JV*W} € <?.([0, m ] ) by 5 i = 0, tjv = m , 2i ** = E m J ' « = i , • • •, -^ v — i , £ i (5.62) 2 / - 1 a J = £ M J ' ^ = 2 , . . . i V . J'=I Then for each i , S j + i < £j and S j + i — > 0 so that L(m) G £^([0 ,771]) . Thus (5.61)-(5.62) defines a bijection between £-^([0,771]) and %m,N-We now break up the sum over walks w in (5.60) according to the intervals on the right of (5.60). Doing so we obtain E ww oj : 0 —y x \w\ = m = E E W ^ E W(U2N-1)* W l , . . . , « 2 J V - l W l : 0 - ^ U l . W2JV-1 : U 2 N - 2 -> x (5.63) \ui\ = s2 - Si = S 2 - Si J V - 1 i V - 2 n E ^ M I I E w(oj2j+ly 1 = 1 W 2 i : « 2 i - l -> « 2 i -? = 1 W 2 J + 1 : « 2 j « 2 j + l 1 | w 2 i | = *i — si+l \W2j+l\ = Sj+2 — tj 87 Then under this scheme, n™L.0 Z ^ e T ^ becomes / m i - l \ E w ^ n E wiRi^) n E ^ ( ^ J ) » Ro€T0 l < i < 2 J V - l : V i^.mjGTL,.^ .) j = l Ri,}£TUi(j) j mi ^ 0 (5.64) where Wj(mj) = itj ( u ; 2 j v - i ( m 2 A r - i ) = a;) and the product over i ensures that if some si = then we do not count the tree emanating from this vertex twice. Similarly the term rjj=i h ^ ] = U.S=i ^ n a , ^ } becomes (/{m^O} + I{mi=0}I{Ri,mi=Ri-1,rni_1}) x N-2 (5.65) -^{« O ni i2 , m 2 ^0 } J r {R2N -3 , m 2 A r _3nR2JV- l , m 2 A r _ 1 #0} II ^{/J 2 (- l ,m 2 ,_ x nK 2 i+2 ,m 2 , + 2 5*0} * (=1 Note that (5.65) contains no information about Rij for 0 < j < mj. Lastly we have that the second line of (5.60) becomes 2 J V - 1 / \ n n W * M = « • (5-66) i = l y i < * < t < m i - l J Combining (5.60) with (5.63)-(5.66), and writing uo = 0, « 2 N - i = x we have that (5.60) is equal to E E E W(Ro) II ( E WiRi^U x u rneHm,N RoeTo 1 < i < 2JV - 1 : xRi^i^Tui ) mi ^ 0 ( J { m i ^ 0 } + I{mi=Q}I{Ri,mi=Ri-hrni_1}) x N-2 I{RonR2,m2^<l)}I{R2N-3,m2N_3r\R2N-l,m2N_1¥:<l>} II ^{#21-l,m2|_ t n i J 2 J + 2 , m 2 i + 2 #0} X / , \ E E w(^) n j i w « = i } W i : ->• « i 5 = 1 ^ - 6 ^ 0 ) \ l < s < t < m ; - l y \ui\ = mi J (5.67) The last line of (5.67) is n2^i-1 hmi(ui — by definition. For any collection of trees {Ri,mi '• 1 < * < 2iV — 1} for which (5.65) is equal to one (i.e. nonzero) we choose v\ G Zd, i = 1,..., N as follows. 2 J V - 1 n 88 (a) I{R0nR2m2^<li} = 1 if and only if there exists a wi G Zd such that v\ G RoC\R2,m2. This means that RQ G 7O(VI ) and i?2,m2 € Tu 2 (ui) . (b) Similarly / { B a j v _ 3 i r a 2 W _ 3 n f l 2 J V _ l i m 2 J V _ 1 # 0 } = 1 i f and only if there exists a vN G Z d such that vN G # 2 jv - 3 , m 2 W _ 3 F l . R 2 ;v - i > m 2 N _ i • This means that i?2Ar-3,m2jv-3 e TU2N_2(VN) and R2N-i,m2N-.1 e Ti(wi). (c) For each i G { 3 , . . . , 2N - 5} such that % is odd, J{fl J i „, .nrt j +3 i m . + 3 ? 40} = 1 i f a n d only if there exists U i ± 3 G Zd such that «j±3 G Ri,mi Fi jRi+3 ,m i + 3 - This means that i ? j , m i G T u ^ V i i i ) and i ^ + 3 ,m i + 3 € T u i + 3 ( u i ± 3 ) where i + 3 is even. Now if mi = 0 (in particular this forces i to be even) then hm(ui — «;_i) in (5.67) is nonzero if and only if m = In addition 1^ =R1_1 } = 1 if and only if Ri,mi = -R/-i ,m;_i- By the above construction we have that vi G Ri,mr and vi+2 G Ri-itmi_1, i.e. vi,vi±2,ui G Ri,mr For T = i ? / ; m , let denote the backbones in T joining the specified verticies. Then there exists a unique wi G T such that Tui^y , Fl Tu.^y.. 2 — Tui^+ujr Collecting the above statements we have that (5.68) N-2 E n I E IX tf0€7b 1 < z < 2N - 1 : V i^.mj 6TUi (-T{m;^ 0} + ^ { m i ^ ^ H i . m ^ - R i - l . m ^ i ) ) X I{R0nR2,mn¥=9}I{R2N-3,m2N_3^R2N-l,m2N_1 #«>} IT A«2(-l,m2,_ j n R 2 (+2 , m 2 i + 2 5*«) ^ E E V F ( ^ O ) £ W ' ( i ? 2 W - l , m 2 N _ 1 ) x U RoeT0(vi_) R2N-\,m2N_1£'Tx(yN) I:m;=0 H | , m ( e 7 i , ( iM ,111^2) n ( : m, ^ 0 m ( + 1 7^ 0 E ^(iw \ ^ R i , m , e 7 i ( ( u ^ ) •^ {i even} "I" E ^ ,m,) I {I odd} ^ , 7 7 1 , 6 7 ^ , ( ^ ^ 3 ) y (5.69) 89 Now observe that E f l e r , ^ ) W(R) = p(y2 - y\) and E W{Rt,m)<Y, E ^(^) E w(**) E ^(^) «/,m,er u,(i;M^i^2) ™; RieTUl(wi) R2erwi(vt) R3eTw,(vi+2) = y P(WI - U[)p(vi - Wl)p(vi±3 - Wl). 2 2 Wl (5.70) This completes the proof of (5.54), and hence Proposition 5.1.7. • 5.4 D i a g r a m p ieces In this section we first prove Proposition 5.1.4 assuming the following two lemmas, which we prove later in this section. L e m m a 5.4.1. Let k G {1,2,3,4} and r<*) G {0, l}k be such that k + £ f = 1 n < 4, then n 25"V-fl2— — £ « W < C m ^ , ^ E , a n d s u p 4>%(x)<Ca4jf2^-. 0<|z|<x/mX \x\>yML m 2 (5.71) L e m m a 5.4.2. / / the bounds (3.33) hold for 1 < m < n and z G [0,2], then for all z 6 [0,2], I £ {1,2, 3,4}, q G {0,1}' and m C G Zl+ such that £ = m < n + 1, |*S„ MOIIOO < C a 2 E 9 i / ! 2 m E g \ and l l s^o .0||i < C a 2 £ * m * > (5.72) P r o p o s i t i o n (5.1.4). £ e U G {1,2,3,4}, a n d fc G {0,1,2,3,4}, q G {0,1}' a n t i f G {0, l } f c 6e SUC/J that k + Y,i=i ri ^ 4- Lefrh^ G Z + a n d m = £ j = 1 m j . // the bounds (3.33) hold for 1 < m < n and z G [0,2] then for all m < n + l and z G [0,2], a n d 1*2(0 * $ > l l o o < m S « + E r , < T 2 ( E « + E r , ) C ^ _ i _ j ( 5 . 7 3 ) m 2 l-So^ olli^ CmE*^ *^. (5.74) Proof. Firstly (5.73) with fc = 0 and (5.74) follow immediately from Lemma 5.4.2. We must therefore prove (5.73) with fc > 1. 90 By definition Hs^o.^o * 8^>H°° i s e q u a l t o su^x2~2us%)^i)(x ~ ")<!>%)(*>) which is equal to x a ; | w | > - \ / m L | u | < v m L < sup <f>%(u') E i ^ ^ - ^ + ^ i , ^ ^ E ^5>(u) l«'l> V m L | « | > V m T 1 | u | < v ^ L (Jcr'Z.'j < = 1 ffl! m 2 (5.75) where we have applied Lemma (5.4.1) and'Lemma (5.4.2) in the last step. Collecting terms we get the result. • Let [x] — \x\ V 1. In order to prove Lemma 5.4.1, we need the following convolution proposition which is proved in [11]. Proposition 5.4.3 ([11] Prop. 1.7(i)). Iffunctions f', g on Zd satisfy \f(x)\ < ^ and \g(x)\ < ^ with a>b>0, then there exists a constant C depending on a,b,d such that r % , if a > d [x]a+b-d i if a <d and a + b> d. (5.76) 5.4.1 Proof of Lemma 5.4.1 We prove the result in two stages. We first prove that k 3 For k = 1 we have from (1.13) that 4% (X) ^ E # ( 2 - i / ) M d - 2 j - 2 E n ' ( 5 - 7 7 ^ j'=0 L J and '(x) < CI*. + j ^ r 2 < E (5.78) tf'C*) * T ^ p r < E L j ( 2 - , ) g ] d _ 2 j _ 2 - (5-79) 91 Which verifies (5.77) for fc = 1. For > 1 we have u 1 •-I fe 1 ^—v ^—v (_/ IT—-v L/ ~ L?"( 2 -")[u]« '- 2 J- 2 ri ^ L ^ - " ) ^ - « ] " - 2 » - 2 E U n i * - i c • 1 , (5-80) - E E rj(2-j/) r.n(2-i/) E r,/lrf-2j j=0 n=0 1 fe-1 ^ E E c c £(j+n ) (2+i/) r r l d - 2 0 + n ) - 2 E t i r i ' j=0 n=0 Ir'J where we have used Proposition 5.4.3 with the fact that fc + r% < 4 in the last step. Wi th a different constant, (5.80) is bounded by y — ^ — (5-8i) ^L7 (2+") [x ] '* -2 i -2Ef=iri as required. Therefore we have fe Q E ^)(^)^E E # (2 -« ' )Md -2 j -2£'-i 0 < | x | < v / m L j = 0 0 < | x | < v / r n £ fc r»r / ^ - n 2 ^ + 2 £ r i fc (5.82) - X^(2-J/) which proves the first bound of Lemma 5.4.1. Similarly, (fe) fe fe £ r = E T j , . . „ ~ D - 2 J - 2 I > , : j=o Ld-3u-2l^rim 2" < C ^ E O - ^ - ^ d - 2 f c - 2 £ > j j _ m 2 (5.83) which proves the second bound of Lemma 5.4.1. • 92 R e m a r k 5.4.4. Observe that the only information about p(x) that we used to prove Lemma 5.4-1 (and hence Proposition 5.1-4) was (1.13). This will become important when we estimate more complicated diagrams in Chapter 6. 5.4.2 Proof of Lemma 5.4.2. In this section we prove Lemma 5.4.2 by induction on I. For I — 1 we use induction on m. For I = 1 and m = 1 we have h\(x) = (pcD(x) and hence HMoo < ^ = | I M i < C . (5-84) Using the fact that D(x) = 0 for \x\2 > dL2, sup | z | 2 / i i (x ) < C L 2 T d ^ Ca2B2, (5.85) x Li and by (3.28) X x This proves the result for the case / = 1, m = 1. The case / = 1 and m = 2 is dealt with similarly using \x\2h2(x) < CY \u\2D(u)D(x - u) + CJ2 \x — u\ D(u)D(x — u). (5.87) u u For I = 1 and m > 2 we use the inequality hm(x) < p(0)(,pc YuD(u)hm-i(x — u) (which holds trivially by replacing the factor Tlo<s<t<m-2[1+t/^] b y Ili<s<t<m-2[1+ Ust] in the definition of tm-2) so that C62 \\hm\\oo < \\hm-l\\oo < \\hmh < | | / l m - l | |oo < K. (5.88) mi Using \x\2 < 2(|u| 2 + |a; — u\2), we have sup |x | 2 / i m (x ) < o- 2||/im_i||oo + sup |a; - u'\2hm-i(x - it') x x-u' Ca2B2 Cu2B2 ( 5 > 8 9 ) < ^ + d 1 d - 2 This proves the result for I = 1 and all m < n + 1. 93 m 2 ' For / > 2 we have s^o^o _ 2-« 5 mi>«i Wi(m2,...,m;)i(<?2,---9f)v I I ^ O ^ o l U < l | s f f i , , J l o o | | s f ~ 2 1 , ) . . . , m j ) , ( m , . . « ) l l 1 C<7 2 < ? 1 /3 2 m'» C g r 2 m i ^ o - ^ ^ i 2 r < 0 . v / 0 . 2 v ' < d ffl! as required. Similarly if m i < y , l l a ( 0 II < HsW II l l 5m('),^0ll°° ^ l l 6 mi , 9 i l l l l l 6 ( m 2 ) . . . , m i ) , (g 2 , . . . g , ) l loo d m2 (5.91) < m2 as required. This completes the proof of the first bound of Lemma 5.4.2 for all I. For the second bound of Lemma 5.4.2, we have II W II <r II ( x ) II II II l |Sm(0,gtO 111 ^ H S m i , g i l l l l l 5 ( m 2 , . . . , m i ) , ( 9 2 , . . . g ( ) l l l < Ca2qi mqi Ca2 ^=2 * m £ U n (5.92) < Ca2^=iqirnPi^q\ as required. This completes the proof of the second bound of Lemma 5.4.2, and thus completes the proof of Lemma 5.4.2. • R e m a r k 5.4.5. Observe that the only information about hm that we used to prove Lemma 5.4-2 (and hence Proposition 5.1-4) was I I U o o ^ , I I U I i < C , (5.93) mi and when some qi / 0 we also used • sup \x x m2 x This will become important when we estimate more complicated diagrams in Chapter 6. 94 Chapter 6 Diagrams for the r-point functions In this chapter we prove Proposition 4.3.2, and Lemmas 4.3.3, 4.4.1 and 4.2.1. Note that since we have proved Proposition 3.4.1 in Chapter 5, one output of the inductive approach of Appendix A is that the bounds of equation (3.33) hold for all n. As a result, the conclusions of all the Lemmas and Propositions of Chapter 5 hold for all n. Another result of the inductive approach is that £ c = 1 (see Lemma 3.5.2). In this chapter £ = 1 and hence it does not appear. Proposition 4.3.2 is proved in Sections 6.1 to 6.4 using the lace expansion on a star-shaped network and the results of Chapter 5. Lemma 4!3.3 is proved in Section 6.5. The other results are proved in Section 6.6, also assuming the results of Chapter 5. 6.1 P r o o f o f P r o p o s i t i o n 4 .3 .2 . For N > 1, recall the definition of TT^AU) from (4.27) where Sjt has at least one of M\,.M2,M3 nonzero (we defined 7rg(u) = p(0)/^_gj). P r o p o s i t i o n (4.3.2). There exists a constant C independent of L such that for TV > 1 andq€ {0,1}, £ \uj\2q^(u) < N^N2a2\\M\UqBN(M), (6.1) 95 where u = (ui, 112,113) G Z 3 d and BN(M) = (C82~%y x 3 1 3 1 1 II r , , , - ^ + E r E E £ J [ M i ] ' £ [ M ^ ^ [ M j -mA*? [Mk + m}] (6.2) We prove Proposition 4.3.2 assuming Lemmas 6.1.1, 6.1.2, and 6.1.3. Lemma 6.1.1. For q G {0,1}, when Mt = 0 but M ^ 0, As stated in Chapter 2, laces on a star-shaped network of degree 3 can be classified as cyclic or acyclic. Let (resp. £ ^ ) denote the set of cyclic (resp. acyclic) laces and define 7t^f(x) and KM(X) to be the contributions to ^ ( ^ ) from acyclic and cyclic laces respectively so that when none of the Mj = 0, TT^(X) = TT^(X) + 7c$(x). Figure 2.6 shows a basic cyclic lace and a basic acyclic lace with 3 bonds covering the branch point. Lemma 6.1.2. For q G {0,1}, £ \UJ\^{U) < N\M\\ti\W (cP2-*)Nil—K=*. (6-4) uezu i=i iMi\ 2 Lemma 6.1.3. For q G {0,1}, £ N 2 9 * £ ( £ ) <7V3(iV2a2||M||0O)"(c7^-^)iV uez3d 3 X 3 1 1 1 ^ ; T -, r , d - 6 ^ y r , , , d-6 . 7d^ i=1[Mi] 2 j^imjKMj Wi -mj) 2 [Mk+mj] 2 (6.5) Proof of Proposition 4.3.2. If Mi = 0 for some / (but M / 0 ) then from Lemma 6.1.1 we have V | ,2 N(^.NHN2a^M\U9(C02^)N > \Uj\ TT.-AU) < "5 7 r 1 M ( E L M , ) " < ^ ( i W U M l U ) " ^ 2 - ^ ) ^ [M, - M ^ [ M , 2 - Mh]^[Mtl + M , 2 ] ^ ' 96 where ly^l and Z2 # /. Summing over / this is trivially less than N^N^MUnCp-^BN'M), • (6.7) as required. Otherwise Mi ^ 0 for all I, and the result follows from Lemmas 6.1.2 and 6.1.3 using E ^ E + E i«>f*£(s)- (6-8) S U M • Before proving Lemma 6.1.1 in the next section, we introduce a Lemma which allows us to replace one or more lines (correpsonding to p's and /im's) in a diagram with different quantities, in such a way that we can estimate the resulting diagrams without resorting to more inductive proofs such as in Section 5.2. Lemma 6.1.4. Given homogeneous Junctionals F : Em —>• K+ and fi : E —> R + , suppose that whenever fi(ai) < hi, we have F(a) < K. Then for scalars ai > 0, the bounds fi(a*) < aibi imply F(a*) < 111=1 a i -Proof By homogeneity, F(d*) = F p , . . . , ^ L J J a i . ( 6.9) Also by homogeneity, fj£)= < ^ for each* (6.10) which implies that F (fj-, . . . , < K by hypothesis. This completes the proof. • In most cases we will use Lemma 6.1.4 with each aj being either hmi(ui) or p(ui) and F being a diagram (i.e. a large convolution of /im's and p's). In fact Lemma 6.1.4 provides an alternative method of bounding the q = 1,2 cases of Proposition 5.1.1. 6.2 P r o o f o f L e m m a 6.1 .1 . Without loss of generality M 3 = 0. By (2.12), <S4 is the interval (i.e. a star-shaped network of degree 1) [ -M 2 , Mi] of length M 1 + M 2 . Consider the lace L = { -M 2 Mi} illustrated in Figure 6.1. Breaking up the walk corresponding to the backbone into 97 0 M, 0 Mf M , Figure 6.1: The single bond lace on [—M2, M{\ and the corresponding [0, M i + M 2]. two subwalks we can show that the contribution to J2xi,x2 7r(M1,M2,o)(xiix2) from this lace is less than or equal to hM2{x2)hMl{xi)p{-2){xi ~ x2)- (6.11) X\,X2 Using translation invariance we can rewrite this as YhM2(u)hMl(x-u)p{2)(x). (6.12) x,u Comparing this to the contribution to (4.29) from the lace on the right of Figure 6.1, YhM1+M2(x)p{2)(x), (6.13) x we see that the only difference is the replacement of / iM 2 +mi(^i) i n (6.16) by J2U hM2(u)hmi (ui - u). Now consider the lace L = {siti,s 2t 2} where s\ = M 2 , t\ — m\ + m 2, s 2 = mi, r.2 = Mi on the left of Figure 6.2. This lace divides the interval [—M2, Mi] into subintervals, one of which contains 0. Using the same method as in the proof of Proposition 5.1.7, but breaking up the walk corresponding to the subin-terval containing the root into two subwalks, we can show that the contribution to 52xi x2 7r(M1,M2,o)(xi,X2) from this lace is less than or equal to £ Y hM2{x2)hmi{ui)hm2(u2-ui)hM^_(m^^ X\,X2 Ul ,U2 (6.14) Using translation invariance we can rewrite this as Y Z\2 hM2{u)hmi{ui-u)hm2{u2-u1)hMl_{mi+^^ X\ U,U\,U2 (6.15) 98 -Mj 0 m, mfm2 M , 0 Mj+m, Mfmfm2 Mj+M, Figure 6.2: A lace on [ - M 2 , M i ] and the corresponding [0, M i + M 2 ] . Comparing this to the contribution to (4.29) from the lace on the right of Figure 6.2, X E ^ A f 2 + m 1 ( u i ) n m 2 ( u 2 - « l ) / l M i - ( m i + m 2 ) ( a ; l - ' " 2 ) p ( 2 ) ( w 2 ) p ( 1 ) ( ^ l - ^ l ) 5 (6-16) Xl U l , U 2 we see that the only difference is the replacement of / i M 2 + m i ( " i ) in (6.16) by Y.uhM2(u)hmi(ui-u). In general, assuming M 3 = 0, the diagram arising from any lace on [—M 2, Mi] is bounded by the opened diagram of (5.9) that arises from the equivalent lace on [0, M i + M 2 ] , except for the replacement of at most one term hmi (ui) by a term of the form (hm * hmi-m)(u) in M^*\ Note that this m is fixed by M i and M 2 (i.e. it is not summed over). Proposition 5.1.4 states that the bound on.a diagram piece does not depend on the degree of the convolution of hmi (provided that degree is less than 4). Thus by Proposition 5.1.7 we have the same diagrammatic bounds for E n ^ ^ A ^ 1 ! ' 1 ! ) a S for Y r n £ U M l + M 2 N & U ^ , b , y Y , x M S \ a ^ X ^ ) - B u t from (5.16), £ *M i ,M 2 > o ( z i>S2) < £ sup^A4A r )(a,6 )a;,y) < N d - 4 ) * i . * 2 ' ' menMl+M2,Na'b'y * [M x + M 2 ] 2 (6.17) which satisfies the claim for q = 0. For q = 1 observe that \XJ\2 < 2N2Ei=i_1 luj,i|2> where the Uj^ are the displacements of the hmi along the backbone from 0 to Xj (there are at most 2N — 1 of these). The resulting diagrams are the same as for the q = 0 case except that one 99 hmi(uj,i) on the backbone from 0 to Xj has been replaced with \ujti\2hmi(uj,i)- In view of Remarks 5.2.1 and 5.4.5, the only bounds on h m that we used to bound the diagrams without the factor \ujti\2 were | |/im i | |oo < a n d l l^mJIi < C. Since m,j < ||M||oo, (5.5) with I = l,k = 0 and (5.6) with / = 1 imply that sup„ \u\2h(u) < C°2WM\\™P2 and £ u \u\2h(u) < Co^MW^. We now apply Lemma 6.1.4 to get that the diagrams with the extra factor of | bounded by c 2 H-MHoo times the bound for the diagrams without the extra factor. Therefore when M 3 = 0, Y\uj\2^(u)< a 2 | |M | | 0 O (2iV) 2 Y supX;^(a,M,!/) u rne'HM1+M2,N a ' ' V x (ce2-*)N (6-18) <a2\\M\U2N)2-± L_. [Mi + M2} — Similarly for M i = 0, and M2 = 0. Since (6.18) is smaller than (6.3) this completes the proof. • 6.3 P r o o f o f L e m m a 6.1.2 In this section we prove Lemma 6.1.2, which gives a bound on the contribution to Y^x \x\2Q7r^(x) fr°m cyclic laces. We first consider the cyclic laces L G £^(<S^) containing only 3 bonds. There are multiple cases to consider, depending on how many bonds have common endvertices. For example, one needs to consider the number of those bonds that have the branch point as one of its endpoints (see the second row of Figure 6.3). • Consider the case where none of the three bonds in the lace L have the branch point as an endpoint. Without loss of generality the branch point associated to branch 1 has its other endvertex on branch 3 as in the first lace in Figure 6.3. Then for each i G {1 ,2 ,3} there exists 1 < rij < Mj that is the endpoint of e j + i (the bond associated to branch i + 1) on branch i. If nj < M j for all i then by first breaking the sum over UJ G ft (3) (x) into the M sum over three walks ojj G ftc(i)(xj), and then each of these into two further subwalks (using the same methods as in the proof of Proposition 5.1.7), it is (3) easy to show that the contribution to Ylx ^ ( x ) fr°m * m s l a c e L 1S bounded by 3 Y £ II hmi (uj)hMj-mj (xj ~ Uj )pV>(xj+1-Uj), (6.19) X u j = l 100 Figure 6.3: Some cyclic laces containing only 3 bonds Figure 6.4: A basic cyclic lace L containing only 3 bonds, its corresponding diagram F(L) and its decomposition into 3 subdiagrams, F\(L), Fi{L), Fs(L). where we use the convention that £ 4 = x\ (see Figure 6.4). We use the expres-sion "opening up" a diagram informally to mean that we drop the restriction that two specific lines have a common endvertex and take the sup over the displacement of their endvertices. For example, the diagram corresponding to 2~2X 0, x, 0) (see Definition 5.1.5) with both ends opened up (i.e. opened up at 0 and x) would be svcpby 2~2X M^(0, b,x,y). Opening up the diagram (which we denote F(L)) expressed in (6.19) at the vertices rij, Equation (6.19) is bounded by 3 (6.20) = s u p S E hni(u3)hMj-nj{xi ~ UJ)P{2)(xJ ~ bj), 3=1 h* 101 which is a product over 3 separate diagrams, F\(L), F2(L), F3(L), each corre-sponding to MJ^(0 , bj,Xj,0) with an extra vertex on the backbone (see 5.7). Suppose now that m\ — M\ (this is possible depending on the relative size of the Mi). Then two of the bonds have an endpoint at the end vertex of branch 1 and the contribution to Ys (x) from this lace L' is bounded by ^ ^ f t M , ^ ) ^ -Xi)p{-2\X2 ~Ui)hn2(u2) x u X hM2-n2(%2 - U2)p^(x3 - U2)hn3(u3)hM3-n3{x3 ~ U3)p^(ui - U3) < sup - x1)p^(x2 - b2)hn2(u2) bi,b2,b3 s a x hM2-n2(x2 - u2)p^(x3 - b3)hn3(u3)hM3-n3{x3 - u3)p(2)(«i - h) = supXX/lMl(a:i^^ 1 _ a ; i ) P ( 2 ) ( w i a 3 fcl Xl Ul X IISUP XI hnj{Uj)hMj-nj{Xj ~ Uj)p^(Xj - bj) 3=2 bi xj,uj = SW£>YhMi{x\)p^){xi -bx) 6 i xi x JJsup £ hnj{uj)hMj-nj{xj - Uj)p(-2)(xj - bj) 3=2 bi Xj,Uj (6.21) Once again this is a product of 3 separate diagrams F\(L'), F2(L'), F3(L'), each corresponding to M^. (0, bj,Xj,0) with an extra vertex (two on backbones and one on a p). • Consider now the case where one or more of the 3 bonds has the branch point as an endvertex. The diagrams arising from such laces depend on how many of the 3 bonds have this property, and each case is treated slightly differently. We present the most complicated case, where all 3 bonds in the lace L have the i i n r i g u i c u.o. i n c t u i i m u u u u n I U Z_*x from this lace L is bounded by branch point as an endvertex, as in Figure 6.5. The contribution to Ys n)J (x) ^J^)^)^ - w)p(w - z)pV>(x2 - z)pW(x3 -z)H hMi(xi), (6.22) x w,z j=l 102 plus two other terms (see Figure 6.5) of similar form arising from the possible shapes of a lattice tree containing 4 fixed vertices. Equation 6.22 is the first diagram in Figure 6.5 and is bounded by sup ^ ] ^ / ) ( ! o - ai)p ( 2 )(a;i - w)p{b2 - z) 61,62,63 g w,z 3 x pM(x2 - z)pW(x3 - 63) IJ hMi(Xi) = sup Y p(w ~ h)p^{xi ~ w)hMl{xi) b l 11,1x1 x sup Y P(°2 - z ) / > ( 2 ) ( X 2 ~ z)hM2 [x2) 62 x2,z (6.23) x s u p £ p ^ ( x 3 - h)h,M3(xz) = s u p £ p ( 3 ) ( x i - h)hMl{xi) x sup]T p ( 3 ) ( x 2 - b2)hM2{x2) 6 2 X 2 X S U p Y P ( 2 ) (^ 3 - 63)^3 (^ 3)-6 3 x3 Again this is a product of 3 separate diagrams F\(L), F2(L), Fz(L), each corresponding to MJ^(0, bj, Xj, 0), two of which have an extra vertex on some p. The two other terms give rise to the same bounds up to permutation of the indices. We have already bounded the contribution from diagrams with an extra vertex in Section 5.2.1. By (5.45) we have, 3 r R 2 - ^ E*M(*)<nTr< (6-24) s j=i Mj 2 which satisfies (6.4) with N = 3 and q = 0. Similarly by (5.49) we have, 3 x i=i Mi 2 &i Mj 2 KWMUoHcp-^ffl 1 (6.25) 3=1 M-2 d - 6 ' 3 103 2 Figure 6.5: The diagrams arising from a lace where all three bonds associated to a branch have the branch point as one of their endpoints. which satisfies (6.4) with N = 3 and q = 1. Therefore we have proved Lemma 6.1.2 for N = 3. At this point we know how to bound the diagrams arising from cyclic laces containing only three bonds. A cyclic lace L that contains N > 3 bonds has N — 3 additional bonds that do not cover the branch point. As such, each of the additional N — 3 bonds has both endvertices strictly on some branch j. Suppose that the number of additional bonds on branch j is Nj — 1, so that ]Cj=i Nj = N.- We perform the same operation of breaking up the diagram F(L) at the branch point and opening the diagram at each rij to get three separate diagrams F\(L), F2(L), F3(L), each (except in some degenerate cases that satisfy stronger bounds) corresponding to M^'\o,bj,Xj,0) with an extra vertex. This can be proved explicitly by induction on Ni, AT 2 and A T 3 . The degenerate cases are when rij is the endpoint of more than one bond for some j. By (5.45) we have, 2 ^ M ( Z ) < 2^ 11 o — * NUN2,N3 : J = l Mj 2 = (6.26) <iv3(^-^)wn4e. i = i M , 2 104 which satisfies (6.4) with q = 0. By (5.49) E i X J , . * S w < E t ^ ^ n ^ i ^ ' x NUN2,N3: 1 = 1 M{ 2 j^i Mj2 ZNi = N (6.27) i=i Mj2 which satisfies (6.4) with q = 1. This completes the proof of Lemma 6.1.2. • 6.4 Proof of Lemma 6.1.3 In this chapter we prove Lemma 6.1.3. We prove the Lemma by considering sep-arately the contribution to Trxj-(u) from laces with two bonds covering the branch a*N _ point and from laces with three bonds covering the branch point. We write ir^(u) aN . a+N for the contribution to (u) with two bonds covering the branch point and 7r $ (u) for the contribution to n^(u) with two bonds covering the branch point. 6.4.1 A c y c l i c laces w i t h 2 b o n d s c o v e r i n g t he b r a n c h p o i n t In this subsection we prove the following Lemma Lemma 6.4.1. For q € {0,1}, *N . .. ., . ., . . . . -.. . .. f &u\N 3 , , (6-28) x , d -6 . ' , d -4 ' =1 [Mi] 2 j^imj^MjWj-mj] 2 [Mk + rrtj] 2 where k 7 ^ Proof. As in the case of the cyclic laces, our strategy is to decompose the resulting diagrams into subdiagrams (3 in general) that we have already bounded in Chapter 5. . Consider the acyclic lace L G C?a containing only two bonds. A n acyclic lace contains a special branch with the property that there is only one bond covering the branch point with an endpoint on that branch. Without loss of generality we suppose that the (although there may be more than one) special branch determined by the acyclic lace L is branch 3, and that the bond e% associated to branch 3 has 105 Figure 6.6: A n acyclic lace containing only two bonds, its associated diagram and the decomposition into subdiagrams. its other endpoint on branch 2. We let m denote the endpoint of on branch 2 so that 0 < rn < M2. In addition we suppose that 0 < m < M2, so that the lace appears as in Figure 6.6. It is easy to show that the contribution to X^x 7 1 "^ (^) fr°m this lace is bounded by Y/~2hMdXl)hm(U2}hM*-m(X2 -U2)hM3{X3)P^(x2 - X i ) / 9 ( 2 ) ( u 2 - X 3 ) X « 2 <SUpYz72hm(u2)hM2-m(x2-U2)hMl(xi)hM3{X3) b2,b3 g U 2 xpW(b2-x2)p{2){h-x3) (6.29) = sup E hm(u2)hM2-m(x2 - u2)p^{b2 -x2) ^2 X2,U2 ' x sup E hMl{x\)hM3{x2)p{-2)(h - x3). b3 Xl,X3 Using translation invariance on the second term, this is a product of two subdia-grams, M ^ ( 0 , bj,Xj,0) with an extra vertex (that we bounded in Section 5.2.1) and M ^ + M s ( 0 , b j , X j , 0 ) with hMi+M3(x) replaced with {HMX *h,M3){x) (which we bounded in Section 6.2). Using (5.45) and (6.17) and summing over the permuta-tions of branch labels we have S ^ w s E E ^ n r ^ ' ( M 0 ) „ - 6 Z . « i=i &i M l ' Wi + Mt] * where k ^ This obeys the bound (6.28) with N = 2, q = 0., Similarly using (5.49) and (6.18) we have E " 2 M « < ^ iuES^fX/C% <6'31) i=l j#i [Mj] * [Mi + Mk] > which obeys the bound (6.28) with N = 2, q = 1. 106 For general acyclic laces L € for which only two bonds cover the branch point, we again suppose that the special branch is branch 3, and that the bond e3 associated to branch 3 has its other endpoint on branch 2. As before we let m denote the endpoint of e3 on branch 2 so that 0 < m < M2. Let e denote the other bond covering the branchpoint. We suppose that L has N\ bonds (other than the ones covering the branch point) strictly on branch 1 and Nj — 1 bonds strictly on branch j, for j = 2,3 respectively. Thus 2 + Ni + N2-l + N3-l = N. We also let mi denote the first vertex (from the branch point) strictly on branch 1 that is an endvertex of some bond in L. The reader should refer to Figures 6.8 to 6.11 when digesting the bounds that follow. We bound the Feynmann diagram F(L) for L by doing the following: 1. We define F\ to be the part of the diagram consisting of the backbone cor-responding to the interval from mi to M\ of branch 1, together with any p obtained from a bond with both endvertices on branch 1 (such a bond must have both endvertices strictly on branch 1, otherwise it would be a third bond covering the branch point). In the degenerate case that mi is also an endpoint of e, the p incident to the backbone at mi is also considered part of F\. We open up F\ at mi . Note that if mi = M\ then F\ is defined to be empty (compare with the N = 2 case). Note further that (except in the degenerate case already discussed) the convolution of two p's obtained from bond e is not considered part of Fi, and thus F\ contains either an extra vertex on the backbone (if the endpoint of e on branch 1 is not the endpoint of any other bond) or on a p (if the endpoint of e on branch 1 is the endpoint of some other bond). 2. We define F3 to be the backbone corresponding to branch 3, together with the backbone corresponding to the interval 0 to mi on branch 1 and with any p derived from a bond with an endpoint strictly on branch 3. In particular we take the p * p obtained from bond e3 as part of F3. We open up the diagram F3 at m (leaving a extra vertex on F2). 3. We define F2 to be the backbone of branch 2 along with any p corresponding to a bond with an endpoint strictly on branch 2 (except for the bond 63). We open up the diagram F2 where it meets F\ so that the p * p corresponding to e is part of F2. Note that the only properties of the acyclic lace L (that has 2 bonds covering the branchpoint) that are important when constructing of F\,F2 and F3 are the bonds that cover the branchpoint, and more specifically, whether or not the endpoints of 107 x2 x l 0 X 3 Figure 6.7: Examples of acyclic laces with only 2 bonds covering the branch point, and their decomposition into opened subdiagrams. those bonds on branches 2 and 3 are also endpoints of some other bonds. To help the readers understanding of this construction we give 3 figures giving examples of the different possibilities which may arise depending on e and e3. As in Figures 6.7 and 6.8, this leaves us with 2 (if m i = M i ) or 3 subdia-grams (in general Fi(L) and ^ ( L ) contain an extra vertex), that we have already bounded. By (5.45) and (6.17) and summing over the possible locations of m i and 108 permutations of branch labels, <N E < E E E ^ T T T ^ -ueZ3d NI,N2,N3 i=l jjti [Mil 2 ^ (Cfr-*)Ni (c^)Nk X E m <Mj [Mj - mj] 2 [Mk + mj} d - 4 2 &i [Mi] 2 m ^ M , - [Mj - mj] 2 [Mk + mj] 2 (6.32) which satisfies (6.28) with q = 0. Similarly by (5.49) and (6.18) E Kf4(-) <N^\\M\UN3 (Cp*-*)" 3 ( 6 - 3 3 ) X E 777^6- E E , d - 4 ' 2 j t i m j K M j iMJ ~mj] 2 [Mk + rrij] 2 which satisfies (6.28) with q = 1, and completes the proof of Lemma 6.4.1. • 6.4.2 A c y c l i c laces w i t h 3 b o n d s c o v e r i n g t he b r a n c h p o i n t In this section we bound the contribution to 7r from (acyclic) laces that have 3 bonds covering the branchpoint. The idea is similar to that of acyclic laces with 2 bonds covering the branchpoint, but the contribution from non-minimal acyclic laces requires careful treatment. In particular for non-minimal laces we need the following two definitions and lemmas. We refer to these lemmas as 4-star lemmas, and they are proved later in this section. Define 9MUM2(XUX2, w) = E E E nmi{ui)hMl-mAxl ~ ul) m\<Ml 7712 < M 2 « 1 > « 2 (6.34) x p ( 2 ) ( u 2 - ui)hm2(u2 - w)hM2-m2{x2 + W - U2). This can be seen diagrammatically in Figure 6.9. 109 0 Figure 6.8: A n acyclic lace with only 2 bonds covering the branch point, and its decomposition into opened subdiagrams for which we have existing bounds. The branches are labelled 1 to 3 from left to right. Figure 6.9: On the left is a so called 4-star diagram of (6.34) for the case I = 2 which is shown in Lemma 6.4.2 to obey the ||.||i and ||;||oo bounds of the diagram on the right times a factor C/3 d . 110 M-m Figure 6.10: On the left is another called 4-star diagram which is shown in Lemma 6.4.3 to obey the | | . | | i and H-Hoo bounds of the diagram on the right times a factor Lemma 6.4.2. The 4-star diagram gMi,M2{xi,x2,w) satisfies the following bounds supsupgMuM2(xi,x2,w) < CB d —j—j, X l 1 2 M* M 2 2 YsuP9MuM2(x-i.^x2,w) < OB2'-* d 5 M i 2 (6.35) gMuM2{Xl,X2,w) < CBZ d j - , 1 1 x2 M* Xl x2 For the second 4-star Lemma we let b(x) = ^{x=o} + 1 , 2 - ^ 1 ^ - 2 • Clearly \x\ < \y\ implies that b(x) > b(y). Note from (1.13) that p(x) < Cb(x), and from Remark 5.4.4 that the only information about p that we used in bounding the diagrams was that p(x) < Cb(x). Lemma 6.4.3. (-< a2 sup Yi Y h m ^ h M ~ m ^ x ~ « ) P ( 2 ) ( « - U ) P ( v -y)p(z -v)< -y)P2~^, X m<M u,v Ml Yi YI Y L h m ^ > h M - m ^ x ~ U ) P ( 2 ) ( U - u)p(v - y)p(z - v) < Cb(z - y)P2-6d . x m<M u,v (6.36) We prove the following Lemma, assuming Lemmas 6.4.2 and 6.4.3. Lemma 6.4.4. For q e {0,1}, •aez3d 3 x x l (6.37) x z\Z r , r i ± z 6 YI YI 771 7HI771 ,d-4' i=i[Mi\ 2 j#m.<M.[Mj - mj] 2 [Mk + mj] 2 111 M, M, i Figure 6.11: Basic acyclic laces with only 3 bonds covering the branch point, and their decomposition into opened subdiagrams. where k ^ Proof. We begin with an acyclic lace L € Cl containing only 3 bonds, all of which cover the branch point. We suppose that the special branch is branch 3 and that the bond ez associated to branch 3 has its other end vertex on branch 2. In general (see for example Figure 6.11) this means that each of the bonds associated to branch 1 and 3 have an end vertex on branch 2. We let m 2 < M 2 denote the first vertex (from the branch point) strictly on branch 2 that is the end vertex of some bond / in L. We wil l also assume that no endvertices of the 3 bonds coincide. When some such endvertices do coincide we must use a decomposition similar to what follows with adjustments as we did for the cyclic laces. Consider the first lace of Figure 6.11. Let m 2 < M 2 denote the first vertex (from the root) strictly on branch 2 that is the endvertex of some bond / in L. Here m 2 is an endvertex of the bond associated to branch 3 and the endvertex of the bond associated to branch 1 is therefore at some m with m 2 < m < M 2 . It is easy 112 to show that the contribution to J^s ^ (x) fr°m t m s ^ a c e * s bounded by E E / l m i ( U l ) / l M i - m i ( a ; i ~ u^)hm2(us)hm-m2(U2 - U 3)/lM 2-m(^2 - U2)flM3[xz) £ u X PW(XI - U2)P (2)(X2 - K l ) p ( 2 ) ( x 3 - 7 J 3 ) <SUp E hm1{ui)hMl-rnl{xi - U i)p ( 2 ) ( z i - bi) SUp E hm-m2(u2 ~ u)hMl-m{x2 - U2)p^(x2 - 62) X u,b2 u2,x2 X SUp E hm2(u3)hM3(x3)p(-2Hx3 ~ 0.3)-b s U3,X3 (6.38) This is a product of three diagrams, two of which contain an extra vertex and the other with hm2+Mz replaced with hm2 * /IM3- The other lace of Figure 6.11 gives a similar product. By (6.17) and (6.18), and summing over the permutations of branch labels we have E < E Trhor E R „ ( 6 - 3 9 ) where ^ This obeys the bound (6.37) with N — 3, q = 0. Similarly using ( 5 . 4 9 ) and (6.18) we have E tM(u) < ^ I I M I U E E R ^ r / f " % ' ( 6 - 4 0 ) ffeZ3d < = i j ¥ i [Mfc] 2 [Mj - mj] 2 [Mj + mj] 2 which obeys the bound (6.37) with N = 3, q = 1. For general L € £ ^ with 3 bonds covering the branch point, if L is a minimal lace (see Definition 2.1.7) then we proceed as before with m2 < M2 denoting the first vertex (from the branch point) strictly on branch 2 that is the endvertex of some bond / in L. We leave it as an exercise for the reader that by breaking the diagram at m2 and 0 we obtain a product of three diagrams that we have already a+N bounded, giving a bound on the contribution to J2uez^ \uj\2q n M(U) from minimal laces of N3(N2a2\\M\Uq(cp2-^Y ° 1 v v _ _ J l _ (6-4 1) i=i [Mi] 2 j # m j < M [Mj-mj] * [Mk + mj]2 113 Figure 6.12: A n application of Lemma (6.4.2) to remove the bond associated to branch 2. i Figure 6.13: A n application of Lemma (6.4.3) to remove the bond associated to branch 2. Therefore we are left to prove a bound of the form (6.37) for the contribution from non-minimal laces. We wil l argue that such a lace has a bond that we can "remove" in such a way that the resulting diagrams are diagrams arising from an acyclic lace V G C^~l (with two bonds covering the branch point) that we have already bounded, together with an extra factor of 82~'*. There are many different cases to consider, depending on which bond (e2 or e i ) is removable and how many endvertices of that bond are an endvertex of some other bond in L. We wil l present the argument for the three cases where e 2 is removable and leave the others as an exercise. From this point we assume that e 2 is a removable bond. Case (0). Suppose that neither of the endvertices of e 2 are the endvertices of any other bond in L. Then we use Lemma 6.4.2 to remove the bond e 2 and obtain the extra factor / 3 2 _ ^ , as in Figure 6.12. This is a non-trivial consequence of Lemma 6.1.4 and so we give further explanation. However this explanation is one of the most notationally difficult parts of this thesis, so we don't give every detail. Removing the bond e 2 from the lace L leaves an acyclic lace L' — L \ e 2 € CN~l 114 Figure 6.14: Another application of Lemma (6.4.3) to remove the bond associated to branch 1. with two bonds covering the branch point, which we analysed in Section 6.4.1. Recall that we bounded the contribution to ]C«eZ3d luil297r -^(^ ) fr°m s u c h laces by breaking up the diagram F(L') for each V at m\ and 0 into three subdiagrams Fi(L'), F2(L') and Fz(V). The bounds on those diagrams relied only on the bounds p(x) < b(x), ||/im||oo < ^ \ a n d | | r i m | | i < C when q = 0, 2 2 and in addition the bounds supx \x\2hm(x) < a mC^ and J2x\x\2a™(x) -ml Cma2 when q = 1. Let JVj denote the number of bonds that contribute to diagram Fi in this decomposition of F(L'). Let m i be defined as in Section 6.4.1 as the first vertex from the root on branch 1 that is the endvertex of some bond in L' that has an endvertex strictly on branch 1. Either the endvertex m * of e 2 on branch 1 is greater than m i or less than m i (it is not equal to m i by definition of m i and the fact that neither of the endvertices of e 2 are the endvertices of any other bond in L). We can write an explicit bound for the contribution to Y l x n N f ( x ) fr°m * n e lace L in terms of a diagram F(L) consisting of various convolutions of p's and h m ' s . In particular that diagram contains a term p^(u — u') obtained from the bond e 2. We break up this diagram at m i and obtain a product of two subdiagrams, which we denote by Fi(L') and F'(L) if m * < m i and Fz(V) and F'(L) if m * > m i . We consider only the case m * < m i , as the proof of the other case is very similar. When m * < m i (see Figure 6.15) the diagram Fi(L') is the same diagram we obtain when estimating F(L') and is bounded by s u P a i , 6 i Z ) x i M^lS>'ni(ai,h,xu0), where rh e 'HM1-m1,Ni, and nx denotes the location of the extra vertex. As in Figure 6.15, F'(L) is the diagram Fs(L') with the first factor hk(v — v') of the backbone being replaced by a diagram F ^ L ) ^ — w). Thus F'(L) is bounded by sup a 3 ] 6 3 M^'*(a3,b3,x3,0), where M^'* is the dia-115 Figure 6.15: A non-minimal lace L and the subdiagrams Fi(L') (the bottom left subdiagram in the second figure) and F'(L). gram M^3^ with one specific factor hk(v - v') being replaced by the diagram F'(L)(v - v'). Furthermore /^(LXu - v') is itself the diagram F2(L') with one of the first three factors h[(u — v!) on the backbone being replaced by gk,i(u — u',v—v', v'), and with an extra vertex at n2. Therefore F^LXu—v') is bounded by s u P a 2 ) 6 2 M^™(a2,b2,X2,0)(v - «'), where M^™(.)(v - v') is the diagram M^2\») with one of three factors hi(u — u') being replaced by 9k,l(u — u',v — v'iv'), and with an extra vertex at n2. A + N It follows that the contribution to Ylx ^ M ( X ) fr°m non-minimal acyclic laces such that: the special branch is branch 3, e 3 has its other endvertex on branch 2, e2 is removable, and has no endvertices in common with any other bond, and m* < m i is bounded by c E E E E SUPY^T'^M^) NI,N2,N3: rni<M1m1&'HM1-rnl,N1ni<M1-niai' 1 xi x E sup£M^ 3 ) ' * (a 3 ,&3,Z3,0) . (6.42) Here M~^^'*(a,3,63,£3,0) denotes the diagram M^\a3,63,a:3,0) with the first factor hm(v — v') of the backbone being replaced by J2 E s u p £ M ^ 2 ) ' N 2 > 2 , 6 2 , * 2 , 0 ) ( < , - ^ ) , (6.43) rn2eUM2,N2n2<M2a2'b2 x2 and Mjj 2 ) ' n 2 ' 9 (a 2 ,b 2 ,x 2 ,0)(v - v') denotes the diagram M £ 2 ) , n 2 ( a 2 , b2,x2,0) with a specific factor hi(u — v!) being replaced by gk,i(u — u',v — v',v'). 116 We prove that (6.42) is bounded above by cf-'-f Y. E ( C ^ 2 - ? ) " ' ( C / 3 2 - T ) * (Cfi-i)"' NUN2,N3: m ^ M i [ ^ 1 - ^ l ] 2 M 2 2 [ M 3 + mX] 2 £ JV; = JV - 1 (6.44) which satisfies the bound (6.37) of Lemma 6.4.4) with q = 0. By (5.45) we have Y E S U P E M m ( o i , 6 i , a ; i , y i ) < — -3=5-M M r m i , N l » < M i - ™ i n i ' 6 1 * * i [ M i - m i ] 2 (6.45) By Remarks 5.2.1-5.4.5 and (6.17), the bounds sup t,_ lI/ hi(v - v') < and 2~2v-v' hi(v — v') < K, together with bounds on the rest of the lines in F3(L') imply that E sup £ M ^ ( a 3 , 6 3 , x 3 ,0 ) < f(C ~^*}2i- (6-46) m € ^ M 3 + m i , i V 3 a 3 ' b 3 *3 [ M 3 + mi] 2 Therefore by Lemma 6.1.4, to prove that (6.42) is bounded by (6.44) it is enough to show: sup YI ^pYMmN2),n2,9(^b2,x2,0)(v-v') v~v' meHM2,N2n2<M2a*'b* x2 ^ d ~6 (6.47) 7-2 k M 2 2 and r(N2),g,n, E E E s u p £ M ^ ( a 3 , 6 3 , x 3 , 0 ) ( , - , ' ) v-v' me'HM2,N2n2<M2a2'b2 x3 M 2 2 (6.48) Again by Lemma 6.1.4, to show (6.47) it is sufficient to prove CB2 61/ CB2 sup sup gktl(u -u',v- v', w) < -r^CB2'^ (6.49) v—v' u—u' ™ ' and sup Y 9kAu ~ u'^v ~ v > M < ^TCP2~^- (6.50) 117 Similarly, to show (6.48) it is sufficient to prove E , / / \ o 61/ C B2 sup gkj(u - u,v - v ,w) <CBZ 1 — 5 - , - (6.51) , u—u' * and 61/ (6.52) v—v' u—u' But (6.49)-(6.52) are exactly the statements of Lemma 6.4.2. This proves that (6.42) is bounded above by (6.44) as required. Case (1). If exactly one endvertex of e 2 is also the endvertex of some other bond (by definition of a lace, in the case we are considering here the endvertex of e 2 strictly on branch 2 could only be the endvertex of e3) then we proceed as in case (1) except that we use Lemma 6.4.3 instead of Lemma 6.4.2 to remove the bond e 2 and obtain the extra factor /32~~^, (see Figure 6.13). Case (2). Finally suppose both endvertices of e 2 are also the endvertices of other bonds in L. Then (exercise left for the reader) e\ is a bond with the properties that at least one of the endvertices of e\ is not the endvertex of any other bond in L, and L \ e\ is a lace. Then, depending on whether or not one endvertex of e\ is the endvertex of another bond in L, we use Lemma 6.4.2 or 6.4.3 to remove e\ and obtain the extra factor B2~ ~^. We have now proved that the contribution (up to permutation of branch labels) to ^2g^^\x) from non-minimal acyclic laces with 3 bonds covering the branch point is at most M1 2 m 2 < M 2 lM2 ~ m 2 ] 2 [M 3 + m2\ 2 For the q = 1 case of Lemma 6.4.4 we use \XJ\2 < 2N'Y^,2N _ 1 \uj,i\2 (tms gives the TV 2 factor) where the Ujj denote the displacements along the backbone of diagram Fj(L'). If the extra factor \UJJ\2 occurs on a part of the diagram F'(L) where F\(L') and -F 2(L') are joined by gk,i then we can use Lemma 6.1.4 to include a factor | u | 2 on one of the lines in the 4-star lemmas, and proceed to get the extra factor cr21| Af || O Q . Otherwise the extra factor p-2||iVf ||oo comes by applying Lemma 6.1.4 to the diagram Fi(L') where the |uj,/,| 2 is attached. This proves that the contribution (up to permutation of branch labels) to Ylg\xj\2ir^(x) from non-minimal acyclic laces with 3 bonds covering the branch 118 point is at most i V 3 ( A T V | | M | U ( C / 3 2 - ^ ^ £ L-—Z=E- L - i z I . (6.54) M x 2 m 2 < M 2 [ ^ 2 - m 2 ] 2 [M3+m2] 2 This completes the proof of Lemma 6.4.4. • Proof of Lemma 6.4.2. For the first bound, note that either mj < y or M j - mj < ^ . Breaking up the sums over m i and m 2 according to these restrictions gives rise to 4 terms. One such term is ]T E YI hmi(ui)hMl-mAxl ~ Ul)p(2)(u2 - U l ) X h m 2 (U2 ~ w)hM2-m2 [x2 + W - U2) . M i m i < ^ m 2 < M i U 1 (6.55) X S U p / i M 2 - m 2 ( « 2 +UJ - U 2 ) ( h m i * p ( 2 ) * / j m 2 J (ttf) u2 ^ ' <—i"— j 2^ 2^ d=4<°p d—T—r» M x 2 ^ ^ ^ ( ' " i + m j ) 2 M ^ M / where we have used 5.5 with I = 1 and k = 0 with the fact that M j — m.j > ^ on the / i M i - m ; ' s and with Z = 2 and A; = 2 on the convolution of /i's and p's. By similar arguments we get the result for the other 3 terms which proves the first bound. For the second bound we again split the sums over m i and m 2 to leave us 119 with 4 terms. The most difficult to bound is £ s u p Y X X hmi{ui)hMx-mAxi ~ ui)pW(u2 - ui) Xl . M l m 2 < ^ I « l . « 2 x hm2 (u2 - w)hM2-m2 (x2 + w- u2) -^ "zCSUP X X X hM1-m1{xi ~Ui)p{2){u2 - ^ l ) Af, ! Z2 X l ^ M i ^, M i K i ,Uo X / i T O 2 (^2 - w)hM2-m2 (x2 + W - U2) CB2 ^ L < J - S U P 2^ hM2-rn2 (X2 +W - U) X2 ( s u p E E zC hMi-mAxi - ui) p(2\u2-ui)hm2(u2-w) (6.56) A/f, i i , i . r . A,f, ^ M i ,ito \ . m 2 < Y \ " l l > C / 3 ™ < j SWp 2_^hM2-m2{X2 +W - U) M* u x2 x sup Xl XI X P(Xl ~ « l ) P ( 2 ) ( « 2 - « l ) / l m 2 ' ( « 2 - ™ ) m 2 < M l « l , « 2 < ^ - s u p , £ £ p ( 3 ) ( n 2 - xi)hm2{u2 - w) '1 m 2 < - 2 L J < C / 3 2 ^ C / ? 2 ~ T C / ? 2 sup X M x 2 3 : 1 ^ ^ M r m / < -^KCB'-ir, M x 2 where we used Proposition 5.1.4 in the penultimate step. The third bound follows from the second by symmetry and taking the sup outside the sum. 120 For the fourth bound we see that E E E E hmiMh-Mi-rmixi - u i )p ( 2 ) (u 2 - Ui) Xl,X2 m\<M\ 77X2 <M2 U\,U2 X hm2 (u2 - U))hM2-m2 {x2 + w - U2) ^ C Y Y Y hmi(ui)p^{u2 ~ Ul)hm2(u2 - w) m i < M i m2<M2 u\,u2 <C E Y Y hrnAuijiiMx-mAxi - u i )p ( 2 ) (u 2 - ui)hm2(u2'- w) ^6'57^ mi<iW " i m 2 <JVf 2 u i , U 2 m 1 < M i m 2 < M 2 [mi + m 2] 2 <C/32-^". • Proof of Lemma 6.4.3. Firstly since \z — y\ < \v — y\ + \v — z\ < 2(\v — y\ V \v — z\) we have J]p(i)-«)(2V(«-yV(2-«) V v . \ v - y \ > ^ . v : \ v - z \ > ^ < c E p{2) (v - u^z - y}p(z -v) + C Y p{2) (v - u")p(v - yWz -< Cb(z - y) (p(3) (z-u)+ p ( 3 ) (y-u)), (6.58) Therefore for the first bound, s u p E YHM^HM-M^X'~^YP^^~U)P(vv)p(z'~ v) X m<M u v <Cb(z-y)swp E YHM^HM )(pW(z-u) + pW(y-u)) <C^b{z-y) £ ^ < ^ - y ) ^ , Mi .M m 2 Mi m<f where we have used Proposition 5.1.4 and the fact that either m > y O r M - m > 4f 121 m<M u ' ' (6.59) For the second bound, we again use Proposition 5.1.4 to get, Y Y Yhm^hM~m^x ~ U ) Y P ^ ^ V ~ u^p^v ~ y^p^z ~ v^ x m<M u v <Cb{z-y)Y Y Y h m ^ h M —m(x U x m<M u <Cb(z-y) Y Yh^u){p(3)(z-u)+p^(y-u)) ( 6 - 6 0 ) m<M u a2- — <Cb(z-y) £ ^ < C b ( z - y ) ^ . m<M m 2 • 6.5 P r o o f o f L e m m a 4.3.3 We now prove Lemma 4.3.3, the companion of Proposition 4.3.2. Recall the defini-tion of BN(M) from 6.2. Lemma (4.3.3). There is a constant C independent of L such that Y Y N3BN(M) < C ' 0 2 J , and (6.61) ( 10-r f Y Y N'WMWooBNiM) < N M <n VO H o c 2 , tfd^io ( g 6 2 ) log||n||oo, i / d = 1 0 . Proof. Summing over ./V first gives the factor CB2 *d , for small enough 8. Summing over each Mj separately we have, Y -ir<HLr> (6-63) M:Mj>nj Mj 2 M 2 and 3 ( 10-rfyo E i w i o o n 4 ^ < I H I o ° 2 ' (6-64) M<n j=iMj2 [log||n||oo, i f d = 1 0 , as required. This verifies the Lemma for the first component of BN(M). 122 For the second component of BJV (M ) (ignoring the sum over j) we sum over M i separately to get V ^ - V _ _ J L M ^ ^ M , 2 m 2 < M 2 [ M 2 - m 2 ] 2 [ M 3 + m 2] 2 " l ^ ^ 1 1 { 6 - 6 5 ) [«i] 2 M2,M3m2<M2 [M2 -m2} 2 [ M 3 + m 2 ] 2 We leave it as an exercise for the reader to show (by summing separately over m 2 < ^2 a n d m 2 > ML) that E E 77; T^TT, r e r ^ - ( 6- 6 6) M 2 l M 3 m 2 < M 2 [Afe - WI2] 2 [ M 3 + m 2] 2 Furthermore E i = ± E M:M2>n2 Ml 2 mt<M2 W2 - m 2 ] ° '2 6 [ M 3 + m 2 ] V <c y y y 1 1 M 2 > r a 2 M3 m2<M2 [M2 ~ rn2\ 2 [M3 + m2]~ S C E E E 1 1 , d - 6 r _ „ , d - 4 M 2 > n 2 M 3 mo<Mz t M 2 - m 2 ] 2 [ M 3 + m 2] 2 — < K 2 ^ 2 . 1 1 + c E E E •' fc.- ^ M 2 > n 2 M 3 m 2 > ^ 2 . t M 2 - m 2 ] 2 [ M 3 + m 2] 2 — ^ Z\Z / E 77T T d - 4 2 > n 2 [M2] 2 M 3 m , t M 3 + m 2] 2 M 2; + C E E E d - 6 I , , d - 4 M 2 > ^ M 3 \ M 2 > m 2 [M2 - m2] 2 y [ M 3 + m 2] 2 n 2 m Q > ^ M3 [Ms + m2] 2 2 ^ [ m 2 ] 2 2 m2>-g- 1 ' J "-2 (6.67) Similarly we get E ~l=fi E 77; ,<j=fi TT; , d = 4 < ~ E F - ( 6 - 6 8 ) M : M 3 > n 3 M i a m 2 < M 2 [^2 - m 2] 2 [ M 3 + m 2] 2 n 2 2 After permuting the labels 1,2,3, this verifies the first claim of the Lemma for the second component of B^(M). 123 For the second claim we need to show that E ll^ ooll—1=!- E r„ 1 lfcfiri> 1 ^ { M < n M1 2 m 2 < M 2 [M2 ~ m2] 2 [Af3 + m2] 2 i|n||oo2 , if d + 10 log ||n||oo, if d = 10. (6.69) If 11 Moo 11 oo = M\ this follows easily by summing first over M i and using (6.66). If llMHoo = M2, as in (6.67) we get a bound of c E ; ^ b E E 1 Ah [M2]"^ M 3 m 2 < M 2 _ [ M 3 + m 2 ] V (6.70) + C E E E m2 m 2 M 3 \M2>m2 W2 ~ m2] 2 / [ M 3 + m 2 ] 2 and the result follows by the same methods that we used for (6.67). Similarly we get the result if ||M||oo = M 3 . After permuting the labels 1,2,3, this verifies the second claim of the Lemma for the second component of Bj^(M), and thus proves the Lemma. • 6.6 P r o o f o f L e m m a 4.2.1 In this section we prove the three bounds of Lemma 4.2.1, and Lemma 4.4.1. F ix a'skeleton network jV(a,n), with a £ E r and recall Definition 2.1.1, where b is the branch point neighbouring the root of jV. Let M. C j\f(a,n). If Ust € {—1,0} for each st, then trivially for any A C E ^ , n [ 1 + < n i 1 + u «] , (6-7i) so that in particular for any finite collection of disjoint sets G j C E^vi, n [i+u,t]<Yi n [i+^i- (6-72) ste^M i steGi We wil l use these bounds frequently without explicit reference. Before we proceed with our analysis of certain error terms appearing in Lemma 4.2.1 we quickly verify a trivial result, Lemma 4.4.1, where #M is the number of branches in M.. Lemma (4.4.1). There exists a constant K, independent of L, M. and K such that for any network M tM(K) < K*". (6.73) 124 Proof. Label the branches of M, 1,. . . , #M a n d write r i j for the length of branch i. Then the vertices of M. can be relabelled by (i,rrii) where i is a branch and 0 < rrii < r i j is distance along the branch i. Note that branch points of M. wil l receive multiple labels. Using #M J[[l + Ust]<\{ n ll + Ust], (6-74) steM ' »=i st e A^J = 0 < s < t < ni we have that #M £ « M ( y ) < / » ( o ) ^ n & < W ' ( 6 - 7 5 ) y-£.1&(#M> *=1 2/i The result now follows from Proposition 5.1.4 with / = 1 and k = 0. • 6.6.1 Proof of the first bound of Lemma 4.2.1 Recall that <p^f(y) > 0 was defined in (4.18) as E n E (n t1+^) (1 - n t 1 + ^ ) > (e-? 6) where Uat is given by (4.15). In this section we prove that E ^ ( ^ E T A ( 6- 7 7) y eeE ne 2 Let Me denote the branch of M corresponding to edge e of a and let lZe,e' = {st € TZ : s G A/"e, * G A/'e'}- We claim that when Ust G { —1,0} for all st, i - n [ i + ^ i < E h - n t i + c / ^ < e, e' e BJV" : me <ne Me n A/"E' = 0 m e/ < n e/ (6.78) e,e where the sum over e, e' is a sum over pairs of edges of a that do not have an endvertex in common (which can be expressed as Me r\Ne> =0). To verify (6.78), observe that each of the quantities 1-Y[[l + Ust], 1 - [I [1 + U8t], - U i e > m e W t < ) , (6.79) steiz steiie'e' 125 are either zero or one. Suppose the left hand side of (6.78) is non-zero. Then there exists some st G 1Z with Ust = — 1- By definition of 1Z, st covers two branch points of j\f so that st G TZe'e> for some e, e' that do not have a common endvertex. For this e and e', we have 1 — YlsteHe'e' [1 + Ust] = 1 and the first inequality is verified. Now for fixed e, e', if 1 — Ylsteiiee' ^ ^s*] * s n o n ~ z e r o the*1 there exists st G 1Ze'e> with Ust = ~1- But s = (e,me), t = (e',mei) for some me < ne, mei < nei so that for this me and m e ' , — f(e,me),(e',m^) = 1- This proves the second inequality. Examining the second quantity in (2.4) when Ust G {—1,0} for all st we have, o < n [i+ust}(i-H{i+ust\\ st£Etf\1l \ st£Tl ) < E E n [ i + ^ i e.e'GBjv-: me <ne st£EM\Tl Me n Mei = 0 m e/ < nei * E E [-^ :;<]n n e , e ' 6 % : m e < ne /#e,e' s,teMf: (6.80) .A/e n A/"e/ = 0 rrv < n e ' 0<s<<<n^ x n [i+c/^ n [i+c/^ s,t£jVe •• s,teMe • 0 < s < t < m e m e < s < t < ne x n [ i + ^ t ] n [ i + o * t ] , 0 < s < t < mei me < s < t < nei where we have used (6.72) in the final step. Breaking up w (in 6.76) at every branch point and at (e,m e) and (e',mer) and applying inequality (6.80) we obtain E ^ ( y ) < P ( o ) 2 R - 2 E E E II M w ) V V e, e' e : me < ne /#e,e' Me n A/"e/ m e/ < nei xYhm^u~Ve^hne~rne(ve(y) + ye-u) ^6-81^ X hme, (v! - Vei{y))hne,-me, (Ve' + Ve'iv) ~ u')p^{u - u ' ) , where ve(y) = Ylf^eVf a n < ^ t n e notation / -we denotes the set of edges in a on the path from the root to edge e (not including e). Rearranging sums we get that 126 (6.81) is bounded by a constant times E n E M * / ) e , e ' £ Bjv = / ^ e > e ' A/ e n A/„, S U P E E hm* (U ~~ v)hne-me (v + ye~ u) (6-82) X v,w x hme, (v! - w)hne,^me, (w + ye> - u')p{2\u - u'). By translation invariance, the last two lines of (6.82) are equal to the sup over z °f Ylxi x2 9ne,nei(xi,X2,z), one of the quantities that we bounded in Lemma 6.4.2. However we now need to prove a stronger bound than that appearing in Lemma 6.4.2. Break up the sums over m i into the two terms m i < ^ and m i > ^ and similarly for m 2 . Then we are left with 4 terms, one of which is E E hme{u\)h ne-me X l , Z 2 , « l , « 2 m e > Se. " V > -f-x hme, (u - w)hne-me(xi - u)p ( 2 ) (u - v!) = £ £ hme(ui)hrne,(u - w)p(2\u - v!) r; 1 % (6-83) X E ^ n e - m e O ^ l - « l ) £ ^ n e , - m e , ( a : 2 + ~ U2) X\ X2 < E m e > ¥ [ m e + m e , ] V where in the last line we have applied Proposition 5.1.4 multiple times. By first summing over the minimum of m e and m'e this is bounded by a constant times meC(32~% ^ meiCfi 2~T-> ^ [m ej 2 ^ [ m e / ] V e " 2 (6.84) C/3 2~^ C/3 2 -^ d - 8 1 d - 8 2 ^ , 2 e' The other 3 terms give the same bounds by symmetry. 127 We may now sum over each y/ in (6.82) separately, and using Proposition 5.1.4 with I = 1 and k — 0 gives £ « # ( * ) < £ C0 2 - ^ d - 8 ' 2 (6.85) y e£E n e ' where the constant also depends on r. This proves the first bound of Lemma 4.2.1. • 6.6.2 P r o o f of the t h i r d bound of L e m m a 4.2.1 Recall that <fij/(y) was defined in (4.23) as M(V)= £ W(u)U E W & ) E ^ A (m ) )n^((AT\5 A (m)) i ) , u££lM(.y) seAf RseT{oj(s)) meHnb i = l (6.86) where Unb = ({rh : 0 < m, < nu i = 1,2, 3} fl {rh : 0 < m,j < n{ - 2, i € G}) \ %nb (6-87) and Unb = {rh : 0 < mj < -± i - 1,2,3} n { m : mj < m - 2, t G G}. (6.88) o As in Lemma 4.3.4, |<%/-(y)| is bounded by 3 C E E E I I E ~ U')*AA- ) mena. fi (6.89) <^EE E ^ w E f i E ^ - ^ V G W , i= l Vi where j\ff = (j\f \ <S4), and yVi denotes the vector of displacements associated to the branches of j\f~ (determined by v, y, and the labelling of the branches of AA). Summing over the Uj and y and using Lemma 4.4.1 this is bounded by oo 3 <?E E E ^ n ^ -^ I m e ^ j fi «=i oo = c £ £ £4ro N=lm€Hrib fi oo 3 3 rR2-% <E"3E E B„W)<T, 0 (6.90) i=1 m:mj>?j-d - 8 ' 128 applying Proposition 4.3.2 and Lemma 4.3.3 in the last line. This verifies the third bound of Lemma 4.2.1. ' • 6.6.3 Proof of the second bound of Lemma 4.2.1 Recall the definition of 4>\r{y) in (4.20). In this section we prove that Xl^(y)l<c£4r- (6.91) j = i n . It follows immediately from the definition of <fi%-(y) that ojefijviy) serf Rser(ui(s)) (6.92) where Z\f is denned in Definition 2.1.1 and is only nonempty if Af contains more than 1 branch point (r > 4). In particular recall that graphs in £ b^ contain no bonds in 7c. We use an approach similar to that of [22] to analyse Let G(N) C {2,3} be the set of labels of branches of jV incident to b and another branch point of j\f. For F C G and e G F, let be be the other branch point in jV incident to branch j\fe. Let Ep/j- = {r G £tf '• for every e G F, Ab{T) contains a nearest neighbour of be} Then, £ n > = E E n>- E n^ > (6.93) (6.94) res w ster where some of these sums could be empty if G ^ {2,3}. Thus, E n>« < E F C G(Af) F ^ 0 E (6.95) Note that if r = 4 then one of £ b ^ ^ or £b3y ^ is empty and £ b 2 is empty. This may also be true for r > 4, depending on the shape a. .Define Fp C F to be the set of bonds st G F such that st is the bond in F associated to e at b for some e G F, or 129 Figure 6.16: A n illustration of the construction of a lace from a graph on some Af in the case 61,62 G -Aj\f(T). The first figure shows a graph T on a network Af. The remaining figures highlight the subnetworks <SF(T) for F — {2}, {3}, {2,3}. • st is the bond in V associated to e at 6E for some e € F and be € Ab(F), or • s,t G Afe for some e E F. Let <SF(F) be the largest subnetwork of Af covered by Tp C T . Clearly r | s F ( r ) = Tp is a connected graph on SF(T). For each e E F , <SF(F) by definition contains a nearest neighbour of 6E in Af, and may contain be itself. Since TF contains at most one bond that covers 6E, if be G SF(T) then it is not a branch point of SF(T). Moreover if F = {2} or F 1 = {3} then 6 is also not a branch point of SF{T), and hence <SF(F) is a network with no branch point (of course it contains at least one branch point of Af, namely b). If F = {2,3} then 6V(r) may be a star-shaped network of degree 3. Fix Af,F. Write S ZZF if <S C M is a star-shaped network with the following properties: (a) for every e G F, S contains a vertex v that is adjacent to a branch point 6E of Af, and (b) S contains no branch points of Af other than 6 and be, e G F. Such star-shaped networks are exactly those for which there exists T G GJ/1 such that S = 6V(r). Define C*S to be the set of laces L on S such that 1. For each e in F, if 6E G <S then there is exactly one bond sete G L covering branch point (of Af) be ^ b, and that bond has s or t strictly on branch Afe-130 2. If F = {2} or F = {3} then there is exactly one bond in covering b, while if JP = {2,3} there are at most 2 bonds in covering b. 3. L contains no elements of 7c (i.e. no bonds which cover > 2 branch points of A O -Then recalling the definition of L r from Definition 2.1.4 we have E n ^ = E E n>« r6£bF,M ^er serf • r e £* » , : **er -F,M SF(V) = S • E E • E E \JUst .steL .steL E n ^ r e s ' t ' e r \ L 5 F(r) = 5 , L r F = L E n r e g~n'con •. ster\L h r = L (6.96) E H". E r* eg -V. S,M\S " n u«, stev* 5 F ( L u r * ) = s where $SM\s = iF 6 : f o r e v e f y s t G r> [ s G 5 , t € ^ \ 5 ] o r [ i . e 5 , a e ^ \ 5 ] } . (6.97) Now note that for any set of sets of bonds % with the property that there exists some N G N and {sji;} € U, i — 1,... AT such that every element of % is a subset of { s \ t i , s N t N } , we have Jlren Uster ust = Ti^eiA1 + ust]- Let Cf* be the set of laces in C*s consisting of exactly N bonds. Then (6.96) is equal to E ( - D " E E iV=l s\zFM Lec1** LsteL n stec(L) n . st n s e s,t e J\f \ s •. SF(LUst) = S,st£K (6.98) 131 If Ust € {—1,0} for each st, then each quantity involving Ust in (6.98) is nonnegative and we have E II"- < E E E N=l SCFJVLeCg* n -Ust .steL n i i + u ° t ] stec(L) n n t1+u't\ st £ E ( ^ s ) i si £ TZ (6.99) where A ^ s is the number of disjoint components (j\f\S)i oiAf\S. This quantity is bounded above by the sum of four terms (corresponding to the 4 possible branches incident to b2 and 63 if F = {2, 3}) each of the form 00 / n e+(n e/-l)\ m / n e - l \ E I I E E n E h N=l \ e € F m e = n e - l / mi=0 \ e £ { 2 , 3 } \ F me=0J £ - n n II i1 + Ust} stec(L) n M e ( ( A A \ S | ) ' 0 e : 0 < s < £ < n^m)") (6.100) where e' denotes one of the two branches (other than e) incident to be, is the star-shaped network defined by (2.12), and (j\f\S^)'% denotes the fact that part of branch J\fei is being removed if me >ne. In addition ne(rh)'1 is the length of branch e of (j\f \ <S4)'\ Since the analysis does not depend on the e', we ignore the fact that there are 4 such terms from this point on. Combining (6.95), (6.99) and (6.100) we have that Y^rze^ l i s t e r Ust is 132 bounded by a constant times oo / n e + ( n e / - l ) \ n i / n e - l £ EII E £ II E i x F C G(jV) N = 1 \ e e F me=ne-l j mi=0 \ee{2,3) \F m c=0y F # 0 E steL - n n <=1 e 6 ( J v V £ ) « n [i+c/^ **ec(L) n M € ( ( J v - \ $ £ ) " ) * : 0 < s < t < rie(m)") (6.101) Putting this back into (6.92), the sum over laces on the star-shaped network gives rise to the quantity n^i*) and the final product gives rise to Chn_^yi(»), with displacements summed over. On the latter on which we use (5.6) with I = 1 to bound | | / j „ r ( , 7 j ) ' i | | i by a constant and we obtain an upper bound on (6.92) of a constant times oo / n « + ( n e , - l ) \ m ( ne-l\ E £ 1 1 E £ n E E * F C G(J\f) N=l \eeF me=ne-l J mi=0 \ee{2,3}\F me-0/ u F^Q v m - n n *• By Proposition 4.3.2 this is bounded above by oo / n e + ( n e / - l ) \ m ( ne-l\ c E EWlI E E II E F C G(J\f) N = l \eeF me=ne-l J mi=0 \ee{2 ,3} \Fm e =0J F ^ 0 (6.102) < C 2—1 d-8 (6.103) F C G(jV) e€F • n e ' F ^ 0 Since the remaining sums are finite, this establishes the second bound of Lemma 4.2.1. • This completes the proof of Lemma 4.2.1. 133 Chapter 7 Convergence to C S B M . In this chapter we relate the convergence of the r-point functions (as in Theorems 1.4.3 and 1.4.5) to convergence of/x„ G MF(D(MF(Rd))) (defined by (1.17)) to the canonical measure of super-Brownian motion (CSBM). The fact that C S B M is the weak limit of certain branching random walk models is a standard result in the theory of measure-valued processes, and we take such a result as our definition of C S B M in Section 7.1. In Section 7.2 we restate The-orem 1.3.1 and briefly discuss some related results. In Section 7.3 we prove a general result (Proposition 7.3.3) that relates convergence of finite-dimensional distributions to convergence of certain functionals and the existence of certain exponential mo-ments for the limiting measures. We conclude in Section 7.4 by proving Theorem 1.3.1 and noting that Theorems 1.4.3 and 1.4.5, together with the convergence of the survival probability (which in general need not be a probability measure) implies convergence of the finite-dimensional distributions of our model to those of C S B M . 7.1 The canonical measure of super-Brownian Motion In this section we indirectly define the canonical measure of super-Brownian motion as the weak limit of a branching random walk model with critical branching. 7.1.1 B r a n c h i n g R a n d o m W a l k We describe a particle model (branching random walk) where we label particles by multi-indicies as in [27] and some of the references therein. The construction we describe here is somewhat nonstandard but is done to resemble the construction of our lattice tree model. A particle is described by a G . / = U~ = 0 Nt 0 > 1 ' " - m > = { ( a 0 , . . . , a m ) : a, G N , m G Z+}. (7.1) 134 We start with a single particle, and set ao = I for all a. Let \(ao,... ,am)\ = m be the generation of a and write a < a a — (ao,... ,a%) for some i < \a\, i.e. if a is an ancestor of a. If a — (ao,..., a m _ i , a m ) and a = (ao, • • •, « r a - i ) then we say that a is the parent of a and a is the a^ child of a. F ix M G Z + and let Y be a random variable with Y(u) G { 0 , 1 , . . . , M}, such that E[Y] = 1 and 0 < E [(Y - E[Y})2] = 7 < o o . Let {Ya : a G 1} be i.i.d ~ Y random variables. Let Go = {(1)} and for each m G N we define Gm recursively as follows. At time m~, each particle a = (ao, • • •, am-i) £ G m - i gives birth to Ya children (ao,.-.,am-i,l),...,(ao,...,am-i,Ya), (7.2) and immediately dies. We let Gm be the set of particles alive at time ra. Note that each particle a G Gm satisfies \a\ — m and has a unique parent a G Gm-i by definition. Clearly if Gm = 0 then Gn = 0 for every n > m. A well known result due to Kolmogorov (see [27] Theorem II. 1.1.(a)) states that mjP(Gm + 0) -> 2, as m -» oo, (7.3) and therefore P ( n ^ = 0 { G m ^ 0}) = 0 so that G = U ^ = 0 G m is almost surely a finite set. We call a set G that can be constructed in this way a geneology. Let Q denote the set of possible geneologies. Since the number of children of a particle is bounded above, the number of possible G m ' s is finite for each m. Therefore Q is a countable set. Given a function D(x) defined by Definition 1.2.4, and a set of particles (multindicies) G we choose a random embedding B of G into M d as follows. Let ClG = {B : G H - » Zd,B((l)) = 0} be the set of maps from G to Zd that map the initial particle (1) to the origin. Then we define a probability measure P' G M i ( f J ) on the set = {(G, B) : G G Q,B G SIG} of embedded geneologies by P'((G,B) = (G*,B*)) = P(G = G*) D(B*(a)-B*(a))I{B.&G,}, (7.4) (a,a)eG* where the product is over all (parent, child) pairs (a, a) in G*. Now given (G,B) G 0, n G N, we define measures X " ' ( G " B ) G MF(Rd), i G N n by xn,(G,B) = C[ y ^ x:y/C2nx£B(Gi) We extend this to all t G M+ by X ( n , ( G ' j B ) = Xn^'B). Finally we define ^ G MF(D(MF(Rd))) by »'n(H) = nC'zP' {(G, B) : {X?'{G'B)}tm+ G H}) , He B(D(MF(Rd))). (7.6) 135 Here, the C[ are some fixed, known constants that may depend on the function D (recall that D depends on L) and 7 . 7.1.2 B R W converges to C S B M In this section (much of which is taken from [27] chapter II.7., but with different notation) we define the precise way in which branching random walk converges to super-Brownian motion. The survival time (often called the extinction time) S : D([0,00), MF(Rd)) —> [0, 00] is defined by S {{Xt}teR+) = inf{s > 0 : Xs = 0 M } , (7.7) where 0M is the zero measure on Rd satisfying O M ( ^ ) = 0. Let D*(MF(Rd)) = {{Xt} e D([0,oo),MF(Rd)) : S({Xt}) > 0, Xt = 0 M V* > S}, Co(MF(Rd)) = {{Xt} € D* : X0 = 0M,X. is continuous}, (7.8) with the topologies inherited from D([0,00), MF{Rd)), C([0,00), MF(Rd)) (the topol-ogy for C is the topology of uniform convergence on compact sets). Note that ,fi'n(S< e) > nC'3P' (xn^B) = 0 M ) = nC'3P ( G [ n e J = 0) > nC'3P (Gi = 0), for n > ^ ( 7 - 9 ) —> 00 as n -4- 00. Let us now also define the finite dimensional distributions of v € MF(D(MF(Rd))). Let R > 1, and t = {ti,.,.,tR} £ [0,oo)R. Let h£ : D(MF{Rd)) -> MF(Rd)R denote the projection map satisfying h^{{X,}) = (X^,... XtR)• Then the finite dimensional distributions of v are the measures vhZ1 e MF{MF{Rd)R) given by vhZ1(H)=v{{X.}:hA\{X.})eH), H £ B{MF(Rd)R). (7.10) Definition 7.1.1 (Convergence in TJ*). Suppose {un : n G N U o o } C Ma(D*),, the set of a-finite measures on D*. We write vn ==> on D* if for every e > 0, un(S > e) < 00, Vn € N U 00, and w • (7-11) vn(», S > e) ==> ^oo( - ) S > e), as n ^ 00, 136 where the weak convergence in the second condition is convergence in MF(D(MF We write vn ==> on D* if for every e > 0, vn(S > e) -> 1/00(5 > e) < 00, (7.12) and for every m G N and t G [0, oo) m , un(S > e) < 00, Vn G N U 00, and -1 «, -1 (7-13) f n / i _ . (•, S > e) ==> Vooh- (•, 5 > e), as n —> 00, where the weak convergence in the second condition is convergence in Mi?(M/r ( lR r f ) m ) . The following Theorem, which states that branching random walk converges weakly to the canonical measure of super-Brownian motion is fairly well known, and a version of it is proved in [27] (see [27] Theorem II.7.3.). Theorem 7.1.2. There exist constants C[,C', C'Z such that the measures p!n defined by (7.5-7.6) for the branching random walk model satisfy (a) for every s > 0 there exists Rs G MF(Mp(RD) \ {OM}) such that for every s>0, p!n{Xs G . , Xs ? 0 M ) =2* Rs{») on MF(RD), and Rs(MF(RD) \ {0M}) = — • 7s (7.14) (b) There exists a a-finite measure No on CQ{MF(R^)) such that u'n ==> No on D*{MF(RD)), and for every s > 0 1. N 0 ( X S G ;S > s) = R8(m), 2. P' ( { X T N , ( G ' B ) } G •\Xs'{G'B) ^ 0M) N 0 ( « | 5 > s) on £>(MF(R*)). Definition 7.1.3. The a-finite measure No on C*([0,00), Mjr(]Rd)) defined by part (b) of Theorem 7.1.2 is called the canonical measure of super-Brownian motion. 7.2 Lattice trees In this section we recall the setup of our measure-valued process and briefly discuss the context of our results. Recall from Sections 1.3 and 1.4 the following definitions: • X?'T G MF(RD) therefore {X?'T}teR+ G D(MF(RD)) defined by X t r = v k l X ^ , a n d ^ = ^ . (7.15) x:Vcr2vnx£Ti 137 F G Mi(7o) defined by F({T}) = W(T) P(0) ' • pn G MF(D(MR(Rd))) defined by Pn(H) = nVAp(0)¥ ({T : {X?>T}teR G }) . Note that for e > 0 and n > i , /J„(5 < e) = / i „ ( X T = O M for all t > e) , = nF^/9(0)P(T : Xn^ = 0M for all t > e) n > nVAp(0)P(T : x g j = 0 M ) n > nF74p(0)P(T = {0}) = nVAp{0), (7.16) (7.17) (7.18) i.e. pn(S < e) /• oo = No (5 < e). Recall however from the previous section that our statements about convergence to C S B M include the condition that S > e. Another way of removing the contribution from processes that have arbitrarily small lifetime is to include the total mass at time e in the expectation, which can be achieved by taking s — e in Theorem 1.3.1, which we restate with our new notation. T h e o r e m (1.3.1). There exists LQ 3> 1 such that for every L > Lo, with pn defined by (1.17) the following holds: For every s > 0, A > 0, m G N, t G [0, oo) m and every F : Mp<(Rd)m —>• C bounded by a multinomial and such that N o / ^ ( ^ V ) = 0, 1. E llnh. 1 Xs{l)F{X) E^ Xa(l)F(X)] , (7.19) and E Hnhr 1 F(X)I{xs(i)>\} F{X)I{Xs(i)>\} (7.20) We show in Section 7.4 that the convergence of the survival probability (to-gether with our results) would be sufficient to prove the following conjecture. Conjec ture 7.2.1. Let pn be defined by (7.17) for the L, D lattice tree model defined in section 1.3, and let d > 8. There exists Lo(d) 3> 1 such that for every L > LQ, on D*. (7.21) fin => N 0 , As in [3], convergence as a stochastic process follows from convergence of the finite-dimensional distributions (Conjecture 7.2.1) and tightness. Tightness for this model is also an open problem and is less well understood at present. 138 7.2.1 I S E Derbez and Slade [7] proved results closely related to Theorems 1.4.3 and 1.4.5. They showed using generating function methods that the scaling limit of lattice trees (sufficiently spread out for d > 8, or nearest neighbour model for d » 8) is integrated super-Brownian excursion (ISE). We describe their results in the form that is most most relevant to this paper. Let TN denote the set of lattice trees containing exactly N vertices, one of which is 0. Remark 7.2.2. Roughly speaking, a lattice tree that survives until time n has, on average, order n particles alive at that time. We infer from this that the total size of such a tree is N w n 2 . Thus scaling space by n~ 2 should be equivalent (in terms of the leading asymptotics) to scaling space by N~~i. For fixed N e Z+ and T G TN, define XN,T = N- E <7-2) where D\ is a constant defined in [7]. Since T contains exactly N vertices, XN'T is a probability measure on Rd. Keeping N fixed, choose a random tree according to which is independent of p. Then XN,T is a random probability measure described by /x^ . Define XN G M i (Mi(M d ) ) by TN{A) = pN({T : XN,T G A}), • A G B(Mi(Rd)). (7.24) with B denoting the Borel sets and M i (E) the space of probability measures on E with the weak topology. Slade [28] shows that the results of Derbez and Slade [7] imply IN T as N —> 0 0 , where the probability measure X G M i ( M i ( R d ) ) is called integrated super-Brownian excursion. This is a statement that for all / G Cft(Mi(E d )) (i.e. / bounded continuous on M n j fdlN -> j fdl. (7.25) Derbez and Slade [7] prove (7.25) for functions of the form 4 » - / e*M<&), (7-26) 139 and Slade [28] shows that this is sufficient to prove weak convergence. To prove their results, Derbez and Slade [7] define for Q,p € C, r > 2, a e S r , and y € Zd(2r~3\ the set T£{y) of trees of skeleton shape a with skeleton displacements yi and the generating functions G T c { y ) = E ' f f c E wv^)- (?-27) n e Z 2 ; - 3 ^ ! T€7V(*,a)(0,iO They then write 2r-3 Gr'"(/?) = Vr~2 TT + Er'aJH), (7.28) ^ f=\ Cn2 + C7 2 ( l - %) \ + C S (1 - 0) P ' C for specific constants C\, C 2 , C 3 . They show that .&'-(«;) is an error term when: • r = 2, 3 for all p < pc and HCIloo < 1> and when • r > 2 for all p < pc and £ = 1-Essentially in [7] backbones were very well understood for r = 2,3 but less so for r > 4. Since n is summed over in the definition of G in 7.27, setting ( = 1 removes all time (backbone length) information from the results of [7] for r > 4. Thus we do not expect Theorem 4.1.8 to follow from the analysis of [7] for r > 4 and at least for r > 4, Theorem 4.1.8 is an entirely new result. The following non-rigorous argument suggests, that Theorems 1.4.3 and 1.4.5 for r = 2,3 may follow from the analysis of [7] without too much difficulty (perhaps with less sharp error bounds). When p = pc, (7.27) implies that the coefficient of Hfj^ (" J in ^'"-(^j) is £ W(T) = a ) ( j = ) . (7.29) y TeTN(n,a){Q,y) Using the fact that for x < a, = ^ YlnLoiaan<^ a s s u m i n g that E is an error term, (7.28) implies that this same coefficient is approximately T / r - 2 2 £ - 3 1 yr-2 2J_^ _ C , This is of the form of Theorem 4.1.8 (resp. Theorem 1.4.3 for r = 2), which was the main ingredient in the proof of Theorem 1.4.5. It is likely that one could adapt this rough argument to get a rigorous proof of a version of Theorem 1.4.3 and Theorem 1.4.5 with r = 3. 140 We describe the connection between ISE and C S B M as follows. Let {Xt} G C*([0, oo ) , Mp(Rd)). By definition of the survival time, S, we have that Xt — OM for every s > S, and Xt / OM for all t G (0, £ ) . If 5 is finite (note that under No, S is indeed finite almost everywhere) then by continuity on [0,5], s u p f X t ( I R d ) < K for some 0 < K < oo. Define a measure n*) = on R d by poo Y ( . ) = / Xt(»)dt. (7.31) Jo Then by the above discussion, Y ( M d ) < fQS Kdt < oo , and we may define a proba-bility measure V on Rd by where for A G tf(Rd) and C > 0, £ = {x E Rd : Cx G A}. Now if we choose Xt randomly according to No ( • |Y ( lR f l ! ) = l ) (which is a probability measure on D(MF(Rd))) then V has law 1. 7.3 F i n i t e d i m e n s i o n a l d i s t r i b u t i o n s In this section we prove some results within the general theory of measure-valued processes. The main result of this section is Proposition 7.3.3. The motivation for proving Proposition 7.3.3 is to obtain a statement about convergence of the measures / j n of (1.17) to No from convergence of the r-point functions (Theorems 1.4.3 and 1.4.5). We use Proposition 7.3.3 in Section 7.4 to prove Theorem 1.3.1 as a consequence of convergence of the r-point functions and in Section 7.5 to show that convergence of finite dimensional distributions of our model follows from convergence of the r-point functions and the survival probability. The applications of Proposition 7.3.3 carried out in Sections 7.4 and 7.5 are also implicitly being used in [20] for oriented percolation and in [17] for the contact process in connecting convergence of the r-point functions to convergence of finite-dimensional distributions. Def in i t i on 7.3.1 (Tightness for finite measures). A set of finite measures F C Mp(E) on the Borel a-algebra of a metric space E is spatially tight if for every n > 0 there exists K C E compact such that sup^p n(Kc) < n. A set F C Mp(E) is tight if it is spatially tight and sup^p fi(E) < oo. L e m m a 7.3.2. If F C Mp(E) is tight, then every sequence in F has a further subsequence which converges in Mp(E) (weak convergence). 141 Proof. Let {/in} C F. If there exists a subsequence /_*„, such that nni{F) —> 0 then we have \ini —> OM by definition (for every bounded continuous /....) and we are done. So without loss of generality there exists no > 0 such that minfin(E) = no-Therefore Mn(*) (7.33) are probability measures. Let n > 0. Since the \in are (spatially) tight there exists K C E compact such that sup n pn(Kc) < nrjo- Therefore s u P p n ( i n = s u p ^ < ™ n n Hn{E) n0 V, (7.34) so {Pn} is tight as a set of probability measures. Therefore there exists a subsequence P n k P & Since {pnk} is tight, {fj,nk(E)} is a bounded, real-valued sequence, and there t1 'oo and • fore has a convergent subsequence /J„*(J5) —»• C > no- So PF\ ~ k fin*(E) —> C > 0 and therefore / j n * —> CPoo G Mp(E) as required. That the full statement of tightness is necessary (i.e. spatial tightness is not sufficient)' for the conclusion of Lemma 7.3.2 is illustrated in Example 7.3.4. Let T denote a Mp(Rd) convergence determining class of bounded continuous functions (j) : Rd C (i.e. un -» v in Mp{R^) if and only if vn{4>) -> v{4>) for every (f> € J7), that contains a constant function, (f)(x) = C? / 0. Hereafter when H G MF (D(MF(Rd))), we interpret nhZl G M F ( ( M F ( M o ! ) ) 0 ) as the measure on the space consisting of only one point x, that satisfies phZl(x) = /z (D(MF(Rd))). The main result of this section is the following proposition. P r o p o s i t i o n 7.3.3. Let a > 0 and fj,n,fj, G Mp (D(Mp every I G Z + and every t G [a, oo)', m G Z+ we /lave Suppose that for 1. there exists a S = S(i) > 0 suc/i £/iai for all 9i < 5, E h-i [e^ <=i < oo, 2. /or every 0 = O n , . ' . . , <fom(} G F^lLi> nil i=ij=i i=ij=i < oo, (7.35) where an empty product is 1 by definition. Then for every m G N and every t G [a,oo)m , pnKZl -> / x / i l 1 in Mp ({Mp(Rd))m). The importance of the I = 0 case in the Proposition is evident from the following example. 142 Example 7 . 3 . 4 . Let u'n G Mp (D(Mp(Wi))) be the measure that puts all its mass (n) on the measure-valued process Xt = for all t > 0, and /J, be the measure that puts all its mass (1) on the measure-valued process Xt = S\ for all t > 0. Next let p,n = fx'n + H i.e. Mn(*) = n8sq_ + St!, / i = ^ i - (7.36) Then E, n U n ^ i =n!=in^1*i(^) = n U n S i ^ ( 1 ) » a n d E, I m, nn^^) i=lj=l "iin^+nn w - o+n n *,-<D-i=ij=i i=lj=l i = l j = l Thus we have tin I mi nn i=lj=l i=lj=l (7.37) (7.38) for every I > 1, rn ^ 0, t, and E^ = eT,8i < 0 0 - However / i n (5 > e) | e £ ***<(*) n + 1 (Vesp. pn {D{MF{Rd))) =n + 1) and (j, (S > e) = 1 (resp. n (D(MF(Rd))) = 1), so that none of the finite-dimensional distributions can converge, and no sub-sequence of \in can converge in Mp (D(MF(Rd))). Note that {p,n}nef$ *s spatially tight but not tight. We prove Proposition 7.3.3 in the form of 5 lemmas. The first, Lemma 7.3.5 establishes tightness of the {anhZl : n G N} for each fixed l,t. Thus every subsequence of the jj,nh? has a further subsequence that converges. The second, * , Lemma 7.3.6 states that any limit point of the {nnh~ : n G N} must have the _ i t same moments (7.35) as / x / i - . The third, Lemma 7.3.7 states that if a certain moment condition holds for every fa G T, we also have that result for all continuous 0 < fa < 1. The fourth, Lemma 7.3.8 says that each subsequential limit point is uniquely determined by certain class of functionals 2 £ . [ e - £ * = 1 ' x ' ' ^ ] , fa > 0 bounded, continuous. Finally, Lemma 7.3.9 says that these functionals are uniquely determined by certain moments of the form (7.35). Taken together they show that since all subsequential limit points have the same moments (7.35), the limit points all coincide, and thus the whole sequence converges to that limit point. Lemma 7 . 3 . 5 . Let pn,/x G Mp (L>(Mj?(IRd))), and a > 0. Suppose that £ ^ [ 1 ] - » £74 [1] < 0 0 and that! for every t G [a, 00), and every <j> G T, , (7.39) 143 Then for each m G Z + and every t G [a, oo)m, the set of measures {/J,nh-1 : n G N} is tight. Proof. The m = 0 case is trivial since jE^f l ] -> -E>[1] < oo. This also gives fj,n (D(MF(Rd))) ->• /J (7J(Mi?(Ed))) < oo so it remains to prove spatial tightness for m > 1. We first prove the m — 1 case. Let e > 0, i > a. Define vn — E h-i[X], and v = E h-i[X]. Then u(Rd) = Lo < oo and applying Fubini to (7.39) we have / fa(x)un(dx) —>• J fa(x)u(dx) for every fa E T hence f n — > IA Therefore there exists no such that for every n > no, un(Rd) < Lo + 1. Since the un are finite, there exists L\ such that un(Rd) < L\ for all n < no. (7.40) Let L = (LQ + 1) A Li and choose M such that < | . Then L > s u p ^ /,-1[X(Md)] > BupE h-i[MIX(RD)>M] =MsupUnhT^XiW1) > M ) . n Dividing through by M , we get that Supunhrl(X(Rd)>M)<^<e-. (7.41) F ix n > 0. There exists K^i C K. d compact such that K ^ - i ) < ^. Fur-thermore there exists KQ C K d compact such that U(KQ) < u(Kti) (e.g. the set . KQ = {x : d(x,K-i) < 1}). Since un ^ u m M/?(M d) and K% is closed, V limsupi/ n(iv" 0 c) < v(K§) < (7.42) Therefore there exists no > 0 such that for all n > no, Uu(KQ) < rf. Also since u\,...,fno-i a r e finite measures there exist Ki C Rd compact such that Ui(Kf) < n Then K = U"=o1 ^ i s compact and supz/„(iv' c) < n. n Now s u p r y i ^ V 1 ( X ( K C ) > t j* ) < s u p £ h -x [ x ( K c ) J n * ' n ' t = supun(Kc) < T]. X{Kc)>r)-i Dividing through by r/4 we get that sup / i „ / i t _ 1 ( X ( X C ) > r/i) < n (7.43) n 1 (7.44) (7.45) 144 Choose 774 = X. Then there exists Kj C M d compact such that s u p / i n ^ 1 .(x(Kj) >±^<±j. (7.46) Choose m > + 1 so that p f r r .< § and let K = p | I.X : * ( * ? ) < 1} f|{X : X(M r f) < M}. (7.47) Now K is (sequentially) compact (see for example in the proof of Theorem II.4.1 of [27]), and K c = (J tx-.X(K<)>±j\\j{x-.X(Rd)>M}. (7.48) Thus, sup/ in/ i r^K 6 ) < s u p 1 ( M tx : X{KD > 1) + s u p / i n V 1 ({X : X ( R d ) > Af}) < s u P f; pnhil (tx : X(K<) >• 1]) + I 00 - 2^ ^37 + 2 - 8 ^ 1 + 2 < 6 ' (7.49) j=m which verifies that the j^nh^1 are spatially tight, for m = 1. For m > 1, and f € [a, oo) m , We have from (7.49) that for each i € { 1 , . . . , m} there exists K; C Mp(W1) compact such that supnnnh^.x{K.\°) < ^ . Let K = Ki x K 2 x • • • x K m . Then K C (MF{Rd))m is compact and (TO \ jJlXiXiGK^} i=i / m < supE / i rA- 1 ( { X : X j G K i c } ) = supYV^. 1 ( K ; c ) < VsupMn/i t . 1 ( K i c > < e, n i=i <=i n (7.50) which gives the result. 145 Lemma 7.3.6. Fix I > 0 and t G [0,oo)'. Suppose that the second hypothesis of Proposition 7.3.3 holds for nn,fj, G MF{D(MF(W1))), for this t and for every v in Mp ({Mp(Rd))1), then for each rh G Zl+ and faj G T, l rrii i=i j=i = E I mi i = i j=i (7.51) Proof. The I = 0 case is trivial, so .we may assume that / > 0. Let pnk h-1 ==> v. Then in particular we have /z„ f c/tr (1) -> v{\). Assume v{\) / 0. Then there exists k-o such that for every k > ko, < nnkh~l{l) < 2/v(l) and we define for k > k0 the probability measures, P ( . ) = 1/(1)' (7.52) Then we have that P7 Then X, P as probability measures. Let XNK ~ PNK and X X and since (Mp is separable, we may assume that XNK and X are defined on the same probability space ( f i , . F , P ) . By Corollary 1 of Theorem 5.1 of [3], F(XN) F ( X ) for every F such that P ( X G VF) = 0, i.e. such that P(T>F) = 0, where Vp denotes the set of discontinuities of F. We apply this to the continuous function F^ : X —> Yli=i IT j=i -^t(^u')-We now show that EF [F$'Xnk) [10], it is enough to show that supfc.Ef» Ef i ^ P O j . By Example 7.10(15) of < oo. But, supEfc fc = s u p £ P fc E (n i=inSi^(^-)) sup fc (7.53) A*nA- (1) < CO, since (Y[\=i YYJ^i Xi(4>i3))2 is also a polynomial of the form appearing in (7.35). Thus we have Ef which implies that Since we also have E, ^hh?[F}m\->Ev[F£X) in the case v(l) ^ 0. M n , ^ 1 [ F $ ( X \ E /j.h (7.54) (7.55) we have verified the claim 146 Consider now the case that u(l) = 0. Then u is the zero measure and we have Ev E , - i ^kht n ; = i n ^ i ^ ( ^ i ) J = 0. By Cauchy-Schwarz, 2 < E ^ k h f [ l2] E ^ - i i=ij=i ( l mt i = i j = i 2-| (7.56) Since 1 is a bounded continuous function and uUk / i - 1 —» O M we have that the first ex-pectation on the right converges to 0. Since sup f c J E^ ^ - I (n!=i l~Ij=i Xi(faj)) 1 < oo we obtain Since also E, I mi n i i w i=lj=l E ,-x I mi i=ij=i -> E h - i 0. i=ij=i . we have that E h-i nL n i^ = o = [nu nri ifies the result. r Lemma 7.3.7. Suppose I > 0, p, p,' € MF ((MF(Rd))1). If E„ l mi i=i j=i = Etf l mi nn^ (^ ) i=l .7=1 (7 .57) (7 .58) which ver-• (7.59) holds (and both quantities are finite) for every (j) S JF^mi then (7.59) holds for every 4> such that for each i,j, 0 < faj < 1 is continuous. Proof. If / = 0 or m,j = 0 then the conclusion is trivial so we may,assume that / > O a n d ^ r a j > 0. Applying Fubini to (7 .59) , using the facts that ^[JlUi lTj=i Xii1)} < oo (by choosing fa — ^ 0, the constant function in J7) and the fa are bounded we have / n n ^ / mi unxiidxij) i=i j=i /» / mt / n n ^ j -=ij=i l mi IJlUXiidxij) i=l j=l (7.60) Since T is a determining class for MF(Rd) one can verify that T^mi is a determining class for MF(Rd^mi) (using the fact that this class of functions,determines the conditional distribution of the nth coordinate given the first n — 1 and proceeding by induction). 1 147 Now l rrii I (Hmi i=ij=i i i k=i (7.61) so the products of <f>ij in (7.60) uniquely determine the measure defined by u(dx) = t^nUi lTj=i Xi(dxij)]. Therefore (7.60) holds for all fc bounded, continuous, so in particular for all continuous 0 < fc < 1. Applying Fubini again we get the result. • L e m m a 7.3.8. Suppose ft,n' G MF ({MF(Rd))m) and assume D0 C (Bb(Rd,R+))m -bp satisfies A f = (Bb(Rd,R+))m. If for all <f> G D0 (7.62) then /_* = //. Proof. If m — 0 the conclusion is trivial as both measures are on the single point space with same total mass, so we may assume that m > 0. We follow the proof of Lemma II.5.9 of [27]. (a) (7.62) holds for every $ e (Cb{Rd ,R+))m. We verify the stronger result that the class £ of $ for which (7.62) holds contains (Bb(Rd, R+ ))m. Let if>n G C be such that (fn % fc Now by dominated convergence (using the fact that /J is a finite measure and dominating by e° = 1), E„ = Eu, \ l im e-T.T=\xMi,n) Ln—>oo lim Eu = lim E, e -EjLl^iWj ,n)j (7.63) = E„ Thus C is closed under bounded pointwise convergence. Since Do o £ by hypothesis this shows that (Bb(Rd,R+))m C C as required. Define e r : ( M F ( M d ) ) m -> 1+ by eAv) = e ~ £7=i ^ (<fe). Now let and U = {# G # 6 ( ( M F ( R d ) ) M , M ) : E^{X)] = ^[*(X)]} ^ o = {ex:<?e(C 6(M d ,R f)r}. (7.64) (7.65) 148 (b) M contains all bounded a{%o) measurable functions. We show that % bp is a linear class containing 1, closed under —>, and that Ho C "H is closed under products. Once we achieve this, we have by Lemma II.5.2 of [27] that W contains all bounded cr(%o)-measurable functions. 1) that % is a linear class is immediate by linearity of the integral. 2) 1 G % by taking <j> = 0 and using part (a). 3) Let $ n G M,$n Then $ G % by dominated convergence since U,\J! are finite measures. 4) Let fi, f2 € Uo- Then f{ = e .^ and / l / 2 = e - E r « ^ ( * i . i ) e - s r - i ^ ( * w ) = e - s r . i ^ ( ^ + ^ » ) = e 0 1 + 0 2 e (7.66) 5) Mo CH was verified in part (a). (c) There exists a countable convergence determining set for (MF(Rd))m. We use the construction of Proposition 3.4.4 of [8] to obtain a countable set V C ( C 6 ( R d , ! + ) ) m such that vn -> V in (MF{Rd))m if and only if ffn($) ->• £(<£) for every ^ G V). Let { ^ i , 92 , • • • } be an enumeration of Q ^ , a dense subset of Rd. For each G N 2 define fi,j(x) = 2 (1 — j\x — qi\) V 0, (7.67) and for A C N 2 define / \ AW = It is an exercise left for the reader to verify that i,j < m A 1 . (7.68) Vo = {gm:meN,Ac{l,...,my}cCb(Ra), (7.69) is a countable convergence determining set for MF(Rd). It follows that V — {(4>i,..., 4>m) : <pi G Vo U {0}} is a countable convergence determining set for (MF(Rd))m. Define G = a(e$:$£V). (7.70) 149 (d) B({MF(Rd))m) C g C cr(^o), where Q = a(e^ :$eV). The second inclusion is trivial since V C (C[,(Rd, R))m. We claim that Q contains all the open sets in (MF(Rd))m and hence contains B((MF(Rd))m). Define the metric m oo j = l 71 = 1 2n (7.71) where {<^i, fci,... } is some fixed enumeration of VQ U {0}. It is a standard result that Q' induces the topology of weak convergence. Let U be an open set in the topology of weak convergence. Then U is also open in ((MF(Rd))m, g'). Now MF(Rd) is separable so every open set is a countable union of balls BQi(v, r) and therefore to show that U G Q, it is enough to show that Bff(v,r) G g. But Be,(v,r) = { p' E E ^ ^NDLJ,J ~ ^ ^ N D V J ^ < R j = l n= l 271 G ^ (7.72) since an infinite series of measurable functions is measurable. We have now verified that g contains all the open sets of (MF(Rd))m and therefore contains B({MF{Rd))m). (e) Conclusion. We have now verified that B({MF{Rd))m) C g C a(H0). There-fore every bounded continuous function is measurable with respect to a(%o). Fur-thermore we have that % contains all a('Ho)-measurable functions (and in particular all the bounded continuous functions). Since / i = / / if and only if f fdfi = f fdfi' for all bounded continuous / : (MF(Rd))m —> R, we have proved the result. Lemma 7 . 3 . 9 . Let /J, G MF ((MF(Rd))m). Suppose there exists a 6 > 0 such that for all 9i < 6, E „ [ e ^ e i X i ( ^ ) ] < 0 0 _ . . (7.73) Then for every 0 < tpi bounded, continuous, the quantity (7.74) is uniquely determined by the mixed moments (7.77) of X(fc), 0 < fc < 1 continuous, i — 1.... , m. '; Proof. If m = 0 the statement is that £^[1] is uniquely determined by £^[1] (the expectation of the empty product), so we may assume that m > 0. F ix £ = . . . , 4>m) and let x = X(fc). Then for all z G C"*-, oX{<t>)-A E„ E .1=0 (z-x) (7.75) 150 By dominated convergence with (7.73), and using the Multinomial Theorem, we have for ||z||oo < 8 that the right hand side is equal to oo ... oo 1 I=O ' z=o E n e t[(ziXi)ni Y.Tli=l (7.76) which is a multivariable power series in zi,...zm with coefficients that are linear combinations of quantities of the form E„ n L i = l = E„ L i = l (7.77) Now let Halloo < 6 and note that 0 < Xi = Xi(fc) < Xi{Rd) for 0 < fc < 1. Then e ( z + A 2 i ) - x _ gi - - : l im / = l im A Z J - > 0 = l im A z ; - » 0 A ^ g A z j - X i _ 1 / Xiez'xdn Az{ -xAd/j, AziY 1=2 ( A z j ) ' - 2 a | /! (7.78) < lim \Azi\ [ eZ?=ite('j)zjx2e\A*i\xid/Ji •• A*;->0 y = l im \Azi\ / " e C ^ J + ^ + l ^ l ^ + ^ i ^ ^ ^ e - ^ d u . Now £ 2 e~ e : 2 : i < C e so the integral converges for all zsuch that Re(zi) + e + \Az{\ < 6 and Re(zj) < d for all j / i. Thus the limit in the above is zero. Choosing i — 1 we get that for fixed z-\ = (z2,---zm) with HiLiHoo < 5, ipl{z) = f ezxd/j, is analytic in z\ such that Re(z\) + e < 8 for every e > 0, and thus in particular for z\ such that Re{z\) < 0. In particular, <pl{z) is the analytic continuation (in z\) of J ezxdp for ||z||oo < 8 and as such tp1{z) is uniquely determined by the moments (7.77). For 1 < i < m, and fixed Zj such that Re(zj) < 0 for j < i and < S for j > i suppose we have <pl~l(z) is analytic in each Zj in the regions Re(zj) < 0 for j < i and \ZJ\ < 5 for j > i. Then we define fl(z) as follows. As in the last line of (7.78), and using the fact that ZjXj < 0 for j < i, we 151 have e(z+Azt)-x _ e z - £ l im Az»->-0 /e(z zi)-x _ ez-x r A5 < lim [AzA / e ( ^ ) + e + l ^ l ) ^ + E r > i ^ ) ^ a ; 2 e - ^ < i ~ A 2 i ->o y * (7.79) This integral converges for all z such that Re(z{) + e + |AZJ | < £ and Re(zj) < S for j > i. Thus for i* such that Re(z\) < 0 , . . . , .Re(;z;_i) < 0 and fixed \ZJ\ < 6 for j > i the function tp\z) = J e£Sdn (7.80) is analytic in each Zj in the region Re(zj) < 0 for j < i and | Z J | < S for j > i, and is the analytic continuation of,ipl~l\z), as a function of Z{. As such, ipl~l is uniquely determined by the moments (7.77). Therefore we have (pm(z) — /e z xd\i is analytic in each in the region Re(zj) < 0, and is uniquely determined by the moments (7.77). Thus for each z with Re(zj) < 0 for each j, and every </> with 0 < <f>j < 1 continuous for each j, we have that j e ^ d a = f e ' ^ x ^ z ^ d a . (7.81) is uniquely determined by the moments (7.77). Therefore for each </>' such that <p'j is bounded, nonnegative, and continuous we have / dp (7.82) is uniquely determined by the moments (7.77). 7.4 P r o o f o f T h e o r e m 1.3.1 In this section we use Proposition 7.3.3 together with the convergence of the r-point fucntions (Theorems 1.4.3 and 1.4.5) to prove Theorem 1.3.1. The first hypothesis of Proposition 7.3.3 is the existence of an exponential moment for the limiting measure. The following Lemma will be used to verify this hypothesis in the proof of Proposition 7.3.3. L e m m a 7.4.1. For every b > 0 the following hold. 1. For every A > 0, N 0 (Xb{l) = A) = 0. 152 2. For every t e [0, oo) m there exists a S(t, b) > 0 such that for all E n o \ x b ( l ) e ^ 6 i X t i ^ < o o . Proof. By Theorem II.7.2(iii) of [27] we have for b > 0, N0(Xb(l)&A)=Q2 J^dx. (7.83) (7.84) Since also No (A^o(l) > 0) = 0, the first assertion is trivial. The second assertion of Lemma 7.4.1 is also a standard result and can be proved using the Markov property of the local time of the Brownian excursion under Ito's excursion measure, or the fact that No is an entrance law for S B M . We choose to give a direct and elementary calculation relying on the representation of S B M as a Poisson Point Process of excursions with intensity No (see (7.85)). Since -Xo(l) = 0, No almost everywhere, we may assume without loss of generality that t = (ti,...,tm) G (0, oo) m and b > 0, and we set to = b. Then Theorem 11.7.3(c) of [27] implies that for 0* > 0, ESo [ e £r=o^(i)] = e x p | y e S r = o ^ i ( i ) - i d N o ( l / ) | , (7.85) where {Yj}t>o is a super-Brownian motion starting at Jo (i-e. with initial law S$0). Lemma III.3.6 of [27] with Cauchy-Schwarz and with fi = 0i (constant functions) implies that the expressions in (7.85) are finite provided ||0||oo < \\i^C\b where co is some constant depending on m. Therefore for ||0||oo < |^|C Q v b a standard application of the Dominated Convergence Theorem allows us to take differentiation through the integral on the left side of (7.85) and obtain E S 0 O £ £ o * * i ( i ) = ^ [ ^ ( 1 ) ^ ^ ( 1 ) 3=0 (7.86) That this quantity is finite follows easily from the fact that (7.85) is finite. The derivative of the right side of (7.85) is exp Therefore we have d (7.87) o=0 | e £ r = o ^ ( i ) - i d N o H Ei* [n(i)< .££i«i*ii(a) D = 0 E s o [ e £r=o^(D = H{b,t,e) < oo. (7.88) 153 By Fatou's Lemma we have HlbXO) = l im f e E ^ i ^ W 9 0 \ 0 e0oMV - i e«o (^i) _ i d N o W (7.89) = y eE£i*i^(i),^(i) dNo(u). Thus for lo° ^ m^bwe h a v e £ N o [ x 6 ( l ) e E ^ i ^ ( 1 ) l < H(b,t,(f) < oo, (7.90) as required. • Recall the statement of Theorem 1.3.1, where Vp is the discontinuity set of F. In Section 7.2 we restated this theorem using the notation of this chapter as follows. T h e o r e m (1.3.1). There exists Lo 3> 1 such that for every L > Lo, with fin defined by (1-17) the following holds: For every s > 0, A > 0, m G N , t G [0, oo ) m and every F : Mp(Rd)m -» C bounded by a multinomial and such that N Q / I ^ 1 ( P F ) = 0, E ilnhv 1 and 2. EHnhZl XS(1)F(X)] £ ^ - 1 [x.(l)FLY)]'-, (7.91) [^(^K{X.(l)>A}J • (7-92) [E(X)I{x3(i)>\} Proof. Define j««,Ng G MF (D(MF(Rd))) by < ( A ) = f X s ( M d ) d / x n , Ng (A)= [ Xs(Rd)dN0. J A . JA (7.93) That these measures are finite (in fact bounded uniformly in n) follows from the fact that p4 (D(MF(Rd))) = E„n [Xs(l)) EMO [X s ( l ) ] < oo. (7.94) We take T = {eikx : k G Rd} which is a convergence determining class of bounded C-valued functions containing the constant function C?'— 1. Now for all 154 I > 0, rh e Zl+, E l mi / rrii — E,,s t=i j=i i=l j=l I mi (^l)IIII1'.^ ) i = l j = l / rrii ENo 1^(1)1111^^) i=ij=i l rrii i=l j=l (7.95) where even in the I = 0 case, the presence of the factor Xs(\) ensures that the convergence in (7.95) holds by convergence of the r-point functions. When any ti = 0, (7.95) is trivial using convergence of the r-point functions and the fact that ^ —)• 0. The case s = 0 does not hold since i?N 0 [Xo(l)] = 0, while Elin[X0(l)} = nCE a_ n 0. (7.96) By Lemma 7.3.5 the measures {psnhr } are spatially tight. Since they are also uniformly bounded by (7.94), } are in fact tight. By Lemma 7.4.1 we have that (7.97) for all 6 sufficiently small depending on t and s. In view of (7.94), (7.95) and (7.97) we may apply Proposition 7.3.3 with a = 0 to the measures /x n,No to get (7.98) Thus jj,snh-.l(l) NQ/I- 1(1). In particular, there exists rio(s,i) such that for n > no, < / x ^ 1 ( i ) < 2 N ^ r i ( l ) (7.99) Therefore for n > n0 we may define P * f G M i ( ( M F ( M d ) ) m ) by (7.100) 155 Now nsnhZl ==> WahZl implies that P » f ==> P i as probability measures, where P'(*) = ?,HJ-ul- Let Xn ~ i " r and X ~ P° Then we have X n -2+ X , and since ( M p ( R d ) ) m is separable, we may assume that Xn and X are defined on the same probability space (SI, T, P). By Corollary 1 of Theorem 5.1 of Billingsley F(Xn) -2+ F(X) for every F such that f(X € VF) = 0, i.e. such that P?(VF) = 0. We now show that if F is also bounded by a multinomial, then Ef |^P(A ?")j —> r - . 1 Ef L J 121 P ( X ) . By Example 7.10(15) of Grimmett and Stirzaker, it is enough to show < oo. But, that s u p n P P [ | F ( X n ) | \\F(Xn)\2 Ef = Eps _ n,t \F(X)\2' | P ( X ) | 2 Q{xf Mi (i) (7.101) where Q is a polynomial such that | P | < Q. Since sup n h-i Q(X)2 < oo holds by definition of psn and the convergence of the r-point functions we have the result. Thus we have that Ef F(Xn) Ef F(X)] , which implies that E n°nhzl F(X) Em F(X) (7.102) (7.103) Therefore for every function F that is bounded by a polynomial and that satisfies W0(DF) = 0, Define Pi = , if Xs(l) < A { x . ( ( i ) A > , otherwise. (7.104) (7.105) Then P i is continuous except at Xs(l) = A, and is bounded above by j. Thus, Lemma 7.4.1 and (7.104) show that E»nhf [Xs(mF] -> E ^ - i [X.imF], i.e. (7.106) • 156 7.5 A note on convergence of finite dimensional distri-butions Recall the definitions of p,n (depending on L) and No in (1.17) and Definition 7.1.3 respectively. In this section give a brief discussion about how Conjecture 7.2.1 follows from convergence of the r-point functions plus the following conjecture. Conjec ture 7.5.1. There exists an L\ 3> 1 such that for every L > L\, and for every e > 0, fin(S>e)^-N0(S>e). (7.107) C o r o l l a r y 7.5.2. / / Conjecture 7.5.1 holds then there exists L2 such that for every L>L2,e>0,leZ+,rhe Zl+, te [0, oo)' and <f e m i , E, I mi nn^(^) 7{5> E) i=ij=i EN0 / mi nn1'.^ )7)^ } »=ij=i (7.108) where T = {elk'x : k 6 2 Proof. If / = 0 or all m; = 0 then this statement is exactly (7.107). Otherwise fix e, / > 0, rh ^ 0, t, 4>. We may assume t e (0, oo)' since if any t{ = 0 the result is trivial. F ix L > L 2 where L 2 = LQ V L \ , and LQ and L \ are those quantities appearing in Theorem 1.4.5 and Conjecture 7.5.1 respectively. Let r) > 0 be given and write F{XA\4>)) = T]j=i LT i^ Xti(<pij). B y con-vergence (and finiteness) of the Fourier transforms of the r-point functions, we have E^ F2{XAl)) -»• ENO [F2(XA[T))] < C and therefore there exists C 0 = Co(t, rh, 4>) such that s u P ^ n \F2{XA\\)) n L J Choose Ao = Ao(n, Co) sufficiently small so that N o ( X e ( l ) € ( 0 , A 0 ] ) < By (7.107) and (7.92) with F = 1 we have . iin ( X e ( l ) > 0) -> N 0 (Xe(l) > 0), and Mn PQ(1) > A 0 ) ^ No ( X e ( l ) > A 0 ) . Therefore fin (Xe(l) e (0, A0]) -»• N 0 (Xe(l) £ (0, A 0]). It follows from (7.110) that there exists no such that for all n > no, < C 0 . JL 6 C 0 (7.109) (7.110) (7.111) ^(x e ( i)e(o,A 0 ])< (jl-J (7.112) 157 Using I{s>£} = I{Xe(i)>x0} + J{A%(i)e(o,A0]} w e h a v e Vn F(X0)I{s>c}] - ENo [F(X0))I{s> < E,n [F(X0)I{XeW>Xo}] -En0 [F(Xti$))I{Xe(1)>Xo}] (7.113) + E, Vn F(X?(<£))I{xe(i)e(o,\o}}\ + ^ N 0 [F(Xti$))I{xe(i)e(o,x0]} We bound the right hand side of (7.113) as follows. By (7.92), the first absolute value converges to 0 so in particular there exists n\ > no such that this term is less than ^ for n > n\. On the second term we use Cauchy-Schwarz to get Vn \F(Xr)\I{Xe(1)e(0M]} <E, Vn , 1 2 F2(Xf{l)) 2 E^n [/{x«(i)€(o,Ao]}]: < C 0 J L = 1 3 C 0 3 ' (7.114) The third term is handled in exactly the same way. Therefore we have shown that for n > n i , E, Vn F(X0))I{s>e}] - ENo [F(X^))I{s>e} < V, which proves the lemma. Fix e > 0 and define nEn,W0 G MF (D(MF(Rd))) by / z „ ( » , S > e ) , N § ( . ) = N o ( . , 5 > e ) . (7.115) • (7.116) That these are finite measures follows from Theorem 7.1.2 and the fact that / i n is a finite measure for each n. We wish to apply Proposition 7.3.3 with a = 0 to the finite measures/i^,Ng. Using the representation of S B M as a Poisson point process with intensity No, one can show that (for the 6(i) of Lemma 7.4.1) for all ||0||oo < 5, Em hF 1 'o t < OO. (7.117) This verifies condition 1 of Proposition 7.3.3. If Conjecture 7.5.1 holds then the second condition of Proposition 7.3.3 is provided by Corollary 7.5.2. Applying Proposition 7.3.3 to the measures shows that for every m G Z + , a n d every F G [0,co) m , fJ-£nhZl^%hz\ (7.118) which is the statement that fj,n =$• No (Conjecture 7.2.1). • 158 Appendix A Extending the inductive approach A . l M o t i v a t i o n We have already noted in Chapter 1 why we expect a Gaussian scaling limit for our lattice trees model in dimensions d > 8. We have also discussed results of Derbez and Slade [7] in Chapter 7, and in particular how their analysis might be used to verify Gaussian behaviour of the 2-point and 3-point functions. A n alternative method is to attempt to analyse the 2-point function by extending the inductive approach of van der Hofstad and Slade [19]. Suppose we have for every z € [0,2] say, supnfc \fn(k;z)\ < K with /o = 1, and n+l fn+i(k;z) = £ gm(k;z)fn+i-m(k;z) +en+i(k;z), (n > 0 ) . (A . l ) m=l For the following nonrigorous argument we also suppose that pi (A:; 1) = D(k) « D(0) — ^ 2 3 ~ , w here D is defined in 1.2.4, and that em,gm+l « 0 for m > 1. / 1.2 2 \ n Then we have / n + 1 « gxfn and so fn(k) « gi{k)n « (1 - . Thus . The inductive method of [19], which followed on from previous work of van der Hofstad, den Hollander and Slade [14] proves an important result detailing specific bounds on the quantities appearing in the recursion equation ( A . l ) , that ensures that there exists a critical zc ss 1 at which / „ ^ - ^ = ; z c j - —>• e~%s. The result of [19] is applied to sufficiently spread out models of self-avoiding walk [19], oriented percolation [20] and the contact process [16], each of which is believed to have critical dimension dc — 4. In each case the lace expansion is used to derive a recursion relation of the form (A. l ) and the required bounds on the quantities in the recursion equation are shown to hold (provided d > 4) by estimating Feynman diagrams. The required bounds are typically of the form \hm(k,z)\ < c , for some functions hm and power b > 0 that varies from bound to bound. What turns out to be important in the analysis is that 5 = 5 = 2 + d ^ is greater than 2 when d > 4. In unpublished work [18] the authors note that the analysis of [19] should be robust enough to permit extension to certain other models where the lace expansion 159 is applicable, above dc / 4. In particular they outline how [19] might be adapted to analyse lattice trees in dimensions d > 8. While deviating somewhat in the details, our analysis in this chapter (and its application to lattice trees) is based on the ideas of [18]. * In our analysis we introduce two new parameters 8(d),p* and a set B. We will discuss the significance of p* and B when they appear shortly. The most important parameter, 9(d), is taking the place of ^ in exponents appearing in various bounds. As in [19] we require that 9 > 2, and we apply the results of this chapter to lattice trees model with the choice 9 = 2 + We also expect the result to be applicable to other models where the lace expansion is used in the analysis above a critical dimension dc. In such cases the lace expansion for d > dc suggests setting 9 = 2 + ^Y^. In particular for percolation (dc = 6) we would expect 9 = 2 + Note that in the case dc = 4, 2 + ^y^. = ^, which is that appearing in [19]. There is an unpublished version [15] of this chapter consisting of full proofs of the material in [19], adapted to our more general setting, with generally only cosmetic changes (e.g. ^ >-+ 9) required. In this thesis we will state the assumptions and results explicitly, but for the sake of brevity we will present only significant changes in the proof and leave the reader to refer to [19] when the changes are only cosmetic. ' *; Therefore the chapter is organised as follows. In Section A.2 we state the assumptions S,D,EQ, and GQ on the quantities appearing in the recursion equation, and the " -^theorem" to be proved. In Section A.3, we introduce the induction hypotheses on fn that will be used to prove the 0-theorem. We advancement of the induction hypotheses is highly technical and our extension does not require significant, alterations from the analysis of [19]. We therefore briefly discuss the role of 9 in this section and direct the interested reader to [19] for the analysis. Once the induction hypotheses have been advanced the 0-theorem follows, without difficulty. A.2 Assumptions on the Recursion Relation Suppose that for z > 0 and k G [—7r, ir]d, we have fo(k; z) = 1 and • n + l fn+i(k;z) = E 9m(k;z)fn+i-m(k;z) + en+i(k;z), . (n > 0), . (A.3) m = l where the functions gm and em are to be regarded as given. The goal is to understand the behaviour of the solution fn(k; z) of (A.3). 160 A . 2 . 1 A s s u m p t i o n s S,D,Ee,Gg The first assumption, Assumption S, requires that the functions appearing in the recursion equation (A.3) respect the lattice symmetries of reflection and rotation, and that fn remains bounded in a weak sense. This assumption remains unchanged from [19]. Assumption S. For every n 6 N and z > 0, the mapping k H-> fn(k;z) is symmetric under replacement of any component ki of k by — ki, and under permutations of the components of k. The same holds for en(-;z) and gn(-;z). In addition, for each n, \fn(k; z)\ is bounded uniformly in k £ [—TT, ir]d and z in a neighbourhood of 1 (which may depend on n). The next assumption, Assumption D, introduces a function D — Dr, which defines the underlying random walk model and involves a non-negative parameter L which will typically be S> 1. This serves to spread out the steps of the random walk over a large set. An example of a family of D's obeying the assumption was given in Definition 1.2.4 and the remarks following it. In particular Assumption D implies that D has a finite second moment, and we define i=i 3 x (A.4) k=0 x Let a(k) = 1-D(k). (A.5) Assumption D. We assume that fi(k\z) = zD(k), ei(k;z ) = 0. (A.6) In particular, this implies that g\{k;z) — zD(k). As part of Assumption D, we also assume: (i) D is normalised so that D(0) = 1, and has 2 + 2e moments for some 0 < e < 0 — 2, i.e., ; 1 xezd (ii) There is a constant C such that, for all L > 1, Halloo < CL -d a2 = a2< CL2, (A.S) 161 (iii) There exist constants rj, c\,c2 > 0 such that ciL2k2 < a{k) < c2L2k2 (||fc||oo < L - 1 ) , (A.9) a(k) > r) (pHoo > L'1), (A.10) a{k)<2-n (k£[-ir,Tr]d). (A.ll) Assumptions E and G of [19] are now adapted to general 6 > 2 as follows. The relevant bounds on fm, which a priori may or may not be satisfied, are that for some p* > 1, some nonempty B C.[l,p*] and B = B{p*)=L-£ (A.12) we have for every p £ B, \\D2fm(-,z)\\p< d K d A a , \fm(0;z)\<K, \V2fm(0;z)\<Ka2m, (A.13) p m 2 P for some positive constant K. The full generality in which this has been presented is not required for our application to lattice trees where we have p* — 2 and B = {2}. This is because we require only the p = 2 case in (A.13) to estimate the diagrams arising from the lace expansion for lattice trees and verify the assumptions Eg, Gg which follow. In other applications it may be that a larger collection of || • | | p norms are required to verify the assumptions and the set B is allowing for this possibility. The parameter p* serves to make this set bounded so that Pip*) is small for large L. The bounds in (A.13) are identical to the ones in [19](1.27), except for the first bound, which only appears in [19] with p = 1 and 0 = §\ Assumption Eg. There is an LQ, an interval I C [1 - a, 1 + a] with a £ (0,1), and a function K H - » Ce(K), such that if (A.13) holds for some K > 1, L > LQ, z £ I and for all 1 < m < n, then for that L and z, and for all k £ [—TT, Tr]d and 2 < m < n + 1, the following bounds hold: |em(A:;z)| < Ce{K)Bm-e, \em{k;z) - em{0;z)\ < Ce{K)a{k)Bm-9+l. (A.14) Assumption Gg. There is an LQ, an interval / C [1 - a, 1 +.a] with a £ (0,1), and a function K H - > Cg(K), such that if (A.13) holds for some K > 1, L > LQ, z £ I and for all 1 < m < n, then for that L and z, and for all A: £ [—TT, Tr\d and 2 < m < n + 1, the following bounds hold: \9m(k\z)\ < Cg{K)Bm-\ \V2gm(0;z)\ < Cg(K)a2Bm-e+1, (A.15) 162 \dzgm(0;z)\ <Cg(K)Pm-9+1, (A.16) \9m(k; z) - gm(0; z) - a(k)a-2W2gm(0; z)\ < Cg(K)8a(k)l+<'m-e^l+e'\ (A.17) with the last bound valid for any e' € [0, e], with 0 < e < 6 — 2 given by (A.7). Theorem A.2.1. Let d > dc and 6(d) > 2, and assume that Assumptions S, D, EQ and GQ all hold. There exist positive LQ = Lg(d,e), zc = zc(d,L), A = A(d,L), and v — v(d,L), such that for L > Lo, the following statements hold, (a) Fix 7 G (0,1 A e) and 6 G (0, (1 A e) - 7). Then f(-^-Zc)=Ae-&[l + 0(k2n-5) + 0(n-6+2)], (A.18) with the error estimate uniform in {k G R d : a(k/Vvo2n) < ^n~l logn}. (b) V 2 /„ (0 ;z c ) _ ^ , _ 2 „ r i , n(t>~-S\ fn(0;zc) (c) For all p > 1, = va2n[l + 0(8n-6)}. (AJ19) | £ > 2 / n ( - ; z c ) | | P < ^ ^ . (A.20) Lpn2p (d) The constants zc, A and v obey 00 1 = E 9m(0;zc), m = l A _ 1 1- 2^m=l VmKV^c) (A.21) 22m=imgm(0;zc) °~2 E m = l m f f m ( 0 ; ^ c ) ' It follows immediately from Theorem A.2.1(d) and the bounds of Assump-tions E and G that zc = l + 0(/3), A = 1 + 0(8), v = l + 0(B). (A.22) A . 3 I n d u c t i o n h y p o t h e s e s The recursion relation (A.3) is analysed using induction on n, as done in [19]. The induction hypotheses involve a sequence vn, which is defined exactly as in [19] as follows. We set VQ = bo = I, and for n > 1 we define 1 n n b b" = —2E v 2 5 m ( 0 ; z), cn = Y(m- l)<?m(0; z), vn = ——. (A.23) m = l m = l 163 The induction hypotheses also involve several constants. Let 6 > 2, and recall that e was specified in (A.7). We fix 7,5 > 0 and A > 2 according to 0 < 7 < 1 A (0 - 2) A e 0 < S < (1 A (0 - 2) A e) - 7 • (A.24) 0 - 7 < A < e. Here A replaces p + 2 from [19] simply to avoid confusion with p(0) from other chapters in this thesis. We also introduce constants K\,..., K$, which are independent of 6. We define K\ — m.a.-x.{Ce{cKi),Cg{cKA,Ki}, (A.25) where c is a constant determined in Lemma A.3.6 below. To advance the induction, we will need to assume that K3 > Kx > K'4 > K4 > 1, K2> KuZK'i, K5 > K4. (A.26) Here a » t denotes the statement that a/6 is sufficiently large. The amount by which, for instance, K3 must exceed K\ is independent of 6, but may depend on p*, and will be determined during the course of the advancement of the. induction in Section A.4. Let ZQ = z\ — 1, and define zn recursively by n+1 zn+1 = 1-^2 9m{0;zn), n > 1. (A.27) For n > 1, we define intervals In = [ z n - K 1 p n - e + 1 , z n + KlPn-0+1]. „ (A.28) In particular this gives I\ = [1 — K\3,1 + K\B\. Recall the definition a(k) = 1 — D(k) from (A.5). Our induction hypotheses are that the following four statements hold for all z G In and all 1 < j < n. (HI) \zj-Zj-t\KKrfj-0. , : (H2) K - ^ - i | <K20j-d+l. (H3) For k such that a(k) < 7 J - 1 log j , fj(k; z) can be written in the form i fj(k;z)=Y[[l-via(k)+ri(k)], i=l with r{(k) = ri(k;z) obeying \n(0)\,< K z 3 r e + \ \n(k) - ri(0)\ < K30a(k)rs. 164 (H4) For k such that' a(k) > yj 1 log j , fj(k; z) obeys the bounds* \fj(k;z)\ < K4a(k)-Xr6, \fj(k;z) - fj-i(k;z)\ < Kba{k)-x+lj~6• Note that these four statements are those of [19] with the replacement P + 2 4 A ; (A.29) in (H4) and the global replacement (A.30) By global replacement we also mean that 4 9 - 1 , 4 0 — 2, etc. whenever such quantities appear in exponents. A.3.1 Initialisation of the induction The verification that the induction hypotheses hold for n = 0 remains unchanged from the p = 1 case, up to the replacements (A.29-A.30). A.3.2 Consequences of induction hypotheses The key result of this section is that the induction hypotheses imply (A. 13) for all 1 < m < n, from which the bounds of Assumptions Eg and GQ then follow, for 2 < m < n + 1. . . As in [19] throughout this chapter: • C denotes a strictly positive constant that may depend on d, 7, <5, A, but not on the Ki, not on k, not on n, and not on /3 (provided 3 is sufficiently small, possibly depending on the Ki). The value of C may change from line to line. • We frequently assume 8 1 without explicit comment. Lemmas A.3.1 and A.3.3 are proved in [19] and the proof in our context requires only the global change (A.30). Lemma A .3.1. Assume (HI) for 1 < j < n. Then I\.D I2 D 3 In-Remark A.3.2. We were unable to verify [19](2.19) as stated. Instead of [19]'(2.19) we use \si{k)\ < K3(2 + C(K2 + K3)8)8a(k)rs, (A.31) 165 the only difference being the constant 2 appears here instead of' a constant 1 in [19](2.19). This does not affect the proof. To verify (A.31) we use the fact that <l + 2x for 0 < x < 2 to write for small enough 0, \Si(k)\ < [l + 2K30] [(l + \vi - i\)a(k)n(0) + \n(k) - n(0)\) < [1 + 2K30] (1 + CK20)a{k)7rT + ^ (A.32) < [1 + 2 3^/?][2 + C^2/3] < [2 + C(K2 + tf3)0. ii/ere we ftai/e used rTie bounds of (H3) as well as the fact that 6 — 1 > 5. Lemma A . 3 . 3 . Let z G In and assume (H2-H3) for 1 < j < n. Then for k with a(k) < 7 J _ 1 logj, \fj(k;z)\ < eCK3Pe-(i-c{Ka+Ka)p)jaW. (A.33) The middle bound of (A.13) follows, for 1 < m < n and z G Im, directly from Lemma A.3.3. We next state two lemmas which provide the other two bounds of (A.13). The first concerns the || • ||p norms and contains the most significant changes from [19]. As such we present the full proof of this lemma. Lemma A . 3 . 4 . Let z G In and assume (H2), (H3) and (H4). Then for all 1 < j < n, and p > 1, d _d_ LP j2P where the constant C may depend on p, d. Proof. We show that I I * ' * ' ) ^ 5 ^ - ' <A- 3 5> For j = 1 the result holds since j/i(A;)| = \zD{k)\ < z < 2 and by using (A.8) and the fact that p > 1. We may therefore assume that j > 2 where needed in what follows, so that in particular log j > log 2. Fix z G In and 1 < j < n, and define Ri = {ke ["*, *]' a{k) < 7 J " _ 1logj, Halloo < L~ 1}, R2 = {k€ a(fe) < 7J~ _ 1iog;', l l * l l o o > L~ lh Rs = {k G 7T] a{k) > 7J" - 1 logj, Halloo < L~ lh i?4 = {k G ["*, 7T] a(k) > -yj~ _ 1 iogi , I W l o o > L~ 166 The set R2 is empty if j is sufficiently large. Then \\D2m = i:f {D{kf\fj{k)\)P^-d. (A.36) 1=1 JRi ' We will treat each of the four terms on the right side separately. On R\, we use (A.9) in conjunction with Lemma A.3.3 and the fact that D2 < 1, to obtain for all p > 0, <n/_*0e-^^,<_^ (A.37, < 1=1 - ~L c ldjd/2-Here we have used the substitution k[ = Lki^/pJ. On R2, we use Lemma A.3.3 and (A. 10) to conclude that for all p > 0, there is an a(p) > 1 such that , p ddk f _,• ddk <R^ ) ( 2 * ) * -where \R2\ denotes the volume of R2. This volume is maximal when j = 3, so that L Nwi) ' ^^cL =ca~j^ (A-38) \R2\ < \{k : a(k) < < \{k : D(k) > 1 - ^ } | < ( 1 l 2 i i n 2 n , < I 1 \2rT-<i (A.39) S L 7 l o s 3 J \W 111 S ( , 7 l o K 3 - ) ^ L ) * 3 1 3 using (A.8) in the last step. Therefore ori\R2\ < CL~dj~dl2 since ^ - < C for every J , and (/3(A;)2|/j(fc)|)P ^ < CL-drdl2. (A.40) On P 3 and R4, we use (H4). As a result, the contribution from these two regions is bounded above by 4 * D(k)2P ddk ^yf r D (k)xP (27r) d ' (A141) i=3 On # 3 , we use D(k)2 < 1 and (A.9). From (A.9), k <E R3 implies that L2\k\2 > Cj~x log j so that CKl f -J—ddk < CK* fX rd-l-2Xpdr (A 42) 167 Since log 1 = 0, this integral will not be finite if both j = 1 and p > jx> but recall that we can restrict our attention to j > 2. For d > 2Xp, we have an upper bound on (A.42) of CKl rz , CKl / C \ d - 2 X p CKl L p rd~^dr < ° « L ( ° \ < (A 43) 2Xp Jo dr^j9PL2XP[L) ^ j9PLd- (A.4o) j6p l2Xp For d = 2\p, (A.42) is CKl [T i ^ CKl (CjLJj\ CKl ( Cj\ A Now since d = 2Xp, we have that #p = || > 5 using the fact that A < 9. This gives CKP an upper bound on this term of d • Lastly for d .< 2Xp, since A < 9, (A.42) is bounded by 2\p-d V CL'j J as required. On i?4, we use (A.8) and (A. 10) to obtain the bound CKl f ^,.^ddk _ CKl f A / f X , A „ Ct f? where we have used the fact that p > 1 and |D| < 1. Since i f f < (1 + Ki)p, this completes the proof. • Lemma A.3.5. Let z £ In and assume (H2) and (H3). Then, for 1 < j < n, |V2/,(0; *)| < (1 + C{K2 + K3)6)o2j. - (A.47) The proof is identical to [19]. We merely point out one small correction to the first line of [19] (2.35), where a constant 2 is missing. It should read |VM0)|=»1ES«(*V*(0'|. f: (A*) 1=1 however once again this does not affect the proof. The next lemma, whose proof is the same as in [19], is the key to advancing the induction, as it provides bounds for e„+i and gn+i-168 Lemma A.3.6. Let z € /„, and assume (H2), (H3) and (R~4). For k € [—IT, Tr]d, 2 < j < n + 1, and e' G [0, e], £/ie following hold: (i) \gj{k-z)\<K'ABj-e, (ii) \V2g,(O;z)\<KyBj-0+\ (iii) \dzgj(0;z)\<K'8j-6+\ (iv) \gj(k-,z)-gj(0-z)-a(k)a-^gj(0;z)\<K>4^ (v) \ej(k;z)\<K'4Bre, (vi) \e3{k;z) - ej{Q\z)\ < K'4a(k)Bj-d+1. A . 4 T h e i n d u c t i o n a d v a n c e d In this section we advance the induction hypotheses (H1)-(H4) from n to n + 1. For (H1)-(H2) the proofs are identical to those in [19] up to the global replacement (A.30) due to the following observations. Since 9 > 2 and e' < e < 9 — 2 we have that YI Z^pi ^ J2 m e - i - e ' < °°' H ^ (n + 2 - r o ) * - 2 " ( A > 4 9 ) 771=2 771=2 J=7l+2—771 Similarly, convolution bounds used in [19] to verify (H1)-(H3) remain applicable under the global replacement (A.30). The above bounds are also used to advance (H3)-(H4). In addition, in (H3) we require that there exists a q > 1 but sufficiently close to 1 so that ( n + l ) - M o g ( n + l ) x ( ^ + 1 ) ° V M ' V** • (A.50) \log(n + l), (9 = 3), is bounded by (n + 1)~5. This holds since <5 + 7 < l A ( 0 - 2 ) by (A.24). This corre-sponds to [19] (3.43). Other similar bounds required to verify (H3) (corresponding to [19](3.50)-(3.51) and [19](3.58) for example) also follow from 8 + 7 < 1A (9 - 2). To advance (H4) we make the additional global replacement (A.29). Then using the fact that 7 + A — 9 > 0 we have that there exists q' close to 1 so that for a(k) < 7 7 i _ 1 log n, C n> C " nenq'y+\-e - neai<k)x- . ^ - 0 i ; This corresponds to [19] (3.62), and is used to advance the first and second bounds of (H4). In addition we use the fact that A > 2 so that a(k)x~2 < C (recall that a(k) < 2 from ( A . l l ) ) to get ^ < _ c _ The proof of Theorem A.2.1 now proceeds as in [19] with the global replace-ment (A.30). 169 Bibliography [1] R. Adler. Superprocess local and intersection local times and their correspond-ing particle pictures. In Seminar on Stochastic Processes 1992. Birkhauser, Boston, 1993. [2] D. Aldous. Tree-based models for random distribution of mass. Journal. Stat. Phys., 73:625-641, 1993. [3] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. [4] D. Brydges and J . Imbrie. Dimensional reduction formulas for branched poly-mer correlation functions. Journal. Stat. Phys., 110:503-518, 2003. [5] D. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. Com-mun. Math. Phys., 97:125-148, 1985. [6] D. Dawson. Measure-valued markov processes. In Ecole d'Ete de Probabilites de Saint Flour 1991, Lecture notes in Mathematics, no.. 1541. Springer, Berlin, 1993. [7] E . Derbez and G. Slade. The scaling limit of lattice trees in high dimensions. Commun. Math. Phys., 193:69-104, 1998. [8] S Ethier and T. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986. [9] J . Frohlich. Mathematical aspects of the physics of disordered systems. In Phenomenes critiques, systemes aleatoires, theories de jauge, Part II. North-Holland, Amsterdam, 1986. [10] G. Grimmett and D. Stirzaker. Probability and Random Processes (second edition). Oxford University Press, Oxford, 1992. . [11] T. Hara, R. van der Hofstad, and G. Slade. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab., 31:349-408, 2003. 170 T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. Journal. Stat. Phys., 59:1469-1510, 1990. T. Hara and G. Slade. The number and size of branched polymers in high dimensions. Journal. Stat. Phys., 67:1009-1038, 1992. R. van der Hofstad, F . den Hollander, and G. Slade. A new inductive approach to the lace expansion for self-avoiding walks. Probab. Theory Relat. Fields., 111:253-286,1998. . ,. R. van der Hofstad, M . Holmes, and G. Slade. A n extension of the generalised inductive approach to the lace expansion. Unpublished note, 2004. R. van der Hofstad and A . Sakai. Gaussian scaling for the critical spread-out contact process above the upper critical dimension. Electr. Journ. Probab., 9:710-769, 2004. . R. van der Hofstad and A . Sakai. Convergence of the critical finite-range contact process to super-brownian motion above 4 spatial dimensions. In preparation, 2005. R. van der Hofstad and G. Slade. A n extension of the generalised inductive approach to the lace expansion. Unpublished note, 2002. R. van der Hofstad and G . Slade. A generalised inductive approach to the lace expansion. Probab. Theory Relat. Fields., 122:389-430, 2002. R. van der Hofstad and G. Slade. Convergence of critical oriented percolation to super-brownian motion above 4+1 dimensions. Ann. Inst. H. Poincare Probab. Statist., 39:413-485, 2003. R. van der Hofstad and G. Slade. The lace expansion on a tree with application to networks of self-avoiding walks. Adv. Appl. Math., 30:471-528, 2003. M . Holmes, A . Jarai, A . Sakai, and G. Slade. High-dimensional graphical net-works of self-avoiding walks. Canadian Journal Math., 56:77-114, 2004. E . Janse van Rensburg. On the number of trees in Zd. J. Phys. A: Math. Gen., 25:3523-3528, 1992. J . Klein. Rigorous results for branched polymer models with excluded volume. J. Chem. Phys., 75:5186-5189, 1981. T. Lubensky and J . Isaacson. Statistics of lattice animals and'dilute branched polymers. Phys. Rev., pages 2130-2146, 1979. 171 [26] N . Madras. A rigorous bound on the critical exponent for the number of lattice trees, animals, and polygons. Journal. Stat. Phys., 78:681-699, 1995. [27] E . Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics, no. 1781, Ecole d'Ete de Prob-abilites de Saint Flour 1999. Springer, Berlin, 2002. [28] G . Slade. Lattice trees, percolation and super-brownian motion. In Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten. Birkhauser, Basel, 1999. [29] H . Tasaki and T. Hara. Critical behaviour in a system of branched polymers. Prog. Theor. Phys. Suppl, 92:14-25, 1987. [30] W . Werner. Random planar curves and Schramm-Loewner evolutions. In Lec-tures on Probability Theory and Statistics, no. 1840, Ecole d'Ete de Probability's de Saint Flour 2002. Springer, Berlin, 2004. 172
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Convergence of lattice trees to super-brownian motion above the critical dimension Holmes, Mark 2005
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Title | Convergence of lattice trees to super-brownian motion above the critical dimension |
Creator |
Holmes, Mark |
Publisher | University of British Columbia |
Date Issued | 2005 |
Description | A lattice tree is a finite connected set of lattice bonds containing no cycles. Lattice trees are interesting combinatorial objects and an important model for branched polymers in polymer chemistry and physics. In addition they provide an interesting example of critical phenomena in statistical physics with similar properties to models such as self-avoiding walks and percolation. We use the lace expansion to prove asymptotic formulae for the Fourier transform of the r-point functions (quantities which count critically weighted trees containing r fixed points) for a spread-out model of lattice" trees in ℤ[sup d] for d > 8. Our results therefore provide additional evidence in support of the critical dimension d[sub c] = 8. The spread out model allows bonds between vertices x,y ∈ ℤ[sup d] with x-y ∞ ≤ L, providing a small parameter L [sup –d/2] needed for convergence of the lace expansion. We extend the inductive approach (to the lace expansion on an interval) of van der Hofstad and Slade [19] to prove convergence of the Fourier transform of the 2-point function (r = 2). We then proceed by induction on r, equipped with the lace expansion on a tree [21]. The asymptotic formulae for the r-point functions imply convergence of certain expectations of the spread out lattice trees model formulated as a measure valued process, to those of the canonical measure of super-Brownian motion. Appealing to the hypothesis of universality, we expect that the results also hold for the nearest neighbour model. Our results together with the convergence of the survival probability would imply convergence of the finite-dimensional distributions of our process to those of the canonical measure of super-Brownian motion. Convergence of the survival probability remains an open problem. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2009-12-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0079636 |
URI | http://hdl.handle.net/2429/16894 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2005-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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