Convergence of L a t t i c e Trees 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 T H E S I S 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 REQUIREMENTS FOR T H E D E G R E E OF D o c t o r of P h i l o s o p h y in. T H E F A C U L T Y OF G R A D U A T E STUDIES (Mathematics) T h e U n i v e r s i t y of B r i t i s h C o l u m b i a July 2005 © Mark Holmes, 2005 Abstract 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 for d > 8. Our results therefore provide additional evidence in support of the critical dimension d = 8. The spread out model allows bonds between vertices x,y 6 Z 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]. d d c 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. A p 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 1 viii Introduction 1 1.1 Background and motivation 1.1.1 Combinatorics and statistical physics 1.1.2 Probability and measure-valued processes 1.2 The model 1.3 A measure-valued process 1.4 The r-point functions 1 2 4 6 10 13 2 The lace expansion 2.1 Graphs and Laces 2.1.1 Classification of laces 2.2 The Expansion 17 17 23 25 3 The 2-point function 3.1 Organisation 3.2 Recursion relation for the 2-point function 28 28 29 3.3 3.4 3.5 35 37 44 4 Assumptions of the induction method Verifying assumptions Proof of Theorem 1.4.3 . The r-point functions 4.1 4.2 48 Preliminaries Application of the Lace Expansion iii . 48 54 4.3 Decomposition of 57 4.4 4.5 4.6 Bounds on the £ . Proof of Theorem 4.1.8 Proof of Theorem 1.4.5 60 62 65 5 Diagrams for the 2-point function 5.1 Definitions and Notation 70 5.2 Proof of Proposition 5.1.1 5.2.1 Diagrams with an extra vertex General Diagrams Diagram pieces 5.4.1 Proof of Lemma 5.4.1 5.4.2 Proof of Lemma 5.4.2 74 81 82 90 91 93 5.3 5.4 6 Diagrams for the r-point functions 95 6.1 6.2 6.3 Proof of Proposition 4.3.2 Proof of Lemma 6.1.1 Proof of Lemma 6.1.2 95 97 100 6.4 Proof 6.4.1 6.4.2 Proof Proof 6.6.1 105 105 109 122 124 125 6.5 6.6 6.6.2 6.6.3 7 70 of Lemma 6.1.3 Acyclic laces with 2 bonds covering the branch point Acyclic laces with 3 bonds covering the branchpoint of Lemma 4.3.3 of Lemma 4.2.1 Proof of the first bound of Lemma 4.2.1 Proof of the third bound of Lemma 4.2.1 Proof of the second bound of Lemma 4.2.1 Convergence to C S B M . 7.1 7.2 . 128 129 134 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 7.3 Lattice trees 7.2.1 ISE Finite dimensional distributions 137 139 141 7.4 7.5 Proof of Theorem 1.3.1 A note on convergence of finite dimensional distributions 152 157 iv Appendix A 159 A . l Motivation A.2 Assumptions on the Recursion Relation A.2.1 Assumptions S,D,Ee,Ge A.3 Induction hypotheses A.3.1 Initialisation of the induction A.3.2 Consequences of induction hypotheses A.4 The induction advanced Bibliography 159 160 161 163 165 165 169 170 v List of Figures 1.1 1.2 1.3 1.4 Example of a lattice tree Example of a lattice tree Particle picture of a lattice tree . Examples of shapes . . . 1 7 8 15 2.1 2.2 2.3 2.4 2.5 2.6 A graph on a network Af and its associated subnetwork A A graph containing a bond in 1Z Graphs and laces on star-shaped networks Construction of a lace from a connected graph Minimal and non-minimal laces Cyclic and acyclic laces 18 19 20 23 23 24 3.1 A lattice tree as a backbone and mutually avoiding branches 4.1 4.2 4.3 4.4 A lattice tree and its corresponding network shape r-point error terms The relation £ - rj = r + 3 Examples of degenerate shapes 5.1 Diagram pieces Jl4!' and A 5.2 5.3 5.4 A typical opened diagram Diagram for N = 1 A typical Feynman diagram . . 6.1 The single bond lace on [ - M , Mi] and the corresponding [0, M i + M2]. 98 6.2 6.3 A lace on [ - M , Mi] and the corresponding [0, M i + M ] Some cyclic laces containing only 3 bonds 99 101 6.4 6.5 Diagrammatic bound for a basic cyclic lace A degenerate cyclic lace and its diagrams 101 104 6.6 6.7 Diagrammatic bound for a basic acyclic lace Diagrams for acyclic laces with 2 bonds covering the branch point . . . . 29 51 56 64 66 = 1 72 mijm2 , 73 85 86 2 2 2 vi '. 106 . 108 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 . Ill 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 6.14 6.15 6.16 Application of Lemma 6.4.3 . . . Another application of Lemma 6.4.3 Decomposition of a diagram from a non-minimal lace Construction of a lace from a graph that covers multiple branch points vii 114 115 116 130 Acknowledgements This work was supported by the following awards from the University of British Columbia: K i l l a m Predoctoral Fellowship, Josephine T . Berthier Memorial Fellowship, 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. MARK The University of British Columbia July 2005 viii HOLMES 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 Background and motivation A lattice tree in Z 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 d 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 l be the number of n-bond (nearest neighbour) 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 n dimensions, and a standard subadditivity argument then shows that Ifi —)• A > 0 as n —>• oo. The bounds C i n - C 2 logn n < A l n < C 3 „ ^ A n j ( !) 1 were proved in [23] and [26] respectively. Using the notation f(x) ~ g(x) to mean linx^oo = 1, it is widely believed that l ~C\ n - , n 1 (1.2) 6 n where 6 is called a critical exponent for the model. Critical exponents convey information 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, p (x) = 2~lreT(x) P^ T(x) is the set of lattice'trees containing the origin and x and # T is the number of bonds in T . The function p (x) is a power series with i the coefficient of p being the number of lattice trees containing 0 and x consisting of AT bonds. This power series has nontrivial radius of convergence p = j, at which it is believed that p(x) changes from having exponential decay in \x\ for p < p to power law decay C a S P c(x) ~ |d-2+,,' M -> °°> (L3) T w n e r e p v N c c P l < p II 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 — p 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 c 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 d 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 d = 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 2~2 y z Ppc( )ppc(y ~ )Pp ( ~ v)Pp { ) implies mean-field critical behaviour for the susceptibility xip) = Y Pp( )The 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 c c x x z X z c c x x 3 s a m e 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]. F i x a probability space (O, T, P) and suppose that Xi are independent identically distributed real valued random variables with mean 0 and finite variance a , 2 and let S n = 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, =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, ==» fj, if and only if for every bounded continuous / : R —)• R, f / dfj, —> J f du. We use the notation E [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 =^4» H E^JfiXn)] -> Ep[f(X)], for every bounded continuous / : R -»• R To n n n (1 n 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, [e ] -> ^ [ e ] (or more traditionally E[e ^] -> E[e ] = e V) to that of the Gaussian. In other words the functions {fk(x) = e : k G [-TT, 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[X +\\T ] — 0) and the central limit theorem may still hold. ikXn i f c X ik ikz - n lkx n n Setting So = 0, the collection {5„} >o is a random walk on R, and writing t > 0, defines a real-valued stochastic process {X^} >o that is right n Xf = t continuous with left limits for each n, i.e. {X™} >o G D ( R ) . Define probability measures /j, on the Borel sets of D(R) by ^„(«) = P({X^} >o € •)• Another fundamental result in probability states that n ==>• 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 converges 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 t n t n 4 of the finite dimensional distributions and tightness. The {ti } are tight if for every n e > 0 there exists a compact K C D(R) such that sup„/j, (K ) c n < e. Convergence of the finite dimensional distributions by definition means that for every m G N , t G [0, o o ) , and every bounded continuous / : K m E, n [f{Xl,..., XI)) m -> K , -+ E [f(B B )]. w tl (1.5) tm To verify (1.5), it is enough to show convergence of the corresponding Fourier transforms 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 X j ) of the process also have mean 0, {X^} >o is a martingale (ElX^^T^} = X™ where = cr({X"} < )). 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 branching 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 Y = Y particles at independently and randomly chosen neighbouring vertices and then dies instantly. It can be described by a measure-valued process X where for each fixed time n, X is a finite measure (X e My(M )) on the Borel sets of R with X (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. W i t h appropriate scaling of space, time, and mass, critical branching random walk converges weakly (i.e. convergence in the space of measures on D(Mjp(K ))) to a measure-valued process X called super-Brownian t s t a n n d d n n FL! t 5 motion (SBM). This is of course a statement that fi' ==>• No for some measures n fx' € MF(£)(M p(M )) and some other measure No called the canonical measure of d n J super-Brownian motion ( C S B M ) . Tightness of the measures u-' can be verified using n martingale methods. Now the support process {At}t>o, where At is the support of the measure Y = J X ds of a S B M X has Hausdorff dimension 4 A d and has no self 0 t s s 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 ( M i ( M ) ) which describes the distribution of the total mass of C S B M d (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 ourselves 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-i uviVi}j v 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 i n 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 , . . . } . D e f i n i t i o n 1.2.3. 1. For x E Z d let T = {T : x <E T}. Note that this set always includes the single x vertex lattice tree, T = {x} that contains no bonds. We also let T (x) = {T G y T : x G T}, and often write T(x) for To(x), the set of lattice trees containing y 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 ifT n 7 ^ 0. 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,.. .,x -i) G Z ( ) and fi G Z ^ we we write x £ T „ i / i ; 6 T /or eac/i i and define 7fi(x) = {T G 7o : x G T „ } . d _ 1 r _ 1 r n i 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] 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]. d D e f i n i t i o n 1.2.4. Leth be a non-negative bounded function o n R which is piecewise continuous, symmetric under the Z -symmetries of reflection in coordinate hyperplanes and rotation by | , supported in [— 1, l ] , and normalised , h(x)d x = d d d d 1 d 8 1). Then for large L we define h(x/L) 22 .ih{x/L) x& Remark 1.2.5. Since Yl ez h(x/L) d x to ~ L using a Riemann sum approximation d ^ h(x)d x, the assumption that L is large ensures that the denominator of d d (1-6) is non-zero. Since h is bounded, ~Y^ ^%dh,{xjV) ~ L also implies that d x \\D\\oo<§- . ' d We define a = 2~2 \x\ D(x). 2 The sum Y^, \x\ D(x) 2 r X x (1.7) 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~ Ir_ ]d(a:) for which D(x) = (2L + l)~ d d 11 I[-L,L] nZ ( )d d x Definition 1.2.6 (L, D spread out lattice trees). Let fi/j = {x € Z : D(x) > D 0}. We define an L,D spread out lattice tree to be a lattice tree consisting of bonds {x,y} such that y — x £ VLDThe 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 . W , (B)= D(y-x), {x,y}eB P D P (1.8) with W D(0) = 1. If T is a lattice tree we define P) W , {T) P D = W , (B ), P D T (1.9) where BT is the set of bonds ofT. Definition 1.2.8 (p{x)). Let (x)= Pp YI T£T(x) 9 pMn w (i.io) Clearly we have 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 p at which 2~2 Pp( ) converges for p < p and diverges for p > p . 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. p x c x c c T h e o r e m 1.2.9. Let d > 8 andfixv > 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 PPA ) = 2(|a.| X a v \ !)d-2 1 ( £( ° \^\ \ + x V r f - 8 ) A \ 2 / L l)((d-8)A2)^ J + ° \{\ \ X V \ 2 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(-) = W (-), and p(x) = p (x). PctD (1.12) Pe Hara, van der Hofstad and Slade [11] also proved that p p{0) < 1 + O (L~ ) and 2+U c ^"^^V'cTv.rj 1 • ' (1 13) where the constants in the above statements depend on v and d, but not L. 1.3 A measure-valued process Let M p ( R ) denote the space of finite measures on R with the weak topology. For D d each i, n € N and each lattice tree T , we define a finite measure XT n,T X = ^l £ g e M F ( R ( 1 D by ) 1 4 ) where 6 (B) = I B for all B e #(M ). The constants C\,Ci depend on L and d and will be stated explicitly later. Figure 1.3 shows a fixed tree T and the set Tj for i = 10. For this T , the measure X ^ assigns measure ^ to each vertex i n the set d X xe 1 Tialy/Cin = {x : yJC^nx € Tin.}. We extend this definition to all t € R t by Xt' T = Xlnl • 10 ' (1-15) Thus for fixed n, T and t > 0, we have { X n , T t } € D(M (R )). d F 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 probability measure P on the countable set 7o by P({T}) = £)= *T , Er£ p( , so that (T) .jg c T ( L l g ) P(0) Lastly we define the measures /x € Mp(D(M (R ))) d n F Hn(H) = C n P ( { T : {X?' } € H}) , T 3 tm+ by H€ B(D(M (R ))), (1.17) d F where B(E) denotes the Borel cr-algebra on E and C3 is another constant that will be stated explicitly later. We expect that /j, ==» No, where No is the canonical n measure of super-Brownian motion. This convergence, which we will 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" / 0M) (where 0 M denotes the zero measure) e 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 finitedimensional distributions we would require the asymptotics of the survival probability P ( T > 0). Without the survival asymptotics we prove Theorem 1.3.1, which n 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^ } denotes a process chosen 1 according to the finite measure u, and {^t} denotes super-Brownian excursion, i.e. n 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 is called a, multinomial if Q(X) function F : M p ( R ) r f m is a real multinomial in {Xi(l),..., X (l)}. m m —> M A -> 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 defined n by (1.17) the following holds: For every s,X > 0, m G N , t G [0, o o ) m and every F : M ( M ) d C m F bounded by a multinomial and such that N Q ( X - G Vp) = 0, ( [F{Y?)Y?{1)] (.1) (2) i j ^ F(yP)/ n { r £ ^E ( 1 ) > A } and [F{Yr)Y {\) N o s n o (1.18) (1.19) F{Yr)I {i)>x {Ys } The factors in Theorem 1.3.1 involving the total mass at time s, are essentially two ways of ensuring that our convergence statements are about finite measures. 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 and (j) : M —> C we define /j,(<j>) = f 4>dp. In particular, D RF Rd (1.20) c For k G [ - 7 r , 7 r ] , let (j> {x) : Z -> C be defined by <f> (x) = e ' . prove the following Lemma in Chapter 7. ik x D r f k k L e m m a 1.3.2. Suppose that for every r > 2, every k G R ( R - 1 We indirectly ) , D and every t G r-1 E, (1.21) N E fJ-n 0 3=1 Then the conclusions of Theorem 1.3.1 hold. 12 Note that for t € (0, oo) r we have, 1 r-l j=i • j=i r-l CCp(0)n r 1 a T€T j=l xj-.VnChxjeT 0 [ntjl (1.22) (n«M*i)) E p(0)n r - 2 I rer /nC2 CzC[r - l p(0)n r - 2 E E V ^ 2 E W(T) E L n I J (5t) W(T). This suggests that it might prove useful to examine the quantities 2^Ter 1.4 T h e r-point Definition 1.4.1 L £ j functions (2-point function). For ( > 0, n 6 N , and x £ R d tn(x;0 = C We also define t (x) n Definition 1.4.2 (x) ^ C H - E W(T). TeTn(x) we define, (1.23) = t (x; 1). n (Fourier Transform). Given an absolutely summable function f : l) —> K, we /ei /(A;) = 2~^ e ' f(x) lfc x x (k € [—7r, 7T]'J denote the Fourier transform offIn [19] the authors show that if a recursion relation of the form n+l /n+i(fc;2) = E 9m(k;z)f i- (k;z) n+ + m e i(k;z) n+ (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 z of z such that f (k,z ) (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 c n 13 c case is that which is proved in [19]. In Section 3.2 we show that t (k;() obeys the n recursion relation n+l t i(k;() = Y n+ ^m-i{k-,OCPcD(k)t i- {k;C) n+ m + T? i{k; (), (1.25) n+ m=l where Tr (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 n using information about p(x) and t[(k;C,) for I < m. The quantities 7? _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 _ 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 d = 8 appears in this analysis as the dimension above which the square diagram m m m m c W(o) P (i-26) = Y, p(*)p(y - *)P(Z - y)p(z) x,y,z converges. Ultimately we verify assumptions E and G for our lattice trees model with p p 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 € T (x). Those n trees are already critically weighted by p (a weight present on every bond in the c 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 t n 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 ( = 1. c 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 ( ? ) ( ^ ) f c v W )• + 0 + 0 with the error estimate uniform in jfc € E : k < ° g ( L J ) | d 2 cl wt ? 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 = va 2 14 in (1.14). < Figure 1.4: The unique shape a(r) for r = 2,3 and the 3 shapes for r = 4. D e f i n i t i o n 1.4.4 (r-point f u n c t i o n ) . For r> 3, he N ^ " ) and x e R ^ " ) we define W(T). (1.28) 1 rer f i - 1 (x) 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 denote the set of r-shapes. We make the edges in a 6 S directed by directing them away from the root. — r r 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 and k 6 R ( ) we define « ( a ) € R ( ) 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 r - 1 d 2 r _ 3 d r r-l (1.29) 3=1 .(2r-3) Next, given a and s € M;4- ,(r-i) we define c;(a) € R;4by (1.30) 15 Finally we define = {3 :Z{a) = i}. (1.31) This is an r - 2-dimensional subset of M + ~ ^ . For r = 3we simply have r R (a) 3 = {(s,ti-s t -a):se[0,ti/\t ]}. i l 2 (1.32) 2 It is known [1] that for r > 2, 0 < <i < ti • • • < t _ i and (j>k(x) = e ' , lk x r 2r-3 r-l E N o e ds. 2d (1.33) 0 aez i( ) jR 3=1 1=1 a r For v = 3 this reduces to * l A ' 2 (ki+k ) s 2 2 fc?(t!-.) fc|(t -a) 2 2d (1.34) C?5. 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 ) ^ ) , r > 3, R > 0, and \\k\loo < R, - 1 - L"tj I ^va n 2 = n 2r-c r-2yr-2^2r-3 e-^^r-'ds-r-O ( 4 n 5 (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~ A~ and C = 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 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 are considered separately and are shown to contribute only to the error term of (1.35). X 2 3 r r 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, S A W 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] " perform the lace expansion in a general setting. Such products will appear in formulas for the r-point functions in Chapters 3 and 4. a n a 2.1 G r a p h s a n d Laces Given a shape a € S , and n G N * " 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. 2 - 3 r 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 i x 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 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). r 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 ) = 6. 1 Examples illustrating some of the following definitions appear in Figures 2.12.2. D e f i n i t i o n 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. graph containing no bonds will be denoted by 0. The 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 = »- - the set of graphs on M containing no bonds in Tl. e 18 Figure 2.2: A graph T £ Q{Af) that contains a bond i n Tl. The bond i n 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 tgr[s,*] = s AA (i.e. if every edge of M is covered by some st £ T). Let Q n M set of connected graphs on Ai, and Q ^ 5. A connected graph F £ Q n M , c o n = QJ^ I" 1 denote the GJA'• 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 A (T) be the largest connected V 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 A {$) — v. v 8. Let £tf be the set of graphs T £ Qjj 1 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 let <S (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 . B y definition of our networks Af(a,n), with n £ N , for any T £ Qj/ \ 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 <S (n) of degree 1 may be identified with the interval A A 2 7 - - 3 1 1 [0, n], and we can write S[0, n] for S (n). l Similarly a star-shaped network S (n\, n ) 2 2 19 Figure 2.3: Two graphs on each of S (8) and <S (4,4,7). The first graph for each star is connected. The second is disconnected. The connected graph on <S (4,4, 7) is a lace while the connected graph on S (8) is not a lace. 1 3 3 l of degree 2 may be identified with the interval [—ri2, ni] and we can write <S[—TJ.2, ni]' for (S (ni, 712). Our main interest will be connected graphs on star-shaped networks. Figure 2.3 shows graphs on each of <S (8) and <S (4,4,7). The first graph in each case is connected, while the second is disconnected. 2 1 3 D e f i n i t i o n 2.1.2. Fix a connected subnetwork A4 C j\f'. Let V € Q ^' 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 A4 (i.e. ti is a vertex of branch A4 and ti ^ v). By definition of connected graph, Tg will be nonempty. From T we select the set Yg for which the network distance from ti to v is maximal. We choose the bond associated to branch A4 at v as follows: ,con M e e v ,max e e 1. If there exists a unique element ofFe' max is maximal, then this Sjtj whose network distance from Sj to v is the bond associated to branch M. at v. e 2. If not then the bond associated to branch M at v is chosen (from the elements e Y<v,max f w lose 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. D e f i n i t i o n 2.1.3 ( L a c e ) . A lace on a star shape S = <S (n), with n G N {1,3} is a connected graph L € such that: A A ,A G • If st € L covers a branch point v of S then st is the bond in L associated to some branch S at v. e • 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. D e f i n i t i o n 2.1.4 ( L a c e c o n s t r u c t i o n ) . 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 S e at b, and let b be the other e endvertex of S . e • Suppose we have chosen {s\tl,... Then we define t\ +l , s f i f } and that u ' [ s f i f ] does not cover b . =1 = max{£ G S : 3 s € S , s e e tf such that st G F}, e \ sf +1 = min{s G S : stf e ) G F}, +1 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 b is covered by Li\ [s^tf], e =l and set Lr(e) = {sf «?,...,*?<?}. Next we define (2.2) L r = UeeFLr(e), and given a lace L € C(S) we define C(L) = {steE \L:L s LlJst = 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. 21 P r o p o s i t i o n 2.1.5. Given a star shaped network S = S (n), A connected graph F G Q (S), con A G {1,3}, and a the graph Lr is a lace on S. Proof. B y 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 S , t G S i (where e' = e if 5 = b or t = b). Then st was chosen as the e e bond in F associated to S or S >, so in particular it is the bond in Lr associated to e e S or S i. Now if st G Lr does not cover b then s and t are on the same branch S e e e 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 andF\L Then L con r • = L if and only if L CF is a lace CC(L). Proof. If L r = L, then L is a lace by Proposition 2.1.5. B y 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 S e at b, and s't' is therefore also the bond in L U st associated to S at b. Similarly if st G F \ L does not cover b then there are bonds in e 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 <S. Therefore for any lace V C F, st is e the bond in L' U st associated to <S so st is not compatible with any lace L' C F. e 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 S is in L. Since L is a lace, these are the only bonds in L which cover e b and they are also in L r by definition. Therefore the st G L D (F \ Lr) must satisfy s,t G <S, ,t s e ^ 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^ ^ ^ y The intermediate figures show each of the Lr{e) for e G F},, while the last figure shows the lace LYn n n3 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 D e f i n i t i o n 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 Definitions 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 minimally connected graph F G Q (S) is a lace. con 23 Figure 2.6: Basic examples of a cyclic and an acyclic lace. (b) For any non-minimal lace L G C{S ), 3 the branch point) such that L\st there exists a bond st G L (that covers G £(S) and L\st is minimal. Proof. For (a), let Y G G {S), and let b be as in Definition 2.1.4. Let st G T cover b and suppose s G <S i and t G S where S are branches of <S (we may have e\ = e2). If st is not the bond in Y associated to <S then T\st covers S Therefore if st is not the bond associated to either iS or S then Y \ st covers <S so that Y is not minimal. B y Definition 2.1.3 this is enough to prove (a). con e e2 ei ei ei er e2 For (b), let L G C(S ) be non-minimal. Then there exists st G L such that L\st is connected. B y 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 «S 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 S cannot be a connected graph o n . 5 . • 3 3 3 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 \ Sjij G £(S). D e f i n i t i o n 2.1.9 ( C y c l i c ) . A lace on a star shaped network <S is cyclic if the 3 edges covering the branch point can be ordered as {sktk : fc = 1,..., 3}, with t k and s +i on the same branch for each k (with S4 identified with S\). A lace that is not k cyclic is called acyclic. See Figure 2.6 for an example of this classification. 24 2.2 The Expansion 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 n[i+^j-f n ii[i+ust}= [i+ust])(i-n^+^V( - ) 2 4 Define K(M) = Yl t£E \R.[l + U t]- Expanding such a product we obtain, for each possible subset of \ a product of U t for st in that subset. The subsets of E x \ 1Z are precisely the graphs on AA which contain no elements of 7Z, hence s M s s K(M)= H 't> Y (-) U recx* 2 5 s t e T where the empty product Yl t€$ U t = 1 by convention. Similarly we define s s AM)= n *t Y (-) u TeQj} ' n con s t 2 e ^ If M is a single vertex then J{M) = 1. If S is a star-shaped network of degree 1 or 3 then E n^= E LeC(S) V^g : c on s st€T Ly — £ = E Le£(5)s*6L E E LeC{S)steL VeQ% : on -^r nu n E = ^ OO « r'CC(L)s't'6r' E N=l LeC (S) steL N s't'er\L n s't'eC(L) (2.7) where the second to last equality holds since for fixed L, {r G Qg : 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[ 't'eC(L)[H~Us't'] we obtain for each possible subset of C(L), a product of U t for st in that subset. in s s Recursion type expression for K(Af) Recall that Af = Af{a, n) where a G E and n G N , 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. 2 7 - - 3 r 25 Definition 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. be connected. However if M c 5^- then N\A4 (at most 1 ifr — 2) and we write (Af\M)i, need not has at most 3 connected components 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 U «+ E u veg^\£ ^^er resetter b 3 E n^n E reefer ACS^-. be A n ^ + E I I « E i=i . r e g -n^ (2.8) St) i sH reefrster eri 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^ F L ^ e r , ^s»t» = 1- Defining (AO = E r e ^ Uster U,u we have E K(M) 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 ( m ) with the interval [0, m] and (2.8)-(2.9) reduce to 1 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 mi<ni E J(S (m))]lK((Af\S (m))i) A m <n -2l2 2 A i 2 "13 < 713 - 2/3 26 = 1 + E^(Af), (2.11) where <S (m) is a star-shaped network satisfying A ( {b} ) 5 (m) <S[0, mj] 1 3 , if rh = 0 , if mi ^ 0 for all i , 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 b that is adjacent to b in A/" (so that r i 2 or 7 1 3 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 • 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 Organisation 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 g (k;z)f i- (k;z) m n+ m + e (k;z), n+l with t - ) 3 m = i /o(M = l, h(k;z)=zD(k), 1 ei(*;z)=0. Secondly we must verify the hypotheses that certain bounds on the quantities for 1 < m < n appearing in (3.1) imply further bounds on the quantities g ,e , m m f m 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 Proposition 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, ..., R . N 3.2 R e c u r s i o n relation for the 2-point 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 t {x) = C n £ W(T). TeTn{x) (3.2) Every tree T € T (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 n U = U{R ,R ) st s t = {-]; I 0, ******** (3.3) otherwise. Then r i o < 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, t (x-o n = c E w: 0 • ! (3.4) £ W(R ) £ tfoer o) Ri€T WiRt)--- 0 w( uW YI Rn£T , u n) 29 II i^ + Ustl 0<s<t<n 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 ^)f[ E w ' W W i=0Ri€T (i) w:0->x, II V + Ust]. (3-5) 0<s<t<n u \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 i n £ 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 i + *t] = K{M) = K([0,n]). Hence 1 u n tn{xyO = C Y (")H E W W(Ri)K([0,n]). (3.6) \u\ = n D e f i n i t i o n 3.2.1. For m > 0 we define m * {x;Q =C m ^MjJ Y Y w : 0 ->• x \us\ = m Note that form = 0 this is simply J2 - i Ro€7 0 = 0 W(Ri)J([0,m\). (3.7) i£7L,(i) fi W(Ri) = p(0) ifx = 0 and zero otherwise. D e f i n i t i o n 3.2.2. Let f,g. We define the convolution of absolutely summable functions 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, ,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. y The following recursion relation is the starting point for obtaining a relation of the form (3.1). P r o p o s i t i o n 3.2.3. n t i(x;Q n+ = ^ ( 7 T * C p D * i _ ) ( a ; ; C ) + 7 r i ( x ; C ) + p ( 0 ) ( C P c D * i ) ( a ; ; C ) . (3.9) m c n m n + 30 n Proof. B y definition n+l t (x;0 =C n+l E + 1 E (")H W \u\ = n + l W(Ri)K([0,n + l]). (3.10) . 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 E N + 1 E ^ M l l \u\=n ^ ( ^ [ l . n + l] Bi£T(u(i)) «=0 w:0->x, + l = E w(Ro) E E W w i ) * (3.12) |«i| = i n+l E E CWi^H w :y->x, W(Ri)k[l,n + l), i=l RiGT(w (i-l)) 2 2 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 ye^D CW(OJX)X wi:0-*y, M E =i E C ^(^)fl n |W2| = n • = P(0) E Pc(D(y)t (x-y;0 n y€&D = p{0)p ((D*t )(x). c n 31 W{Xj)K[0,n] 2. The contribution from graphs which are connected on [0,n + 1]: n+l C Y + l W(u,)j[ Y W(R )J([0,n i + l])=-K (x;O (3.14) n+1 |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" E + 1 ^ M l l co-.O^x, |w|=n + l ' n = EEE m=l « v E W(Ri)J2J[0,m]K[m i=0 RieT m=l uii) E c o-»u, / m r ^ i ) nE V^O^ieTL^i) i : + l,n + l] \ J M X / |cui j = m E C^(^2)X |w | = l . 2 £ C N _ M W > 3 ) ( fi E ^)W[ra + 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 i this is equal to: +m+ EEE t = m l u E CWCi) Ml wi:0->u, v V E = 0 / i i G r W{R4)]J[0,m]x "l(i) / / n—m p C.D(<; - u) E c 1^31 C"" W(w ) m 3 m=l n u 7rm II E ^ W K ^ n ~ m l =ra— m n = E EE \ - u)t - {x ( OVcC,D[v u; n v m -v\Q = 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 ^m(fc;C) „ , P(0) ^(or m A p(0)c „ f^n^n-mjk-X) ,^Wl^C) , , n x rvi/'"^ . 0 ^ " ~7(oT~ ~7^ ( ) + (3.17) We now massage (3.17) into the form (3.1) required for the analysis of A p pendix A . D e f i n i t i o n 3.2.4. Forfixed( > 0, define 1) z = p(0)(p . %) fo(k",z) = 1, fi(k;z) c 3) = g\{k\z) = zD(k), and ei{k;z) — 0. For n>2, fn(k;z) = e (k;z) = n p(0) ' g _i(k;z) n p(0) h(k;Q P(0) zD(k) + (3.18) ^n(fc;C) p(0) ' We note from (3.17) with n = 0 that since £()(£) = p(0)/ o,we have to(k) = p(0) x= and p(0) P(0) 33 (3.19) Therefore for n > 2 tu \ „ (k;z)-^ tu e (k;z)=g n n \?[!(*iO + , nn{k;() - . 1 (3.20) M For n > 3 this is e (k;z) = ^- ^°zD(k)^P- + 2 (o) • n ' v P p(0) Lemma 3.2.5. T/ie choices of f , g , e m m M k ' X (3.21) ) p(0) above satisfy Equation (3.1). m 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)f i- {k;z) n+ +e i{k;z) m n+ m=l = gi(k;z)fi(k;z) + g {k;z)f {k;z) 2 0 = zD(k)zD(k) + ^PzD(k) p{0) p(0) 2 + zD{k) ni - + e (fc;z) ^(*> + *^*> o) *i(fc;C) p(0) 3l>(fc) + P(0) (3.22) + p( P (0) * (fc;C) (0) ' = 2 P by (3.17) for n = 1. For n > 2, 71+1 E 9m(fc;2)/n+l-m(fc;2) + e i ( f c ; z ) n+ = gi(k;z)f (k-z) + g {k;z)fi(k;z) n n + g +i{k; z)f {k; z) + n 0 e i(k;z)+ n+ n-l E ^)/i+i-m(fc; m=2 = *D(k)^£PP(0) 7r _i(A;;C) n p(0) zD(k) n - l 7rm-^i(A;; C) E m=2 + ^^-zD(k)zD(k) P(0) P(0) *i(fc;C) + zD(fc) P(0) t _ (k; C) n+1 m P(0) 34 P(0) ^n+i(fc;C) P(0) (3.23) ^P-zD(k) + 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 remaining terms on the right side of (3.17), hence by (3.17) the entire quantity is equal to = f (k;z) n+1 3.3 as required. ' • Assumptions of the induction method 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 f ,9m,e as given in Definition 3.2.4, m m n+l fn+i(k;z) = Y 9m(k;z)f - (k;z) n+1 +e i(k;zy m n+ (n > 0), (3.25) m=l with fo{k;z) = 1. Assumption S. For every n € N and z > 0, the mapping k i-> f (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 e (-;z) and g (-;z). In addition, for each n, \f (k; z)\ is bounded uniformly in k e [—TT, ir] and z in a neighbourhood of 1 (which may depend on n). n n n d 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\ D(x) 2+2e 35 < oo. (3.27) (ii) There is a constant C such that, for all L > 1, Halloo < CL~\ a = a <CL , 2 2 (3.28) 2 L (iii) There exist constants 77,ci,C2 > 0 such that < a(k) < c L k Lk 2 2 2 Cl (\\k\lao 2 2 a(fc)>»/ (Hfclloo > a(k)<2-n < L" ), (3-29) 1 L' ), (3.30) 1 {ke[-TT,n] ). (3.31) d For h : [—n, ir] —> C, we define d (3.32) The relevant bounds on / , which a priori may or may not be satisfied, are that m <-^r, \\D f (-;z)\\ 2 m 2 La \f (0;z)\<K, \V f (0; 2 m m z)\ < Ka m, 2 (3.33) m4 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-> C (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] and 2 < m < n +1, the following bounds hold: e d |e (*;z)| < CeiK)?™,-^, \e (k; z) - e (0;z)| < C e ^ o W K ^ . m m m (3-35) Assumption G. There is an Lo, an interval J C [1 — a, 1 + a] with a € (0,1), and a function K C (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] and 2 < m < n +1, the following bounds hold: g d \9m(k;z)\ < C (K)Bm- -^, d g |V 2 ( 0 ; ^ ) | < C (K)a 0m^, 2 5m g \d g (0;z)\<C (K)8m- -^, (3.37) d z m g < C {K)8a{k) *'m'^', \g (k;z)-g (0;z)-a(k)a- V g (0;z)\ 2 m m 2 1+ g m with the last bound valid for any e' G [0,1 A ( ^ ) ) . 36 (3.36) (3.38) 3.4 Verifying assumptions Assumption S: The quantities f (k; z), n = 0,1,... are (up to constants), Fourier transforms of t (x,Q, which are symmetric by symmetry of D. Hence the / „ have all required symmetries. Similarly 7r (x,£) are symmetric by symmetry of D, so that the quantities g ,e 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 + U ] < 1 in (3.5) we obtain E x 0 < ((p ) p(0) E D^(x) = (CPc) p(fj) , where denotes the n-fold convolution of D(»). Therefore for n>l, \ f (k,z)\ < ^°ffi < (CPcp(0)) = z so that / „ is bounded uniformly in k G [—7r, ir] and z in a neighbourhood of 1 and therefore satisfies the weak bound of Assumption S. n n m n n n st n n+1 x c n+1 n C) n n d Assumption D: By Definition 3.2.4 we have fi(k,z) = zD(k) and e\ = 0. Additionally, 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, g and e could be expressed in terms of the quantities 7 r for m < n. In Chapter 5 we will prove the following proposition. n n m Proposition 3.4.1 (7r 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}, (3.39) where ( = v > p ^ p , the constant C = C(K,d) does not depend on L, m and z, and 0 is the constant appearing in Theorem 1.2.9. We choose v < 1 in (1.13) so that 2 - ^ > 1 and therefore / 3 ^ < L~i. The proof of Proposition 3.4.1 involves reformulating ir 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. 2 - m 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\(k\ z) = 0 and observe from (3.20) 37 that \e2(k;z)\ = P(0) P(0) < + P(0) 7fi(fc;0 P(0) <z < P(0)2^ vr (fc;C) + 2 P(0) (3.40) 2— w h e r e w e h a v e a p p l i e d P r o p o s i t i o n 3.4.1 w i t h |7r (A;;()l < E x l^mO^OI) m a n d have a l s o u s e d p(0) > 1. S i m i l a r l y f o r 3 < m < n + 1, 7ri(fc;C) , 7Tm(fc;C) P(0) P(0) pv r P Wi \ m(k', z)\ — e u < (0) (m-2) f 2 / 9 d 2 + + •zC(K)B'-T p{0)m i d-4 m 2 2 T h u s w e h a v e o b t a i n e d the first b o u n d of A s s u m p t i o n E . I tfollows i m m e d i a t e l y t h a t C'{K)d -^ 2 \e {k;z) - e ( 0 ; z ) | < (\e {k;z)\ + \e (0;z)\) < m m f o r a l l m > 2. m m (3.42) 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 T h u s i t r e m a i n s t oe 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 for Halloo > L~ . l Halloo < m 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 : Z —> 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 d and under permutations o fcoordinates. h(k) - h{0) < coordinate Now ^ V ^ ( O ) 2d + h(k)-h(0)-^ h(0) 2d 2 (3.43) Y,[ccB(k-x)-l-^Y x )h{x) 2 i + i 2d V /i(0) 2 2d B y s y m m e t r y w ehave that 1 - £ which implies \x\ h(x) = ^j-V /i(0) = 2 — 1 2 Yl £ x h x = (3.44) E ) ( ), X H X 2 )H ) X < £ i ( ) i—\ x 2^?i=i(kjXj) h(x). O n t h e o t h e r h a n d i f i ^ j t h e n x . E s X{Xjh(x) = 0, s o t h a t E x ( ^ ' c a n r e w r i t e (3.43) a s h(k) - h(0) -£ 2 X a l s o equals ^cos(fc • x) - 1 + i(fc • x) ^ 2 38 Y, Ef=i(^* *) M )- T h u s w e a ; x h(x) + 2 V h(0) 2 2d : c (3.45) We claim that there exists a constant c, such that for all 77 € [0,1], | cos(i) 1 + \t \ < c i . To see this note that f o r > 1 the left hand side is bounded above by 2 + \t < \t < §rj ?. For \t\ < 1 the left hand side is bounded above by 2 2 + 2 j ? 2 2 2+2j 2(n-l-n) 2n 0 0 t t k ^ k (3.46) < "» 2 k ! ^ where the constant is independent of 77. This verifies the claim. Putting this result into (3.45) we get Jfcp h(k)-h(0) <C^2\(k-x) h(x)\ + 2+2n (3.47) V h{0) 2 2d In particular if we choose 77 = 0 then (3.47) becomes d h(k)-h(o) <C ^ ^ ( ^ ) 2 I M X ) I + %v m 2 (3.48) x j=l <C\k\ Y,x)\Hx)\. 2 Now e (fc; z) — e (0; z) is equal to m m (g -i{k\z) -g -i{0;z)) n ^ p(0) n +g -i(0\z) n + p(0) P«>) (3.49) By (3.47) with 77 = 0, and Proposition 3.4.1 with q = 1 we have that , o - |5fm(fc;C) -S?m(0;C)l < C(i^)fc 2 2 / 3 2 - ^ (3.50) V ,• m 2 Therefore |e (fc;z) — e (0;z)| is bounded above by m m o 8 -f 2 \g -i(k;z) -g -i(0;z m m C{K)B -^ + |^ _i(0;z)|C(if)fc m p(0) p(0) 2 p(0)m 2 2 2 |gm-i (fc; z) - g -i (0; z) I + \g -\(0; z) |fc cr + 2 p(Q) : fc a 2 < + C(.rY)fc / m 2 m ra d-6 2 (3.51) Thus recalling that <7i(fc;z) = zD(k) we have |e (fc;z) - e ( 0 ; z ) | < 2 2 92 C(K)B*-f , - fcV za(fc) + zfc cr +. _ 2~ 2 d P(0) 39 6 (3.52) For m > 3, recall that g -i{k\z) = ^ ~ ) zD(k) m m \g -\{k;z) - o _i(0;2;)| < m m |S?m-2(*; p(0) < 0 C(K)k a B -ir 2 C)|JD(0) + a(fc)|7r _ (0; C)| m 2 C(K)a{k)p -- 2 . which gives fc;C) - 7Tm-2(0; L 2 2 ( ( 0 2 d + d-6 (m-2)T~ ' (3.53) (m-2)~ Therefore for m > 3, |em(A:;z)-e (0;z)| < a{k)0 C{K)R ~ + (m - 2 m , • d-6 2 2 2 +, d-4 1 2 \(m-2) k2„2a zk o (3 -^ . d-4 1 2) 2 2 1 (m — 2) 2 Both (3.52) for m = 2 and (3.54) for m > 3 are bounded above by ' " ] ? C (K ( m 2 PHoo < £ - 1 fc 6 d-6 m2 (3.54) ^ for by (3.29) and the^act that a ~ L (see Remark 1.2.5). 2 2 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)p -ir • 2 < C'(K)/3 -d 2 < p(0)(m - 1) V (3.55) d-4 m 2 which gives the first bound of Assumption G . For the second bound we note that by symmetry the first derivatives of n m and D vanish at 0. Hence for m > 2 \V g (0;z)\ 2 m = zD(k)- , z P(0) fc=o (c{K)B -^a 2 p(o) m 2 m , C{K)P - -it 2 m A 2 & d-6 \ |V 7r _i(0) + 7r _x(0)V D(0)| 2 P(0) C'{K)B ^o 2 d-6 d-4 m 2 2 m 2 2 (3.56) This verifies the second bound of Assumption G . Next for m > 2, we have that g (k;z) m =7f -i(fc;C) ffl p(0) Z l m = ^ " V m £ _ i ( f c ; C A 0(fc) z P(0) m m _ 1 (3.57) where ^ " ^ - T ^ does not depend on z (or (). Therefore \d g (k;z)\ z m Hi:: .! / 5 r _ i ( f c ; C ) \ 5(fc) m yim—l D(k) P(0) 40 P(0) < (3.58) d-6 m 2 which proves the third part of assumption G . Now for ||fe||oo > L -, (3.30) applies and we have that for m > 2, 1 \g (k; z) - g (0; z) - a(k)a~ V g {0; z)\ 2 m 2 m m • *=* m 2 S + 61/ d-6 < a(fc) i m + a { K *=* m2 ) 4 (3.59) ' 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 < £ > we again use the method of [21]. B y the triangle inequality we bound \g {k;z) - g {0;z) - a(k)a' V g {0;z)\ by - 1 2 m 2 m m \k\ g (k;z) -g {0;z) - -^rrV 2d g (0; z) 2 + 2 m m m (l-D{k))o- -^ |V m(0;«)|. 2 2 2d 5 (3.60) Recall that for m > 2, g (k; z) = ^ y C ^ m * D)(k). O n the first term we apply the analysis of the first term of (3.43), to the symmetric function ir * D. Choosing 77 = e' we see that the first term of (3.60) is bounded by m m zO\k\ <'Y\ \ '\(*rn-l*D)(x)l 2+2 X 2+2e (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\ ^'\(n ^*D)(x)\ ¥, 2 1 m I±^+I^= g(x) = \(7r ^D)(x)\ ^, 1 m 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 m2 We now apply Proposition 3.4.1 with q = 0,2 together with the inequality (a + b) < 8 ( a + b ) (obtained by squaring the inequality (a + b) < 2(a + b ) and 4 4 4 2 41 2 2 applying the same inequality again) to get E N l ( ^ - i . * )(*)\ <s(EM K-i(y)lE ^ - v)+ 4 D 4 * \ p y x Ekm-i(y)iEk-yl ^-y)) a; / < C ( £ lvlVm-lfo)l + E l^-l(2/)k ) 4 2/ 4 <o - 4 o C(K)B ~^ < a C(K)B -^ A C(K)^-1f d-8 4 l "F d-8 m d^4 2 2 (3.65) m 2 m2 Note that we have used Remark 1.2.5 to obtain 2~2 \x\ D(x) < Ca with the constant independent of L (it may depend on r). Putting (3.64) and (3.65) back into (3.62) we get r r X Eki 'i^-iwi<f 2+2£ / ^ntis\ni- o^C(K)B —d l d-4 2(l ,') (K \ m 2 a < 2 cw2_6; + C m ) / ? (3.66) d-8 2-f m 2 2 Combining (3.66) with (3.61) gives \k\ 2 C(K)0 -ir(a \k\2)| i . | 2 U + 2 g (k; z) - g {0; z) - ^ " V 5 ( 0 ; z) < 2 m m 2 m r e ' , m 2 (3.67) C(K)3 -^a(k) ' 2 < 2 d-6 i+£ d-6 ; m 2 when ||fc|| < L . 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 - 1 2d a (3.68) 2 x and proceed as for the first term to obtain l-D(k) a* _ \kf ~~2d < c\k\ ' 2+2£ E \x\ '\D(x)\ 2+2e 42 < c\k\ 'L ^ '\ 2+2e 2 l+e (3.69) Together with Proposition 3.4.1 with q = 1 this gives C{K)d -^a 2 (1-D(k))a- 2 2d |V <U0;z)| < (\k\ L ) ' 2 2 2 1+e (3.70) 2 d-6 m 2 G for which satisfies the required final bound of Assumption < 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 Proposition 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 - 7 ) . There exists and S G (0, (1A a positive LQ = Lo(d) such that: For every L > Lo there exist A',v,z depending on d and L such that the c following statements hold: (a) T ( k Zc 1 Al P(0) = Ae - — 2d k' 1+ 0 with the error estimate uniform in {k G K n" D 1 O (3.71) d-8 n 2 : 1 — D(k/Vva n) 2 < 1 logn}. (b) l +O va n 2 r d <r (3.72) (c) for every p > 1, < C d d . d-i L?n p 2 (d) (3.73) 2 The constants z , A' and v obey c 1 = £ 9m(0;z ), m=l 1 + Em=iem(0;z ) A' = Em=l ff™(°^c) c c m E^ V (0;z ) v — — °~ E m = l ? m ( ° ; ^ ) ' 2 = 1 2 5 m m 43 c (3.74) The constants A',v satisfy A' = 1 + O [L 2 j , = 1 + O [L 2 J . Also z v c = 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 ^ * dear s 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 = ~i' Ae + 0 (£) + ( ^ ) C ( _ J ^ ) , +0 ,3,5, with the error estimates uniform in {k G R : \k\ < Cblog(|_nrJ V I ) } . d Lemma 3.5.2. The critical value C, = fy c p Pc 2 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^ j (y^hf^! Ccj is equal to ni r-2T-n' \ = Cc 1 y/va [nt\ z Ae 2dn « ' J \[nt\-^ (3.76) where the error estimate is uniform in k e M :1- D ( d V j L ^ j<7 H " 1 logLntJ }> - (3.77) We claim that there exists a constant CQ such that {k : \k\ < Co log([nrJ)} C H 2 44 n j t . Define G Mk:\\kU<^K I n 3. ) ( 78 By (3.29) and using the fact that a ~ L, there exists C\ > 0 such that for k € G n > t, , Now since WkW ^ < \k\ Co < ^ such that 2 f | - ^ ] < ^ . I - [nt\ (3.79) v^pj 1l v and using the fact that a ~ L, there exists a constant 2 {A;:|A;| <Colog(LnrJ)}cG , . (3.80) 2 n t Then for k < C l o g ( [ n i J ) we have 2 0 1 D I k ^ 2 \ < " < Clk g y/va [nt\ I ~ [nt\ ~ < 7 i o g l Q g(NJ) [nt\ (3.81) log(Ln*J) [nt\ Thus verifies the claim, and thus (3.76) holds with the error estimate is uniform in {k : \k\ < Co log([nrJ)}. Since \nt\ < nt in the first error term of (3.76), and 2 fc [nt] kt 2 e 2dn 2 — e <i 2 we have proved Lemma 3.5.1. • The significance of Lemma 3.5.2 was discussed immediately before the statement 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 p ) and this idea is the basis of the following proof. c Proof of Lemma 3.5.2. The susceptibility, x( ) z = J2 x(z) = YU(0;z) n where £ = HK ' x *n(Q;C) P(0) TeT (0,x) N z P(0)Pc ' 45 is defined as (3.83) Let z' denote the radius of convergence of x( )exists a z > 0 (resp. ( ) such that B y Theorem 3.4.3 there z c c c c^E x wm^A, E (3.84) TeT (x) n so that (3.85) v x 7 Ter„(i) Thus the radius of convergence of x{0 (resp. x( )) Write J2x p( ) = limM->oo £ | x | < M p( ) It follows from Theorem 1.2.9 that J2 P( ) x M. a is Cc > 0 (resp. z ). z x a n c observe that £ = °°- d X T X h u N < (| |vi)«i-» M B s (3.86) K n W x T6T„(0,x) r ' v which implies that £ < 1. c Recall from (1.16) that P{T G T (0,x)) = 3.4.3 states that for every fc, ^ Er6Tn ( 0 n p( ) W ( T ) . Then Theorem (3.87) X Setting = 0 we have A £ P ( T G T ( 0 , *))-+!, (3.88) n and dividing (3.87) by (3.88) gives P(TeT (0,x)) n E P(TGT (0, )) u n -» e Let Z„ be Z -valued random variables defined by P{Z Then (3.89) is the statement that d E for every fc, where Z ~ j\f(0,I,i). every R > 0 we have e . , ^J(<>,«)) = x) = E p^ ( r € r „ (^ o,«))p n E ik-Z This is equivalent to G S(0,i?) 1 —> P (Z G 5(0, J2)). 46 (3.89) 2d. W n x)) tt (3.90) Z , and thus for (3.91) where B(0, R) denotes the ball with centre 0 and radius R in (R , | • |). Choose Ro such that P(Z G B(0,R )) > | . Then there exists an i V = N (Ro) such that for every n > No, Z V 1 P l — ^ e B ^ R o ) ) ^ - . (3,92) , V<7 vn 0 0 0 n Therefore for every n> NQ, \x\<RoV(T vn 2 Applying (3.88) to the denominator, we find that there exists Ni > No such that for every n> Ni, ^ E P(TeT (0,x))>-, i.e. n \x\<RoV(r vn YI P(T&T (0,x))>-. n \x\<Ra\/a vn 2 2 (3.94) Bounding £ r T „ ( o , x ) W ( T ) by p(z) = £ e m Erer.fo,,) ^ follows that In c We also have from (1.13) that, £ E \x\<RoV<r vn 2 - ( l / ( v i ) ^ - c ( L ' f i o ) n - ( 3 - 9 6 ) \x\<RoV(r vn 2 Thus from (3.96) and (3.94), £ < Cn for every n > no- This requires that ( > 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. c 47 Chapter 4 The r-point functions We have shown Gaussian behaviour (Theorem 1.4.3) of the 2-point function with appropriate scaling i n 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 i n Chapter 2, and prove the result, assuming certain diagrammatic bounds. The diagrammatic estimates are again postponed until Chapter 6. 4.1 Preliminaries 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 eT , t = l , . . . , r - l } i and $(x)= (4.1) n| J2 ?)W{ (-) 4 2 T€T (x) fi For T G T(x), let T^ X be the backbone in T from 0 to a;. D e f i n i t i o n 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) TeT B 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^ C\B^ — B^ . 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. Xi Xj 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. B y definition, a branch point that is not the origin must have degree > 3. Definition 4.1.3 (Degenerate bare tree). Forfixedr, 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)\B (n,x). D Clearly from (4.3) we have <n(*) = E E ( ) + E E W{T). (4.4) r W T B£B (n,x)T€T B6B (n,x) c D B D T€T B 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" , x G Z ^ " ) . 1 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, x -\ be the unique branch point in B net-neighbouring x -\. Removing the netpath B^ _ \ B^ , we have that x is a vertex of degree 2 in B* = B \ (B^ _ \ B^ ) and therefore B E B ( ( n i , . . . , n _ 2 ) , {xi,...,£7—2)) contains r — 3 branchpoints, each of degree 3, none of which is the origin. Thus B* is nondegenerate. B y 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 - 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. r r Xr Xr 1 1 x x r r 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. B y induction B* G B ( ( n i , . . . ,n - ), (xi,... ,x -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. r 2 r D e f i n i t i o n 4.1.6. For a fixed shape a G S and n G N ^ 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\,J2Let 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' ( ~ \ we define TM\ ,n)(y) be the set d r_1 of lattice trees T G 75 such that there exists x G Z ( ) , i i G and B G B ^ ( i i , x ) such that 2 - 3 r d 2r 3 t a _ 1. Te T B , 2. B has network shape Af(a,n), and 50 1 o / i I I x3 Figure 4.1: A shape a G £ 4 with labelled edges, and a nearest neighbour lattice tree T G T )&) 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 + y = X]_. M{afi 3 4 3. if the endvertices of netpath Bj are Uj,Vj G M. , where B^ d Uj B^ C Vj then Vj — Uj = yj, for each j = 1,..., 2r — 3. Suppose T G Ttf(aji)(y), with corresponding x, ri, B as i n Definition 4.1.6. Since B has shape j\f(a,n), we may label the netpaths {B\,... ,Z?2r-3} of B ac- cording to the edge labels { 1 , . . . , 2r ^ 3} of a . Let Ei = {j : Bj C £ L * } , and note Xi that Ei is equal to the set of edges i n the unique path i n a from the root to leaf i, defined in Section 1.4. B y definition we have Vj 2~2jeEi = » x a n d T^jeEi i n = 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{ ,n){y)forsome a G S , n G P P a , y G Z ( ~ ) satisfying Y,jeE i ~ - 3 d r 2 r 3 n { *> YljeEi Vj = *> * i ' • • • > ~ suppose T G TjV{ ,H)(y)Let rcj be the vertex i n T corresponding to leaf i of a, i = 1,... , r — 1, and let ri; = | T U 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). n x e 1 r 0 n t n e o t h e r h a n d a X For fixed a G S , ri G P f r _ 1 and x G Z ( d 51 r _ 1 ) we write Y]s" - to mean the sum over {n £ N " 27 : YljeEi j 3 sum over {y £ Z ( - " ) : £ V d ~ i> * = 1> • • • j»" - !}> and n 2r 3 £ n t o m e a n t n e . W = i ' * = 1, •••,>" - 1}- Then x e J S £^cn=£££ E ^cn- (-) 4 5 See Figure 4.1 for a concrete example of this idea. D e f i n i t i o n 4.1.7. For fixed r > 2, a £ E , network shape Af — Af{a,n), and r netpath displacements y — ( y i , . . . yir-z) tM{a,n){y)= € Z ( d E 2 r _ 3 ) we define ^( )- (-) 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 , n £ N " , R > 0, and « £ [ - i i , # ] ( - ) , 2 7 - 3 2r 3 d r F — (53=) - ™ ~ n ^ -(E w (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 ^ _£ *k-x E E^cn ^ ,n)(y) + &(k). ( a e We will 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 = £ = I 2 r-l E j=i r-l K J ' i x = 3 a n 2r-3 d in (4.8) we use 2r-3 y r-l E J ' E J"{'eB,-} = E '' E J' {'eB,-} = J k j=i ;=i ;=i 52 k j=i 7 2r-3 £ y r 1=1 K l = n •& (-) 4 9 where KI was defined i n (1.29). Thus the first term on the right of (4.8) is equal to £ E E E & *£*"<°.*)(fl= EEe**w<«,*)0) E E ^ ( M ) ( « ) - E = 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 tf (kiM) E = nun2) xi,x E e^" ^ " 1 2 2 E^(n,n -n,n -n)(y,Xl 1 n=l 2 2 y, X - y) 2 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!An2)-l E E E ' E n=l xi,x 2 ( M X \~ *~ ^^ W ) + M W y (niAn )-l 2 - AA/"(n,ni-n,n -n)(«l)«2,«3)- E 2 n=l (4.12) Recall from (3.4)-(3.5), and the fact that Q = 1 that we were able to express the critical 2-point function as c 71 i (z)= n E w u u : 0 ->• x, i = n [i+^i. E ( )ji 0 Ri£%(i) (- ) 4 13 0<s<t<n = n using the notation T\i=o £ ^ e " 7 L E W(R )--floer^o) 0 The product nt^+^t] W W rio< <t<n f + *t] to represent 1 (i) U s E ^ ( ^ ) «ne7L(„) IT + (4.14) o<«<*<n 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 i n 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 Z if d CJ is a map from the vertex set of Af into Z that maps the root to 0 and adjacent r f vertices in Af to D(-) neighbours in Z . Let 0*Af(y) be the set of embeddings LV of Af d into Z d 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 {R } j^f, s s€ define as in (3.3), U st = U(R ,R ) s = { t Q j o t h e r w i g (4.15) e 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 E a ( } E s shorthand notation for E •" K o £ 7 ^ ( ) ^(1,1)67^,(1,1) J?(l,2)G7L(i,2) 0 E •R( 2 r _3, n 2 r • _ )67L(2r-3,n 3 2 r «•«) _3) Then, tN(a,n)(y)= E ^)II venial) E W(R ) ]J[1 serf R €T ( beE^S S U + U,}, (4.17) S) 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 R are sums over all branches at vertices s of the embedding CJ, s 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), {R } ecj) s such that the R s tree T £ Tjv'ta.n) 4.2 s are all mutually avoiding lattice trees, uniquely defines a lattice a n d vice versa. A p p l i c a t i o n of the Lace E x p a n s i o n We now apply the expansion described in Section 2.2. Let ifim = E w IT w E M (IT i + ) ( W 1 1 ben / (4.18) 54 Then by expressions (2.4) and (4.17) we can write tM{a,n){y)= £ where K(JV) = ILeT^ \1 ^ Another such error term comes from e Mv)= W{R )K{N)-4%{y), £ ^ w s e e W{UJ)1\ £ shortly that <^(K) is an error term. £ £ renter seA/"fl er(w(s)) ue&M(y) (4.19) S s (4.20) 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 n = (ni,n2,nz) be the vector of branch lengths for branches incident to b and b 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 Un {X) C Z\ and Un {N) B 7b' n = {m:0<m,i<Y, nb * = 1,2,3} n {nl : mi < n - 2, i G G} { %n = ({rh : 0 < m < n b { C l\ by B ' i = 1,2,3} D {m : u < n; - 2, i G G}) \ U^. Note from (2.11) that U Hfi = {m : mi < n i , m < n - 2/2, ra < ^ 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 b tM(y)= £ YI W(OJ)H wen^(jT) 2 W 3 2 3 ^ ^ (m))n^((A '\ S (m)) ) A £ se/\f R eT(w(s)) / i 1=1 men^ 3 A l + rAy) + <f>Ar(y)-4>}v(y}, (4.22) where 3 ^(y)= £ uen {y) M £ ^(^) £ S<EAT R eT{u(s)) J(<S (m))n^((^\5 (m)) ). A i meHn s A i = l b (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 n A n = 1). 2 For r = 3 there is only one branch point, 6, hence <f>tf(y) = Lemma 4.2.1 states that in fact for large Ti—oo 3 4>j\f(y) = 0- = infi<j< r—3 Uj, all the terms 0^-, 2 <jfyf and <f>Jf are error terms, so the main term i n (4.22) is QhT{afi){y) = tAT(a,n)(y) ~ ^//(v) ~ <l>j\f(y) 55 + ^(j7)> ( - ) 4 2 4 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 (fP . The second graph does not contain such an edge but branch 2 is covered so this graph contributes to 4> . In the third graph, branches 2 and 3 are not covered, but r i 2 - 2 > 772,2 = 2 > ^ = | and this graph contributes to 1 F ,7T which is the first term on the right of (4.22). Taking Fourier transforms of (4.22) or (4.24) we obtain (4.25) Lemma 4.2.1. The error terms defined in (4-18)-(4-23) satisfy (4.26) 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. 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 . and rj < r. This permits analysis by induction on r. r j We first define the quantity ^^{u) and then the constant V appearing i n Theorem 1.4.5 in terms of this function. We then state some bounds on the function 7 r ( « ) 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 i n (4.43). These error terms are bounded in Section 4.4. M 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 Z . We define 3d = E M Note that if Mj = 0 then E II Wi^JiS^). (4.27) M is empty unless Uj = 0. In particular ifS^ = {0} is figA/^ 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 n E N=l LeC (S )beL N i i + u ^ b'eC(L) A (4.28) = E(- ) I E W N=I so that for Lec (s^)beL N TT^U) = £ £ E E ^ Lec (s )^en N = 1 b'eC(L) ( - 1 ) * n^{u) where E II W{Ri)H(-U>) Y[[1 + U ]. its* Ri€T(u(i)) A A() M IIH*) n V \>&L \>'ec(L) M (4.29) Note that n^lu) > 0 since —U\, > 0. We also define V ^ E E E^^n^^-^^cEE^wMez^ uez i7ez M M M V *=i 57 (-°) 43 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| %£(£) (4.32) <NHN a \\M\U B (M), 2 2 2 q N uez 3d where u = (ui,U2,us), B (M) N and N = (ce -^' 2 n , d-6 .,ifefi E [Mi]T- ^ E + =1 [Mi]— i = r 1 m .< E M . [Mj - mj] d-4 2 [M k + ] 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^ E 3 ^=1 B N ( M ) < ^ ^ , M:Mj>nj j = 1,2,3, and ^ 1 0 - d (4.34) s |n||oo , 2 E ^ E n^iioo^iv(M) < \ M<n ^=1 \ Given M G TLft we define A/^ b log||n||oo, if d^ 10 if d= 10. = (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 lP -^ T d and v G Z 3 d and 5 4 c J V 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)- Lemma 4.3.4. Let y Vi (4-35) 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 Meu Hb E^M^II^E^^-^/V-iVvi)« 1=1 58 Vi (- ) 4 36 Proof. First from (4.22) and (4.24) we have £ QM «,n){y)= { £ Me«5 t ^(^)^ (M))f[mi- E wen^(f) A seAf R eT^^ a i=i s) (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 Wi^WYPcDivi-Ui) E E W(ui). (4.38) Trivially, n E n w(R )= A seArR eT„ s 3 E a seS^RseT (3) n w(R )ii i=i uis) S i e A E r-R S i W eT . 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( ,n){y) equal to 1S a ( \ E E f E n W(R.)J{S%) E x \ M 3 E I]E^^-^) I I W E W(R )K(jVr-) Si (4-40) 3 = E E ^ M ^ I I ^ E " as required. Given /? G [—7r, 7 r ] 2 r 3 0 ^ we let K = (KI, 6 K2, K3), • and K* denote the vector of Ki, i = 1,2,..., 2r - 3 such that 2 is the label (inherited from M) of a branch of Afj Then 3 iR-y e _ gi^-u JJ iitjivj-uj) iK]-g e 59 e Vj _ (4.41) From Lemma 4.3.4 we have 6 rc Finally we write QAA(K) = where the \[Ae-i ^ +4(K) + ^ + £P(K) n 2 (K) + #(*), (4.43) are defined by £gw= E - 1 ) ) ^ E ^(^)n^(^), E C {1,2,3} E^<D 4(«)=P? E ) \leE b ^ X (^(^-^(oOn^^)' 3 Men \i= nb l 2 = 1 ( i=i 3 V -" Y[Ae-i^ V r MeHn . -.J £Y(K) = D 4 4 4 ) J 2 ^M(O)- £ c The first term is obtained by writing D(KJ) = (l + {D(KJ) - 1)^, the second by writing % ( K ) = ( % ( 0 ) + ( ^ ( K ) - ^ ( 0 ) ) ) and so on. 6 4.4 6 Bounds on the £. 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 except £ m d TTJ^{K) such as those appearing in Proposition 4.3.2. Each term will 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 t (Z)<K* . M M 60 (4.45) 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 * where the constant is independent of L. In particular since 7TQ(0) (4.46) = 1, this proves that V = l + 0{3 -^). 2 L e m m a 4.4.2 (£~ b o u n d s ) . For all K, ^ ) ={ = o ( i 2 K l (4-47) i / d = io, od^lViogiHU), #W =O [ E 4 T V Proof of (4-4V- F ° l> £ E we bound Y$=itjVj(Kj) using Lemma 4.4.1 and ( 4 . 4 6 ) . This leaves us with r \£gm<c E (- ) 4 a n d n°M- E C {1,2,3} Y I M ^ M ^ ^ 49 by constants ( - °) 4 5 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). ( -51) 4 In particular since CL(KJ) < 2 this quantity is also bounded by a constant C. If | | £ j | | o o > L~ , then C < C | | K | | L and we have obtained ( 4 . 4 7 ) for ||KJ||OO > L~ . By ( 3 . 2 9 ) we have for Halloo < L that ' ! L 6 2 2 _ L 1 O ( K J ) < CL K (4.52) L~ . • 2 which proves the first bound for ||KJ||OO < 61 L 2 Proof of (4-48)- We bound the of d with ^M(^) — n = b ^M^ )- ^ n • • • , «2,d, K2,i, (KI,I, • • • , b by a constant and apply (3.48) with 3d instead K 3,i; ••• > to bound the difference «3,d) doing so we obtain <C|« I 6 ^M(O)-%(K ) & E 2 «ez \ \ ^ {u)\. 2 Uj (4.53) a 3d This gives us l4(*)l<C£ l^l E Kll%(^)l2 2 (4.54) Applying Proposition 4.3.2 and Lemma 4.3.3 we obtain I n?6|2_2||^>b||C f » I | K " | c r l l n MOO ie Sl(H) | « | ^ | | M | | o o i V ^ ( M ) < cfi—d <C £ 2 2 5 K | a log||n || , 6 M<n 2 2 6 0 0 b l£tv0 ) if d ^ 10 if d = 10, (4.55) as required. • Proof of (4-49)- We bound each exponential by a constant, leaving (4.56) Men-,, 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 =o and 2 n (4.57) from (4.48) that 0 n \ I o n = k-lVlln'IlL^^ if d ± 10, < 0 4.5 <r n n*iv^ii*ii,\ i f (4.58) d = 1 0 - 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 i n the proof 62 of Theorem 1.4.3) we have by Theorem 3.4.3 and Lemma 3.5.2 that = Vva n 2 Ae-i^+0(^)+0 d-8 n1 | (4.59) ' 2 with the error terms uniform in {K G R : \ K \ < C logrii}. This yields the required result for r = 2. Now fix r and M = j\f(a,n) with a G S and rt G N * , and assume the theorem holds for all r-j < r. B y (4.25) and Lemma 4.2.1, we have that d 2 0 2 7 - 3 r = QAT Vva n 2 K + Vva n 2 V (4.60) '=i n J 2 t Next from (4.43), (4.57)-(4.58) and (4.49), we have that QAT ( ^ 3 ^ ) is equal to V - U Li Ae-i% r 2 2 plus 3 K cind + V va'n o l g ^ ) + 0 ( i " i ^ : i i ) r + 0 (E4 , 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 =pi V va'n %(°V nv E Meu nb 2r-3 Vva n 2 -T/r-3 JJ Ae ^ - 1=1 (4.62) By the induction hypothesis applied to rj < r, we have *2 K* 3 K Vva n 2 = ^ - 2 ^ - 3 TJ * e (4.63) ( + 0 E ,<J=S + 0 \^\ nf2 E s) n where the sums and products are over branch labels of branches in JVJ . If Hfi — $ b then £ l n d = 0. For M G Hn , for every j G {1,2,3} we have ^ b 63 < n jt < n/. This Figure 4.3: A n illustration of the relation 2~2i=i i — ?" + 3 resulting from the decomposition of a network jSf into A/i, when M 6 Tin - The 3 extra vertices generated by this decomposition are indicated. r b 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( i ~ l ) i equivalently YA=I i + 3 (see Figure 4.3) and o r r II 3=1 j K v. ' = y JJ r-3 2r-3 A \ Vva n = r ^ e JJ e ~ r ' 2 i=4 ( E4r 2r-3 n j=l \ (4.64) /2r-3 - +° E 2 r ' 3 n 2 Thus, nj=1 (^) y r- 3 j 4 - V~ r 2r-3 2r-3 JJ 3 UT=i 3 2 Ae~^ 3 is equal to 3 .2 e 2dn /=4 '2r-3 „2„ — I I e 2d" (4.65) •2r-3 -. | „(l-*)\ /trn | 2 n ./=i n, Next using a telescoping sum and the inequality e ° - e 64 6 < C(6 - a) for 6 > a > 0 nil- we see that f(nj-Mj) 2dn g~ 1=1 -n is equal to KJ (TIJ — Mi) fl K n — g~ 2dn 2dn \j<l 3 ~ 2 K M n »ftn,-M,-) (4.66) 2 3 ; 2dn ' 2dn l=i 1=1 Collecting terms and applying Proposition 4.3.2 and Lemma 4.3.3 we have ind £" n I / \yva z o n a-JE^bE E 2dn " '=1 f M K S , 2r-3 + ° 2 E T 3 T i=i n. 3 6 =° E ISI2 . 1=1 K\ n + ° +0 n ' - 'iSpni -"' 3 1 E z=i n '2r-Z 2 r ~ 3 E - i r +° Ei=i i=i n, 1^12^(1-5) urn n (4.67) Since 1 — 6 > 4.6 V 0 these are all error terms, and the proof is complete. • 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. B y (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 will 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 be the set of rooted r 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 1 02 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(fi,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 onlyfinitelymany degenerate shapes for eachfixedr. See Figure 4-4 f or 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 GZ ( d r _ 1 ) then B has the topology of some a G S . For a G S r r consisting of / < 2r — 3 edges and ft G N we define V(a, ft) to be the abstract network shape ( obtained by inserting rij — 1 vertices onto edge j of a, j = 1,...,I. Furthermore for y G 1> we define Tt>{a,n) (v) *° be the set of lattice trees T G To with network dl shape T>(a,rt) such that for each edge i in a with endvertices i\,i 2 (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^cn<EEEE = 2^1^ Note that for any given ri G P T - 1 E ^(a,n)(0). 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, Tr]^ - 1 ^, |&(k)| < C ||n||£ . 3 r 66 (4.70) Proof. Let I = 1(a) be the number of edges in a. Applying Lemma 4.4.1 to V we obtain tv{a,n)$)<K . (4.71) l Therefore, (4.69) implies that E E ^ |&(k)| < S € S fJ^ri r E NI* **" 8 aeE 4 < Crllfill^ . (4-72) 3 r The second inequality holds since X ^ ^ n * * * ~ 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. s a s T h e o r e m (1.4.5). Fix d >. 8, 7 € (0,1 A ^ ) u m o v e r a m o s r and S e (0, (1 A 4=3) - 7 ) . There exists Lo = Lo(d) ~> 1 such that: for each L > LQ, t € (0, and ||/c||oo < -R> co)( ), r > 3, R > 0, r_1 r-2yr-2 2r-Z n A IVwn >*J 2 (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) "'" + (4.74) |r-3 20 n where K = « ( a , k) as described in (1.29). Theorem 4.1.8 may be applied to the first term, giving 2r-3 , „ JJ ^(ir) 2 $r)_ ( k K = E E F r-2 2r-3 A '2r-3 ° E ^ F 2 r - 3 + 0 E 7=1 i«| n]" 2 + 5 + n A r-2 l n 0 Mlo 3 n (4.75) 67 Considering the first error term, note that ^y XI — d-8 d-8 2 n -2» [ntj : J s E d-8 n^[ni\: ^IIL^JIloo flj d-8 2' 2 c 1 E ^ E d-8 1 2 n n A |nt| : 2 (4.76) n,- = m < E m < E d-8 IIL"tJ||oo — 2 m |n*J 2 i+n : c ir-2 loo 2 ir-2 < C I I L n t J I l l o ^ ^ H L n t J I I - + - ^ U LntJ | oo ) 3 n 2 where i n 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 /10-d Q% V we interpret the quantity || |ntj ||oo as log(|ntJ). Thus, since | E | is a finite quantity depending only on r, the first error term in (4.75) is 2 r n-0 r (4.77) 2 n" 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 (4.78) n where we have used (1.29) with n) < (r - 1) Y%Z\ (kjI ) • The third error term is already of the form n ~ 0 (^y) where the constant leEj r 2 depends on t. Thus it remains to show that 2r-3 E -«?/„,>. fi -2*I nlnt\ HA llr: n £ N 2 r _ 2r-3 n^ -» - / ,n-e w J j=l r r 2 J f i i( t t ' i=l 2d ds = O l-j , n" (4.79) 3 for each a G E , where the constant depends on t, r and R. We rewrite the left r 68 hand side as 2r-3 n r-2 n L_ - « ? / „ , x r - 2 r - 3 K , a ) 2 s . n^-jfii'-^* E n e N ~ 2 r (4.80) 3 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 ~ . A s such there are at 2 r most C\n ~ T z 3 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. W i t h i n each cube of side £ we have, for all s in that cube, — «j e 2d Uj n — e" 2d Sj - n =o (4.81) B y a telescoping sum representation (as in (4.66)) this gives us that for all s in that cube, (4.82) j=i v j=i / Using K < (r - 1) YHj=i {kjIieEj) , this verifies (4.79) and hence proves the Theo21 2 • 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 function 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 will 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 Definitions and Notation In this section we introduce some notation and results that we need to prove Proposition 3.4.1. Let ir {x;() be defined by (3.7), with U m ST given by (3.3). Then from (2.7) and writing U t = (—1)(—Ut) we have that for m > 1, s s oo ^ C H C E M ) " N=l £ LeC ([0,m]) N m n E i=0RieT (i) u wmni-ust] steL E UJ:0-+X |w|=m n ( 5 1 ) [I+UM- 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 £ ^( ;o=c m X £ w{ )x U LeC ([0,m]) N u-.O-^x M =m m n ( wni-^ n E 5 2 ) [ 1 + ^ . and from (5.1) we have for m > 1 that Tr (x;() — 2 ^ N = I ( l ) ^ m ( i 0 ^ h \^m(x;C)\ < X)jv ^mO^i 0 - Therefore Proposition 3.4.1 follows immediately (by summing over N) from the following Proposition. — 7r a; a n ( e n c e m 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 ^ m x 2 / , (5.3) 9 where £ = ^ ^ , i/ie constant C = C(K,d) does not depend on L , m, z, N, and z Q 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 h (u) = h (u, C) by mi mi C p (D*t - *D){u), 2 if m i > 2 2 c mi 2 hrrn (u) — (p < D(u), c if m i = 1 [/{ o}, if m i = 0 u= (5.4) where t (u) = p(0)I{ =o}0 u Definition 5.1.2. For qi G {0,1}, nii G Z+ we define s (x) muqi I < 4 we define s ^ (i) = \x\ h (x). For 2qi mi ^{x) to be the l-fold spatial convolution of the s . mi>qi Definition 5.1.3. For n € {0,1}, let 4> (x) = \x\ p(x). For I G {1,2,3,4}, let 2ri Ti (^^(x) denote the l-fold spatial convolution of the <f> , and define </)^(x) = I{ o}Ti x= 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\=i i- If the bounds (3.33) hold for 1 < m. < n and z G [0,2] then for all m m < n + 1, and z G [0,2], I'So * «> * *!8>ll~ < rnZv+X'toVX*™?^, m '71 (5.5) 2 x+y m Figure 5.1: Feynman diagrams for M$ (a,b,x,y), A (a,b,x,y) and A o{a,b,x,y). A jagged line between two vertices u and v represents a quantity h (v — u). A straight line between two vertices u and v represents the quantity p(v — u). miiTn2 mii mi and l4^olli<^™ ^ E < 2 E < ? i - ' (5.6) Definition 5.1.5. Let MM(a,b,x,y) = h {x-a)pW(x + y-b), m (5.7) and A A i u (a,b,x,y) mum2 ^ (h (y-a)h (x-y)pW(b-x), = \ mi m ^ 0, m2 2 ( 2 ) (5-8) We recursively define '(a,^,!/) E ^ A m i | m 2 (a> n,«)M{^ ) 1 ) i m 2 j v _ (« «,a;,y). i ) ) ( 5 9 ) The diagrammatic representation of these quantities appears i n 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 h (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). mi Lemma 5.1.6. Setting M^\a,b,x,y) UQ = a and U2N-1 = %> for every N > @, =X1 £ r2N-i YI hmitn - fH-i) -- Y Vl P(vi ~ b)p(v -(x+ N x y)) x VN 1 <I< mj,m 2N — 2 : ! + 1 ^ 0' (mi,...,m N-3) (a, 6, u, v)A.• m 2 j v - i , m 2 jv_2 (x,y,u,v). 2 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 (5.11) i=l For general N > 2 we let (5.12) 73 and for N = 2 we also define G = {m e ft , : (mi V m ) < m } . 2 TO 2 m Note that for N > 3, i ? ^ U Similarly for J V = 2, ^ U J ^ U 3 (5.13) 2 = Wm^N since m i + m + m i v - + rri2N-i < m. 2 G ^ , 2 2 H ,2- = m 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 14 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 { { 5.2 x- b , if # { m : m^j ^ 0} is odd 2 j x+ y —b , if # { m j : m - ^ 0} is even x + y —a , if # { m 2 j : m j 2 (5.15) 2j 7^ 0} is 2 odd # { m j : m j 7^ 0} is even P r o o f o f P ra;o— p oa s i t i o,nif 5.1.1 2 2 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. C a s e 1: q = 0. Our induction hypothesis is that E s u p ^ M f ( a , M , y ) < ^ . (5.16) 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 a,f>,2/ m ( > > , y) = sup ^ M a fe a.,b,y x supY m{x)p - (x+y-b h ( h (x x m X = sup^/i (rc)p^(a; + 2). m 74 (2) x + a) 2) - a ) p ( x + y - 6) ( 5 ) 17 Applying (5.5) with / = 1, k = 2 and all qi = rj = 0, this is bounded by ^ Jf C as ] d m 2 required. We consider separately the contributions to (5.16) from E N , and in the case N — 2 also the contribution from G . 2 M Now by (5.9) we have sup£>W(a,ft,*,!/) E = E E E mi<^m < f-mim'eH _ 2 2 X S U y P E r ? M x = Y - 1 (> ) U m ( m i + sup^A m 2 ), A r _i a ' m i i m 2 (a,6,n,v) «.« 6 ' > v) U S supJ2 mum (a,b,u,v) E (- ) A 5 2 18 u.,t/ sup X]Mir (t« ,t/ x y) £ m'eK _ m m2 ¥ N £ <^ < _ , > , _ ( m i + m 2 ) < E mi 1) S u 1 M ' ' y A {a,b,u,v)" ( m mum2 «,„ mi (C/3 -^)^2 E 0 > x ? / - ( 1 + m )) ~2 ' d mi 4 2 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 ^} m < 2 a 2 m i E m 2 < 2 | i _ S m i U P E A (a,b,u,v), murn2 ( ) 519 <».& 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 m = 0, m > 0 to get 2 E Y m + i SUpY m m2(a,b,U,v) E m i < f m = 2 2 < f - A u a m sup£/i < 2 m a,b E m i < f < E i , bu > ( u - a)p(v - u)p (6 - t;) (2) m i E 0 < m 2 v sup£/i ( u - a)h {u - w)p (6 - u) (2) m i m2 < f - m i E •E 75 ^ "-L<g^, 2 R (5.20) where we have applied (5.5) with all qi = rj = 0 i n the penultimate step and the fact that d > 8 in the last step . Combining (5.18)-(5.20), we get that E supE^M,^) < ^J?\ [C (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 n for m,2N-2 we get 2 E supE^W^V) =E E E x sup E^(«>>«.«')• ni< f "2< f -ni" 'e H -(n +n ),N2 L 2 i M )1_ a,b,v' i 6 supEA.i,n(aJ,y,«',«) m 1 2 (5.22) 2 , "''^ < X 1 V u Using translation invariance of A , (x,y,u',v) nitV 2 get we proceed as in (5.18)-(5.20) to E supEMf(a,M,!/)<^^, rheFN ^y x . (5.23) rn 2 a as required. It remains to prove the bound (5.16) for the sum over rh G G ^ . Note that in this case m,2 ^ 0 and so M V ( a , 6, x, y) is equal to 2 EP ( 2 ) ( 6 - v)h (u - a)h _ mi m (v - u)h (x - v)p - (x + y - u). ( {mi+m3) m3 2) ^ ^ . 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). B y symmetry of M^\a,b,x,y) and translation 76 invariance we have snpJ2 S Mx y) Y M y t meG2 ' > x a b y <2 Y, Y rn,S™ 2 mi<-2 snpJ2 SHa,b,x,y) M / 1JI3 < t n i mi > < >y a mi : E =2 Y m . < f2 sup^J^tft-^V^-a) <mi: m s m > 2 * >" E = 2 m i m X i < u)h (x - " 2 ~ s u m 3 < mi - v)p - \x + y - u) { 2 m E m '(5.25) u ^ - ( m i + m a ) ^ x X •"' b pEE^ ( -«)^miH ( 2 ) 1 : ni2 > mi u,v ^m-(n»i+m )(« ~ u)h (x - v)p^(x m3 3 6 + y - u), 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 m i < y [ PYp ( ~ ) -( i+ 3)( Y mz m < m i 2 {2) su > b v hm m m ~ ') v I [supY m3( u h x ~ ') v j : 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 ... (m-(mi+m )) m i < f m < m 3 m2 > i 3 : E m AC/til m d-4 2 m " d-4 2 i < E -1=1 j m l m3 m <c ' m i ^ 4 <g d-4 / j m i < f d-6 ' TYl^ 2 77 — (mi < 1 m i + ms) : > m i (5-27) 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 £ n YM < \^,x,y) 2 S rneH ,N a m < N P ' ' b y a H C m x ^ \ ] and 2 (5.28) Y supXUM | ^(a6 ,,*,y)< 2 m€H ,N a m ' ' b y d-6 m x 2 In view of Proposition 5.1.7 with q — 1, these clearly imply Proposition 5.1.1 with 9=1For N = 1, the first statement of (5.28) is sup X y|x + a,b,y o- (C6 -ir) 2 6| /i (a; 2 m x a)p^(a; + y - 2 b) m < N (5.29) d-6 2 Writing p( ' (a: + y — 6) = 2~2 P( ~ °)p(x + y — u) and using \x + y — b\ < 2(\u — 2 u 2 U b\ + \x + y - u\ ), (5.29) is bounded by 2 2 2 sup ^ 4>i(u - b)p(x + y — u)h (x — a) + 2 s u p ^ p ( u — b)(/>i(x + y — u)h (x — a). m a,b,y m a,b,y x x (5.30) Applying (5.5) to each term with I = 1, k = 2, q\ = 0 and exactly one r - = 1, (5.30) 3 is bounded by ^ ^ - e ^ as required. The second statement for N-= 1 is a 2//-»/o2-^\JV sup Y \x ~ a\ h (x - a)pW (x + y - b) < 2 , K m m 1 a,6,y (5.31) 2 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 % ,N into sums over m G E^, m G F£, and when N = 2, also m G G ^ . For 2 m the contribution from rh G E% we write | z | < 2 ( | z ^ | + \ZM| ) 2 (ZA,ZM) = < 2 2 (a; - u, ti - 6) , if #{m j (x + y - u,u - b) , if #{m,2j (x - v,v - b) , if # { m j (a; + y - v, u - 6) , if # { m j m2j 7^ 0} is even and m 2 = 0. 2 2 m 2 j 7^ 0} where is odd and m > 0 2 7^ 0} is even and m 2 m 2j > 0 # 0} is odd and m = 0 2 2 (5.32) 78 Thus E supEl^M^M^y) m€E»> ' > a b <2 x y E P E m£E ' > x,u,v \zA\ A , {a,b,u,v)M ?,~ \u,v,x,y) S U a b 2 { l (5.33) rni m2 y miN + 2 E P E S U AmumziaAUiV^ZMfM^'^iuiViXiy). rh€E» ' ' x,u,v a b V As i n (5.18) the first term on the right of (5.33) is equal to (CP*- ?)' 1 E 2 E <^m <^- mi <2 2 supEM A7ii,m (a,M^)2 /^.o2-^\JV-l jzi m 2 >" «.« 2 m i N {m-{m +m )) a mi l E E < 2 m <2m_ TO2 2 s i -4 2 (5.34) SUpEl^4| Ani,m (a,M,v). 2 2 > u,v a mi b We now proceed exactly as i n (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 a m(CP -*) (5.35) d-4 m 2 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^ ~^(u,v,x,y). We proceed exactly as i n (5.18)-(5.20) except that the induction hypotheses give a bound 2 2 N N 2^ m'p-W , S U N », P \ M\ m> z '<°'>V '{u ,v ,x,y) <a M X U -jze ( - M ( m l+ 2 ) ) m 2 (mi + m 2 ) ) 2 (m - d 4 (5.36) which contains an extra factor of a m compared to that appearing i n (5.18). We now proceed exactly as i n (5.19)—(5.21) to get a bound on the first term of (5.33) of 2 <y m m s — . (5.37) 2 This proves that E su El^Mi P rneE" > > a b y J V ) (a,6,,,y)< * ( 7 2 ( C m 79 2 ^ ) i V . (5.38)' As in the q = 0 case of Proposition 5.1.1, the bound £ s u p ^ l z p M f l a , ^ , ; ; ) ^ fheF» > ' a x b y 2 ^ 1 " , (5.39) rn 2 follows by symmetry. When N = 2, the contribution to (5.28) from m G is easily bounded as i n (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 ^ AL^^ as required. This completes the m2 • A C proof of Proposition 5.1.1 for q = 1 C a s e 3 : q = 2. Our induction hypothesis is that £ u Y\z\ k\ M£\aAx,y) 2 S ™eH , ' ' " Cf 4(C < 2 P x a b y m N )JV (5.40) 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|*||*|M^W,z,y), 2 2 (5-41) we use the expansions \z\ < 2(| • | + | • | ) and | z | < 2(| • | + | • | ) yielding 4 2 2 2 2 2 2 terms instead of the two in (5.33). One such term is 4 ] T sup Y fh£E% '' a b \ A\ \z \ A z (a,b,u,v)M^^ (u,v,x,y), 2 2 A (5.42) 1] mum2 *,u,v 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 Am.^i^b^u^v^z'^^M^'^i^v^x^y), P£ (5.43) N > >yx, ,v a b meE u 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| U| M^(fl 6 x y) 2 2 > fh£E» ^ a y x > > < a 4 ( C ^ / m 2 80 ) 2 - (5.44) The contribution from rh € F% also obeys the bound (5.44) by symmetry, while the contribution from rh € G ^ when N = 2 is handled as for the q — 1 case 2 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. • R e m a r k 5.2.1. Observe that apart from the recursive representations of the diagrams M( ) 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 complicated diagrams in Chapter 6. N 5.2.1 D i a g r a m s w i t h a n e x t r a 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^ \z). We say that a diagram has an extra vertex on some h if it is the same as a diagram corresponding to some except one h (z) i n that diagram is replaced with 2 m mj h i * h - >(z). When we consider the diagrams arising from the, lace expansion on a star-shape of degree 3 we will encounter diagrams with an extra vertex on some p or h . 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. m mj m m We let n denote the location along the branch point where the extra vertex is located. If n = Yll=i i f ° 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 h ). Let M^' (a,b,x,y) denote the corresponding diagram with an extra vertex at n. m r n mi We prove by induction on N that E - P E ^ K ^ ^ ) ^ ^ ^ - M5) E 1 meH , n<m > >y x a m N ™> b 2 For N = 1 the left hand side of (5.45) is E su E(^n P * h - ){x - a)pW(x + y - b) + 2 sup m n 0<n<m > >y x a ' 'y b a b E m(x h - a)p^(x + y - b). x (5.46) 81 Using (5.5) with I — 2, k = 2 and all qi,rj = 0, the first term i n (5.46) is bounded by — o ~ ^ — o o<n<m rn 2 - - ( - ) 5 47 m2 Similarly using (5.5) with I — 2, k = 3 and all qi, rj = 0, the second term is bounded by ^d-1 • Adding these together we get a bound of ^ J[ C which satisfies the c d 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 A m mit or M^,~ \ l 2 In the former case we use the q = 0 result in the proof of 5.1.1 on the M^' ^ part and (5.5) (increasing k or I by one due to the extra 1 vertex) on the -A™ the M^,~ ^' l 2 P a r t - •"• nt n e latter case we use the induction hypothesis on part and (5.5) on the -Aj^ n and rh £ m m 2 part. The contributions from rh £ F£ are dealt with as usual. Similarly we prove (5.49) men , m N n<m ^y x . \ a ™ 2 Note the factor \x — a\ i n (5.49) rather than \z\ or \z\. This is to avoid the situation 2 that could arise of having a convolution of four p's with one of them having an extra factor \u\ on the same diagram piece. 2 This would violate the condition ^ + Yli=i i < 4 i n Proposition 5.1.4. Using \x — a\ instead, we will use path r 2 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 General Diagrams 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 U N-I = > for every N > 2, x 2 2N-1 M^(a,b, ,y)=Y- £ X h (Ui - Ui_i) mi i=l p(vi-b)p(v -(x Ml «2JV-2 £ + y))x N Vl,...,V N n E( p ~ t-i)p( 4f- wi u vi - d I)P( L W - w v ;>2:m;=0 w t IJ 1 <I< : E M {P( ± V 2N - even) + P ( « i ± 3 - ~ ti {l U T Ui)I i odd^j { 2: ( m i , , m _ ) (°> > ' ) m -i I ) 6 2 N U V , m J V - (s, V, «, «)• A 3 2 2N 2 (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 m > 0, m = 0). For N > 3, if m > 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 2 2 2 £ I ™i ( h U\,U2 ~ )m Ul a (2 - i) h u 2 u E( \ p )I U2 ~ )p( \ ~ Vl h v J Vl 2N-1 H . " 3 £ p(vi±3 h m i (ui - Ui-i) i=3 V-2N-2 - ui)p{v - [x + y)) n N E ( p wi ~ i-i)p( 4? u vi - i)p( i w v - i) w i>4:m;=0 VJI V2,...,V N II X 3 <I < 2N - (P^k ~ e v e n > + ( ? ~ ^{l p VL Ul odd}) 2: rnum2{aAui,U2)M\^~^ _ (u ,U ,X,y), = E A m2N i) L 2 Ul,U 2 (5.51) by definition of A m ^0 mi>m2 and the induction hypothesis. This proves the result when 2 83 If m = 0 then by separating the / = 1,2 terms from the first product and 2 I = 2 term from the second product in (5.9) we have that Mm (a,b,x,y) is equal ! M to E E Ul,W2 ( mAui ~ a)h {u - « i ) £ p ( v i ~ )p{ 2 h M 2 b 0 \ Vl- E / JJ mi(Ui -Ui-l) h i=3 U N-2 2 p(u2±2 - w )p(v 2 2 V2, — ,VN 2 . "2JV-1 E-E 1*3 ~ui)p{vi ~ w ) J w 2 N - (x + y)) JJ £ l>4:mi=0 P(™f - « i - i ) p ( « l ± 2 - wi)p{vi - W[) wi even} + p(w/±3 - Ui)I{i 3 < I < 2N - 2 : » n j , m j = £ Ml A + i # (a,6,7Ji,W2)M ^ ( m i ) m 2 m 2 ; v _ (u ,u;2,a;,y), i ) 1 ,«>2 (5.52) and the induction hypothesis. This proves the result when mijjn2 2 ) ( by definition of A m }) odd 0 = 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 rn N-2 and separating the terms I = 2N — 1,2N — 2. 2 =0 „. • 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>| %feC)< 2 £M «MW(O,O,*,O). E 2 rheH ,N x (5.53) x m Proof. We prove the stronger result that *£(*;0< E MW(o 6,x,o). t (5.54) Recall the definition of n^ix; ) from (5.2). r 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 h (x), while straight line between 0 (resp. x) and v\ represents the quantity p{v\) (resp. p(x — v\)). m is compatible with {Om}, so by (5.2) m 74(*;C)=C Y, m ^ W U to : 0 —^ x \LO\ =E E W(R )[-Uom}U^ 1 ^ + U i J=0Kte7L(i) b^Oro =m WiRo) E ^(^m)h^0m]x ( 5 5 5 ) m—1 II E w> II w E r w : 0 -> x |w| = i = t 1 + ^i- b^Om 1 m Note that everything in this expression is non-negative. Now —Uo = I{R nRm^D} so TTm( '-> 0 i nonzero if and only if there exists v € Z such that v € i?o FI i?m and therefore m s x E RoeTo w(Ro) 0 d E R eT m x w(#m)[-tfom] < £ £ v RoeTo(v) E Rm€T {v) ^ x ^ V If m = 1 then the last line of (5.55) is C E u :0 W(u) = ( D(x), . Pc —> H = (5.57) x i as required. For m > 2, n^Omt + ^b] ^ I L ^ - c ^ m - J + *t\ 1 1 85 U a n d l e t t i n g I/i ( P res Figure 5.4: A n example of the Feynman diagrams arising from the lace expansion. A jagged lines from to Ui represents the quantity h (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). mi y ) be the location of the walk UJ after 1 step (resp. m — 1 steps) we have 2 rn— 1 = m <J2J2^ ^^D(x-y )x D 2 2/12/2 . ' (5.58) m-2 c m-2 £ £ W(0J')H W':J/I->2/2 |w'| = m - 2 WWlfll 3=° RjtTw'U) + tfk] b =/i (a;). m Combining (5.55)-(5.58) gives the desired result for A = 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 7 m c E m {sitl,...«Artjv} £C '([0,m]) N E w:0-fx |oj| = m W . AT ^ ) U i = 0 E w(Ri)l[[-u. ] Ri^T {i) a iti i = 1 n [1 + ^ ] . \>eC(L) (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 , is +2 86 compatible with L = . . . , SJV^JV}- Therefore (5.59) is bounded above by m c £ E {siti,...sjv*Ar} w :0->I m e^^dO.m]) N ^ ) I I [ - M ' w( )H E U i = 0 i£%(i) R i = X 1 |w| = m N-l N-2 n [ i + n [i+^n n ti+^n n l>e(si,S2) b6(tjv-i,*jv) i = 1 l>e(si+i,t») i= be(tj,s ) l j+2 (5.60)' whereforb = s i we are using the notation b € (a, b) to mean a < s < t < b. For L = { s i t i , . . . , s j v t j v } . € £ * ( [ 0 , m ] ) we define rh(L) G Z ^ 2 =s -0, m i m AT_i 2 m-tN-i, = 2 m 2 i = *i-«i+i> ^ 2 i - i = - 1 by « t + i ( 5 . 6 1 ) Then m i > 0, m . 2 i _ i > 0 and X ^ i " i i € %m,N- Similarlyforany fh G %rn N we define L ( m ) = { s i * i , . . . ,5JV*W} € <?.([0, m ] ) by 1 m = m s om 2 1 5i = 0, tjv = m , 2i ** = E m « = i , • • •, -^v — i , J' £i (5.62) 2/-1 a J = £ J ' M ^= 2,...iV. J'=I Thenforeach i , S j + i < £j and S j i > 0 so that L(m) G £ ^ ( [ 0 , 7 7 1 ] ) . Thus (5.61)-(5.62) defines a bijection between £ - ^ ( [ 0 , 7 7 1 ] ) and % ,NWe now break up the sum over walks w i n (5.60) according to the intervals on the right of (5.60). Doing so we obtain + — m w w E oj : 0 —y x \w\ = =m E E Wl,...,« JV-l 2 W l W 2 1 W(U2N-1)* 2 N (5.63) - 2 -> x = S - Si 2 iV-2 n 1 W2JV-1 : U - Si JV-1 = E : 0 - ^ U l . \ui\ = s 1 ^ E W2i:« 2 i - l | w i | = *i — 2 ^ M I I -> « 2 i i+l s -? = 1 E w(oj y 2j+l W2J + 1 : « 2 j «2j+l \W2j+l\ = Sj 2 + 87 — tj Then under this scheme, n™L. Z ^ e T ^ becomes 0 m i - l / E w ^ E n Ro€T 0 wiRi^) V^i.mjGTL,.^.) l < i < 2 J V - l : \ E n ^ ( ^ J ) » j j = l Ri,}£T (j) Ui mi ^ 0 (5.64) where = itj Wj(mj) (u;2jv-i(m Ar-i) 2 = 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 (/{m^O} ^ ] U.S=i = + {m =0} {Ri, =R - , _ }) I I i mi i 1 rni becomes ^ n a , ^ } x 1 N-2 -^{« nii2,m ^0} {R2N-3,m Jr O 2 2Ar _3nR2JV-l, m2Ar (5.65) II ^{/J (-l,m ,_ n K i + 2 , m , (=1 _ #0} 1 2 2 x 2 2 +2 5*0} * Note that (5.65) contains no information about Rij for 0 < j < mj. Lastly we have that the second line of (5.60) becomes 2JV-1 / \ n W * M = « n yi<*<t<mi-l i=l • (- ) 5 66 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 u E E W(Ro) rneHm,N RoeTo II ( { m i ^ 0 } + {mi=Q}I{Ri, =R - _ }) I J mi i hrni E ( 1 < i < 2JV - 1 : mi ^ 0 WiRi^U x xRi^i^Tui ) x 1 N-2 I{RonR ,m ^<l)}I{R2N-3,m _ r\R2N-l, _ ¥:<l>} 2 2 2N 3 m2N II 1 ^{#21-l,m _ n i J J 2| t 2 + 2 ,m #0} 2 i + 2 / X , \ 2JV-1 n E y W i : E ->• « i 5 = 1 ^ - 6 ^ 0 ) w(^) n j iw«=i} \l<s<t<m;-l J \ui\ = mi (5.67) The last line of (5.67) is n ^i hmi(ui — by definition. For any collection of trees {Ri, '• 1 < * < 2iV — 1} for which (5.65) is equal to one (i.e. nonzero) we choose v\ G Z , i = 1,..., N as follows. 2 -1 mi d 88 (a) I{R nR ^<li} 0 1 if and only if there exists a wi G Z such that v\ G RoC\R2, . = d 2m2 m2 This means that RQ G 7 O ( V I ) and i?2,m € Tu (ui). 2 2 (b) Similarly / { B Z d a j v _ such that v N T _ (VN) U2N 3 i r a 2 W _ nfl 3 2 J V G # jv-3,m 2 _ 2 W l i m 2 J V 2 2N 1 i f 1 and only if there exists a v G N _ F l . R 2 ; v - i m _ i • This means that i? Ar-3,m jv-3 3 > 2 N 2 1 (c) For each i G { 3 , . . . , 2N - 5} such that % is odd, J{fl „,.nrtj 3 . 40} = + Ji only if there exists that i ? j , m i G Z such that d Ui±3 e 2 e Ti(wi). and R N-i,m -. 2 _ #0} = G T u ^ V i i i ) and i ^ + 3 , m i + 3 «j±3 € Tu i + 3 im 1 i f a n d +3? G Ri,mi Fi jRi+3,m - This means i+3 ( u i ± 3 ) where i + 3 is even. Now if mi = 0 (in particular this forces i to be even) then h (ui — «;_i) i n (5.67) m is nonzero if and only if m = =R _ 1 1 } = 1 if and only B y the above construction we have that vi G Ri,m and if Ri,mi = -R/-i,m;_i- vi+2 In addition 1^ r i.e. vi,vi±2,ui G Ri,mr For T = i ? / , let G Ri-itmi_1, ;m denote the backbones in T joining the specified verticies. Then there exists a unique wi G T such that (5.68) Tui^y , Fl T .^y.. — Tui^+ujr u 2 Collecting the above statements we have that n E tf €7b I 1 < z < 2N - 1 0 : E I V^i.mj 6TUi (-T{m;^0} + ^ { m i ^ ^ H i . m ^ - R i - l . m ^ i ) ) X N-2 {R0 2,mn¥=9} {R2N-3,m _ ^R2N-l,m _ I nR I 2N ^ E U E VF(^O) RoeT (vi_) I:m;=0 H | , e 7 i , ( i M m( n ( : m, ^ m 7^ ( + 1 2N 2N 1 #«>} IT A«2(-l,m ,_ j n R ( + 2 , m 2 1 2 N 2 2i+2 5*«) _ )x 1 N ,111^2) ^(iw E 0 0 3 £ W'(i?2W-l,m R2N-\,m _ £'Tx(y ) 0 X •^{i even} "I" E \^Ri,m,e7i (u^) ^ I{I odd} ,m,) ( ^,771,67^,(^^3) y (5.69) 89 Now observe that E f l e r , ^ ) W(R) = p(y - y\) and 2 E W{Rt,m)<Y, «/,m,er ,(i;M^i^2) E ^(^) E ™; RieT (wi) u P(WI - wi U[)p(vi - Wl)p(vi±3 ^(^) R3eT ,(vi+2) t w - 2 Wl E R2er (v ) Ul = y (**) w Wl). 2 (5.70) This completes the proof of (5.54), and hence Proposition 5.1.7. 5.4 • D i a g r a m pieces 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} be such that k + £ f k = 1 n < 4, then 25"V-fl2— — n £ « W < C m ^ , ^ E , 4>%(x)< jf ^-. Ca a n d s u p 0<|z|<x/mX 4 \x\>yML m 2 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 Z l MOIIOO < ||*S„ C a 2 E 9 i / ! 2 m E g + such that £ = m < n + 1, and l l s ^ o .||i < C a £ * m * > \ (5.72) 2 0 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 i ^ - Lefrh^ G Z r 4 + and m= £ j = 1 mj. // 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], 1*2(0 * $ > l l o o < mS«+Er, < T 2(E« Er,)C^_i_ m 2 + j ( 5 . 7 3 ) and 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 * which is equal to ^8>H°° is e q u a l x ^x2~2u %)^i)( su s ~ ")<!>%)(*>) x a; |w|>-\/mL < t o | u | < v m L E i ^ ^ - ^ +^ i , ^ ^ sup <f>%(u') l«'l> V m L |«|>VmT E ^5>( ) u |u|<v^L 1 (Jcr'Z.'j < m = 2 ffl! 1 (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 Z satisfy \f(x)\ < ^ d and \g(x)\ < ^ with a>b>0, then there exists a constant C depending on a,b,d such that if a > d r%, [ ]a+b-d i if a <d and a + b> d. x 5.4.1 (5.76) Proof of Lemma 5.4.1 We prove the result in two stages. We first prove that k 4% () ^ E X 3j'=0 #(2-i/) Md-2j-2 L J En' ( 5 - 7 7 ^ For k = 1 we have from (1.13) that '(x) < CI*. + j ^ r 2 < E (5.78) and tf'C*) * T ^ p r <E 91 L j ( 2 -,)g ] d _ _ 2 j 2 (5-79) Which verifies (5.77) forfc= 1. For u > 1 we have •-I 1 ^—v ^—v fe (_/ 1 IT—-v L?"( -")[u]«'- J- i ^ ~ 2 i *-i • c - j=0 E n=0 E rj(2-j/) 1 ^EE n=0 j=0 r.n(2-i/) «]"-2»-2EUn , 1 E r,/lrf-2j c fe-1 L/ L ^ - " ) ^ - 2r 2 (5-80) c £(j+n)(2+i/) r ld-20+n)-2Eti i r r Ir'J ' where we have used Proposition 5.4.3 with the fact that fc + step. W i t h a different constant, (5.80) is bounded by y r% < 4 i n the last — ^ — (5-8i) ^L7(2+")[x]'*-2i-2Ef=iri as required. Therefore we have fe E Q ^)(^)^E E 0<|x|< / v mL #(2-«')Md-2j-2£'-i j = 00<|x|<v rn£ / fc r»r / ^ - n ^ £ 2 - + 2 r i (5.82) fc X^(2-J/) which proves the first bound of Lemma 5.4.1. Similarly, (fe) fe = E ,.. „ ~ j=o L -3 - l^ im < T j d u 2 r fe D £ r - J- I>,: 2 2 2" C ^ E O - ^ - ^ d-2fc-2£>jj_ 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) = (p D(x) and hence c HMoo < ^ = |IMi<C. (5-84) ^ Ca B , (5.85) Using the fact that D(x) = 0 for \x\ > dL , 2 2 sup | z | / i i ( x ) < 2 C L 2 2 T d 2 Li x 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\ h (x) < CY \u\ D(u)D(x - u) + 2 2 2 CJ2 \x — u\ D(u)D(x — u). u (5.87) u For I = 1 and m > 2 we use the inequality h (x) < p(0)(,p Y D(u)h -i(x — u) (which holds trivially by replacing the factor Tlo< <t<m-2[ + ^] y Ili<<t<m-2[ + U t] in the definition of t - ) so that m c 1 t/ u m 1 b s s s m 2 C6 2 \\h \\oo < \\h -l\\oo m < m \\hmh mi < | | / l m - l | | o o < K. (5.88) Using \x\ < 2(|u| + |a; — u\ ), we have 2 2 2 s u p | x | / i ( x ) < o- ||/i _i||oo + sup |a; - u'\ h -i(x 2 2 m 2 m m Ca B 2 < - it') x-u' x 2 ^ + d Cu B 2 1 2 ( 5 > 8 9 ) d-2 This proves the result for I = 1 and all m < n + 1. 93 m For / > 2 we have s ^ o ^ o 2-« _ 5 mi>«i W (m ,...,m;)i(<?2,---9f) i v 2 2 ' I I ^ O ^ o l U < l|sffi,,Jloo||sf~ , ..., ),( ,..«)ll 1 ) 2 ^ C<7 C o - ^ /3 ^m 'i m'» 2<?1 mj 2 C g r 2 2r < 1 m 0 . v / 0 . m2i m < ' v d ffl! as required. Similarly if m i < y , l l am('),^0ll°° II < ll ^ HsW l l m i , i l l l l l ( . . . , ) , ( g , . . . , ) l IIl o o 5 ( 0 6 6 9 m2) mi 2 g (5.91) d m2 < 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 l|S W II <r II ( ) II II II m(0,gtO 111 ^ H m i , l l l l l ( , . . . , ) , ( , . . . ) l l l < Ca m Ca ^= * £ U n x S 5 gi 2qi m2 qi 2 mi 92 g( (5.92) 2 m < Ca ^=i rnPi^ \ 2 qi 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 h m that we used to prove Lemma 5.4-2 (and hence Proposition 5.1-4) was I I U o o ^ , mi IIUIi<C, (5.93) and when some qi / 0 we also used • sup \x x m x 2 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 £ = 1 (see Lemma 3.5.2). In this chapter £ = 1 and hence it does not appear. c 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 of 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\ ^(u) 2q < N^N a \\M\U B (M), 2 2 q N 95 (6.1) where u = (ui, 112,113) G Z B (M) = N and 3 d (C8 ~%y x 2 3 3 1 1 1 £J [ M i ] ' £ [ M ^^ r , , , - ^ II + E [ E r E M j -mA*? [M + m ] k } (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 M = 0 but M ^ 0, t As stated i n 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}, £ (cP -*) il—K=*. 2 < N\M\\ti\W \UJ\^{U) N (6-4) i=i i i\ uezu M 2 Lemma 6.1.3. For q G {0,1}, N £ 2 <7V (iV a ||M|| )"(c7^-^) 3 *£(£) 9 2 2 iV 0O uez 3d 3 1 3 X ^ ; T -, r [Mi] i=1 1 , d-6 2 ^y r 7d^ . ,, , d-6 Wi - j) [Mk+mj] m j^imjKMj (6.5) 1 2 2 Proof of Proposition 4.3.2. If Mi = 0 for some / (but M / 0 ) then from Lemma 6.1.1 we have V , > \Uj\ TT.-AU) < r | 2 1 N ^.NHN a^M\U (C0 ^) "5 7 2 9 2 N ( ( E L M , ) " M < ^ ( i W U M l U ) " ^ [M, - M ^ [ M , 96 2 - - ^ ) ^ 2 M ]^[M h tl + M, ]^ ' 2 where ly^l and Z # /. Summing over / this is trivially less than 2 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 S + E i«>f*£(s)- U (-) 6 8 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 /i '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. m Lemma 6.1.4. Given homogeneous Junctionals F : E —>• 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 i m a Proof By homogeneity, F(d*) = F p , . . . , ^ L JJ . ai (6 .9) Also by homogeneity, fj£)= which implies that F (fj-,..., < ^ for each* (6.10) < K by hypothesis. This completes the proof. • In most cases we will use Lemma 6.1.4 with each aj being either h (ui) or p(ui) and F being a diagram (i.e. a large convolution of /i '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. mi m 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 , Mi] of length M 1 + M 2 . Consider the lace L = { - M M i } illustrated in Figure 6.1. Breaking up the walk corresponding to the backbone into 2 2 97 M, 0 Mf M , 0 Figure 6.1: The single bond lace on [—M , M{\ and the corresponding [0, M i + M ]. 2 2 two subwalks we can show that the contribution to this lace is less than or equal to M {x2)hM { i)p - { i h x 2 { 2) (M ,M ,o)( ii 2) 7r J2xi,x 2 x 1 x 2 (6.11) ~ 2)- x x l from X\,X 2 Using translation invariance we can rewrite this as YhM (u)h ( -u)p (x). x 2 (6.12) {2) Ml x,u Comparing this to the contribution to (4.29) from the lace on the right of Figure 6.1, YhM M (x)p (x), (6.13) {2) 1+ 2 x we see that the only difference is the replacement of /iM +mi(^i) i 2 J2 M (u)h 2 (6.16) by (ui - u). h U n mi Now consider the lace L = {siti,s t } where s\ = M , t\ — m\ + m , s = mi, r. = M i on the left of Figure 6.2. This lace divides the interval [—M , 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 subinterval containing the root into two subwalks, we can show that the contribution to 52xi x (M ,M ,o)( i,X2) from this lace is less than or equal to 2 2 2 2 2 2 7r x 1 2 £ Y X\,X 2 2 2 M { 2)h {ui)h (u2-ui)h ^_( ^^ h x 2 mi m2 M m Ul ,U 2 (6.14) Using translation invariance we can rewrite this as Y X\ Z\2 M {u)h {ui-u)h {u2-u )h _ ^^ h 2 mi m2 1 Ml {mi+ U,U\,U 2 (6.15) 98 -Mj 0 m, M, mfm 2 0 Mj+m, Mj+M, Mfmfm 2 Figure 6.2: A lace on [ - M , M i ] and the corresponding [0, M i + M ] . 2 2 Comparing this to the contribution to (4.29) from the lace on the right of Figure 6.2, X E Xl Ul,U ^Af2+m (ui)n 1 m 2 (u -«l)/lMi-(mi+m )( 2 2 a ; l-'"2)p ( 2 ) (w2)p ( 1 ) (^l-^l) (6-16) 5 2 we see that the only difference is the replacement of / i M 2 in (6.16) by +mi("i) Y.u M (u)h (ui-u). h 2 mi In general, assuming M 3 = 0, the diagram arising from any lace on [—M , Mi] is bounded by the opened diagram of (5.9) that arises from the equivalent lace on [0, M i + M ] , except for the replacement of at most one term h (ui) by a term of the form (h * h - )(u) in M^*\ Note that this m is fixed by M i and M (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 h (provided that degree is less than 4). Thus by Proposition 5.1.7 we have the same diagrammatic bounds for 2 2 mi m mi m 2 mi 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), N £ *Mi,M o(zi>S2) *i.*2 ' ' 2 > < £ men sup^A4 (a,6 a;,y) < Ar) ) , ' 'y a * b Ml+M2 N d-4 ) [M + M ] 2 x 2 (6.17) which satisfies the claim for q = 0. 2 For q = 1 observe that \XJ\ < 2N2Ei=i luj,i| > where the Uj^ are the displacements of the h 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 _1 2 mi 99 hmi(uj,i) on the backbone from 0 to Xj has been replaced with \uj i\ h (uj,i)2 t view of Remarks 5.2.1 and 5.4.5, the only bounds on h diagrams without the factor \uj i\ t sup„ \u\ h(u) < 2 and £ ° W \\™P C 2 M 2 \u\ h(u) < Co^MW^. 6.1.4 to get that the diagrams d ll^mJIi < C. = 0 and (5.6) with / = 1 imply that We now apply Lemma 2 u a n mi Since m,j < ||M||oo, (5.5) with I = l,k In that we used to bound the m were ||/i ||oo < 2 mi with the extra factor of | c H-MHoo times the bound for the diagrams bounded by without the extra factor. 2 Therefore when M 3 = 0, Y\uj\ ^(u)< a ||M|| (2iV) 2 2 u supX;^(a,M,!/) Y 2 0O rne'HM +M2,N a 1 (ce -*) 2 <a \\M\U2N) -± 2 ' V x - N 2 [Mi ' (6 18) L_. + M2} — Similarly for M i = 0, and M = 0. Since (6.18) is smaller than (6.3) this completes 2 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 ^( ) fr° cyclic 2Q7r x x m 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 < M j 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 ft (i)(xj), and then each of these into two further c 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 X £ u II i hm ( j) Mj-mj u h j = l 100 (j x ~ Uj )pV>(x -Uj), j+1 (6.19) 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 expression "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 2~2 0, x, 0) (see Definition 5.1.5) with both ends opened up (i.e. opened up at 0 and x) would be svcp 2~2 M^(0, b,x,y). X by X 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) = 3=1 s u p S E h (u )hMj- { i x ni 3 nj * h 101 ~ J)P ( J U {2) x ~ bj), which is a product over 3 separate diagrams, F\(L), F (L), F (L), each corre2 3 sponding to M J ^ ( 0 , bj,Xj,0) with an extra vertex on the backbone (see 5.7). Suppose now that m\ — M\ (this is possible depending on the relative size of the Mi). Then two of the bonds have an endpoint at the end vertex of branch 1 and the contribution to Ys { M2 n2 2 - U )h (u )h -n {x - U )p^(x 2 2 3 2 sup 2 s M2 2 - u )p^(x /lMl a:i 2 2 3 = supXX (^^ Xl a 3 ~ U )p^(ui 3 3 - 2 - U) 3 b )h (u ) 2 n2 2 a x h -n (x 3 M3 3 1 3 fcl n3 - x )p^(x bi,b ,b - - b )h (u )h -n {x 3 n3 3 M3 3 3 1 _ a ; i)P ( 2 ) u )p («i (2) h) 3 ( i w Ul II XI X ~Ui)h (u ) 2 u X h -n (%2 < x -Xi)p - \X2 ^ ^ f t M , ^ ) ^ x ( ) from this lace L' is bounded by SUP 3=2 i b nj{Uj)hMj-nj{Xj ~ Uj)p^(Xj h - bj) xj,uj = SW£>Y Mi{x\)p^ {xi -b ) h ) x 6 xi i x JJsup £ 3=2 i b Xj,Uj - Uj)p - (x h {uj)h -nj{xj nj ( Mj 2) j - bj) (6.21) Once again this is a product of 3 separate diagrams F\(L'), F (L'), F (L'), 2 3 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 branch point as an endvertex, asi in 6.5. Ys i n Figure riguic u . o . The i n c contribution t u i i m u u u u n to IU Z _*x )J ( ) from this lace L is bounded by n ^J^)^)^ - w)p(w - z)pV>(x - z)pW(x -z)H 2 x 3 j=l w,z 102 h (xi), Mi x (6.22) 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 (a;i - w)p{b - z) (2) 2 61,62,63 g w,z 3 IJ x pM(x2 - z)pW(x3 - 63) Mi(Xi) h = sup Y p( ~ h)p^{xi ~ w)h {xi) w Ml b l 11,1x1 x sup 62 Y P(°2 z)/> - ( 2 ) ~ z)h (X2 M2 [x ) 2 x ,z (6.23) 2 x sup£p^(x = sup£p ( 3 ) X SUp 63 X - (xi - x sup]T p 6 2 3 ( 3 ) h)h,M ( z) x 3 h)h {xi) Ml (x - b )h {x2) 2 2 M2 2 YP ( 2 ) (^3 - (^3)- 63)^3 x 3 Again this is a product of 3 separate diagrams F\(L), F (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. 2 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< s j=i Mj 6 24 2 which satisfies (6.4) with N = 3 and q = 0. Similarly by (5.49) we have, 3 x i=i M 2 i &i KWMUoHcp-^ffl M 2 j (6.25) 1 d-6 3=1 M3 2 103 ' 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. A s 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), F (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 and AT . The degenerate cases are when rij is the endpoint of more than one bond for some j. 3 2 3 By (5.45) we have, 2 ^ M ( Z ) < * 2^ N N ,N U 2 3 11 : J=l o — Mj 2 = (6.26) <iv (^-^) n4e. 3 w i=iM, 104 2 which satisfies (6.4) with q = 0. B y (5.49) E i X J , . * S w < t ^ ^ n ^ i ^ ' E x N N ,N : U 2 1 = M 1 3 j^i 2 { Mj 2 ZNi = N (6.27) i=i Mj 2 • 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 separately 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 N . a a+ N (u) with two bonds covering the branch point and 7r $ (u) for the contribution to for the contribution to n^(u) 6.4.1 _ ir^(u) with two bonds covering the branch point. 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 h e 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 . .. ., . ., 3 x .... -.. . .. f &u\N , =1 [Mi] , 2 j^imj^MjWj-mj] (6-28) , d-6 . ' , d-4 ' 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? containing only two bonds. A n acyclic lace a 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 < M . In addition we suppose that 0 < m < M , so that the lace appears as in Figure 6.6. It is easy to show that the contribution to X ^ x " ^ (^) fr° this lace is bounded by 2 2 m 71 Y/~2 d ) ( } *- ( hM X Xl hm U2 hM m - 2) M {X3)P^(x2 X2 U h - 3 Xi)/9 ( 2 ) (u 2 - X 3 ) «2 <SUpYz 2 m(u2)hM2-m(x2-U2)h (xi)hM {X3) 7 h Ml b2,b g 3 3 U 2 xpW(b -x )p {h-x ) (6.29) {2) 2 = sup E ^ 2 2 3 h (u )hM -m(x2 m - u )p^{b 2 2 2 -x ) 2 2 X2,U2 ' x sup E M {x\)h {x2)p - (h b Xl,X h { l 3 M3 2) - x ). 3 3 Using translation invariance on the second term, this is a product of two subdiagrams, M ^ ( 0 , bj,Xj,0) with an extra vertex (that we bounded i n Section 5.2.1) and M ^ ( 0 , b j , X j , 0 ) with hMi+M (x) replaced with {HM *h,M ){x) (which we bounded in Section 6.2). Using (5.45) and (6.17) and summing over the permutations of branch labels we have + M s X 3 3 S^wsEE^nr^' „- .« 6 Z i=i &i M l ' Wi + t] M (M0) * 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% i=l j#i [Mj] * [Mi+ Mk] > which obeys the bound (6.28) with N = 2, q = 1. 106 <' 63 1 ) 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 e associated to branch 3 has its other endpoint on branch 2. As before we let m denote the endpoint of e on branch 2 so that 0 < m < M . 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 + N -l + N -l = N. We also let m i 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: 3 3 2 2 3 1. We define F\ to be the part of the diagram consisting of the backbone corresponding 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 F 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 e as part of F . We open up the diagram F at m (leaving a extra vertex on F ). 3 3 3 3 2 3. We define F 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 F where it meets F\ so that the p * p corresponding to e is part of F . 2 2 2 Note that the only properties of the acyclic lace L (that has 2 bonds covering the branchpoint) that are important when constructing of F\,F and F are the bonds that cover the branchpoint, and more specifically, whether or not the endpoints of 2 107 3 x 2 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 e . 3 As in Figures 6.7 and 6.8, this leaves us with 2 (if m i = M i ) or 3 subdiagrams (in general Fi(L) and ^ ( L ) contain an extra vertex), that we have already bounded. B y (5.45) and (6.17) and summing over the possible locations of m i and 108 permutations of branch labels, <N E < ueZ 3d E E ^ T T T ^ [Mil 2 E NI,N ,N 2 i=l jjti 3 ^ Ni E X ( r-*) m <Mj (c^) Nk Cf [ j - mj] M [M + mj} 2 d-4 2 k &i [Mi] [Mj - mj] m^M,- 2 2 [M + mj] 2 k (6.32) which satisfies (6.28) with q = 0. Similarly by (5.49) and (6.18) E Kf4(-) <N^\\M\UN (Cp*-*)" 3 3 X ( 777^6- E E E j t i m j K M j 2 , i J ~ j] M m 2 - 3 3 ) d-4 ' [Mk + rrij] 2 which satisfies (6.28) with q = 1, and completes the proof of Lemma 6.4.1. 6.4.2 6 • 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 h e 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 i n this section. Define 9M M2( U 2, X U X ) w = E E m\<Ml x p 7712 ( 2 ) <M E 2 mi{ui)h -mA l n l) u (6.34) «1>«2 ( u - ui)h (u 2 ~ x Ml m2 2 - w)h -m { 2 This can be seen diagrammatically in Figure 6.9. 109 x M2 2 + W - U ). 2 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: O n 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: O n the left is another called 4-star diagram which is shown i n 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,M {xi,x ,w) 2 supsupgM M ( i, 2,w) X l < CB x x u 2 d —j—j, 2 M* M 2 1 2 Y P9M M2( -i.^ 2,w) su < OB '-* x x satisfies the following bounds 2 2 u d M gM M {Xl,X2,w) u 1 < CB Z 2 x 1 2 Xl i 5 (6.35) 2 d j-, M* x 2 For the second 4-star Lemma we let b(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 i n bounding the diagrams was that p(x) < Cb(x). x Lemma 6.4.3. sup X Yi Y (-< a2 h m m<M u,v ^ h M ~ ^ ~«)P m x ( 2 ) Yi YI Y L ^ > - ^ ~ h m x h M m x U («- )P( )P U ( 2 ) v -y)p( u -y)P ~^, 2 Ml ( - )p( U -v)< z v - y)p( - ) < Cb(z - y)P - d . z v 2 6 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}, •aez 3d 3 x x x l z\Z , i ± z 6 YI YI 771 7HI771 ,d-4' i=i[Mi\ 2 j# .< .[Mj - mj] 2 [M + mj] 2 r r m M 111 k (6.37) 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 < M denote the first vertex (from the branch point) strictly on branch 2 that is the end vertex of some bond / in L. We will 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 < M denote the first vertex (from the root) strictly on branch 2 that is the endvertex of some bond / in L. Here m 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 < m < M . It is easy 2 2 2 2 2 2 112 2 to show that the contribution to J^s ^ EE / l £ m i( U l ) / l i- M m i( a ; i ( ) fr° x m t h 2 s ^ * bounded by a c e s - U )/lM -m(^2 - U )flM [xz) ~ ^) m (us)hm-m (U2 u m 2 3 2 2 3 u - X W(XI P <SUp E u,b 2 - Kl)p 2 m {ui)h -rn {xi 1 Ml ( 2 )(x - 7J ) l 3 3 -Ui)p h SUp E X U )P (2)(X2 ( 2 ) h -m (u2 ~ u)h -m{x m 2 Ml ( z i - bi) - U )p^(x 2 2 2 - 6) 2 u ,x 2 2 X SUp E hm (u3)h (x3)p - Hx3 ~ 0.3)( 2 2 M3 U3,X3 b s (6.38) This is a product of three diagrams, two of which contain an extra vertex and the other with h M replaced with h * /IM - The other lace of Figure 6.11 gives a similar product. B y (6.17) and (6.18), and summing over the permutations of branch labels we have m2+ z m2 E where (5.49) < E Trhor E „ (- ) 6 R ^ This obeys the bound 39 (6.37) with N — 3, q = 0. Similarly using and (6.18) we have E <^ I I M I U E E t (u) M 3 ffeZ 3 d < = i j ¥ ^ R i [M ] fc 2 r / f " % ' [Mj - mj] 2 [Mj + mj] ( 6 - 4 0 ) 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 2.1.7) then we proceed as before with m < M denoting the lace (see Definition 2 2 first vertex (from the branch point) strictly on branch 2 that is the endvertex of some bond / i n L. We leave it as an exercise for the reader that by breaking the diagram at m and 0 we obtain a product of three diagrams that we have already 2 N a+ bounded, giving a bound on the contribution to J2uez^ \uj\ n M(U) 2q from minimal laces of N (N a \\M\U (cp -^Y 3 2 ° 2 q 1 2 v i=i [Mi] 2 j # _ _ J v m j < M [Mj-mj] 113 l _ * [M + m ]2 k j (- ) 6 41 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 will 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^~ (with two bonds covering the branch point) that we have 2 already bounded, together with an extra factor of 8 ~'*. There are many different cases to consider, depending on which bond (e or e i ) is removable and how many endvertices of that bond are an endvertex of some other bond in L. We will present the argument for the three cases where e is removable and leave the others as an exercise. From this point we assume that e is a removable bond. l 2 2 2 Case (0). Suppose that neither of the endvertices of e are the endvertices of any other bond in L. Then we use Lemma 6.4.2 to remove the bond e and obtain the extra factor / 3 ^ , 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. 2 2 2 _ Removing the bond e from the lace L leaves an acyclic lace L' — L \ e € 2 2 114 C~ N l 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 i n Section 6.4.1. u 297r Recall that we bounded the contribution to ]C«eZ l il ^-(^) fr° ch 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), ||/i ||oo < ^ \ d | | r i | | i < C when q = 0, 3d a m s u n m m 2 and in addition the bounds sup \x\ h (x) < 2 x m a 2 ^ and J2 \ \ ™( ) mC x 2a - x x ml Cma when q = 1. Let JVj denote the number of bonds that contribute to diagram Fi i n 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 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 are the endvertices of any other bond i n L). We can write an explicit bound for the contribution to Y l x f ( ) fr° * lace L in terms of a diagram F(L) consisting of various convolutions of p's and h ' s . In particular that diagram contains a term p^(u — u') obtained from the bond e . 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 P , 6 i Z ) x i ^ ' ( i,h,x 0), where rh e 'HM -m ,Ni, and n denotes the location of the extra vertex. 2 2 2 n N m x n e m 2 M s u lS> ni a a i u 1 1 x As i n Figure 6.15, F'(L) is the diagram Fs(L') with the first factor h (v — v') of the backbone being replaced by a diagram F ^ L ) ^ — w). Thus F'(L) is bounded by s u p M^'*(a ,b ,x ,0), where M^'* is the diak a3]63 3 115 3 3 Figure 6.15: A non-minimal lace L and the subdiagrams Fi(L') (the bottom left subdiagram i n the second figure) and F'(L). gram M^ ^ with one specific factor h (v - v') being replaced by the diagram 3 k F'(L)(v - v'). Furthermore / ^ ( L X u - v') is itself the diagram F (L') with one 2 of the first three factors h[(u — v!) on the backbone being replaced by g ,i(u — k u',v—v', v'), and with an extra vertex at n . Therefore F^LXu—v') is bounded 2 by s u M^™(a ,b ,X2,0)(v P a 2 ) 6 2 2 the diagram M^ \») - «'), where M^™(.)(v 2 - v') is with one of three factors hi(u — u') being replaced by 2 9k,l( — u',v — v'iv'), and with an extra vertex at n . u 2 A + N It follows that the contribution to Ylx ^ M ( ) fr° non-minimal acyclic laces such that: the special branch is branch 3, e has its other endvertex on branch 2, e is removable, and has no endvertices i n common with any other bond, and m* < m i is bounded by m X 3 2 E c E NI,N ,N : 2 x E E SUPY^T'^M^) rni<M m &'HM -rn ,N ni<M -ni ' ai 3 1 E 1 1 l 1 1 1 xi sup£M^ '*(a ,&3,Z3,0). 3 ) 3 (6.42) Here M~^^'*(a,3,63,£3,0) denotes the diagram M^\a ,63,a: ,0) 3 with the 3 first factor h (v — v') of the backbone being replaced by m E J2 sup£M^ rn eUM ,N n2<M2 ' a2 b2 2 2 2 2 ) ' N 2 (6.43) > ,6 ,* ,0)(<,-^), 2 2 2 x 2 and M j j ' ' ( a , b , x , 0 ) ( v - v') denotes the diagram M £ ( a , b ,x ,0) with a specific factor hi(u — v!) being replaced by g ,i(u — u',v — v',v'). 2) n2 9 2 ) , n 2 2 2 2 2 k 116 2 2 We prove that (6.42) is bounded above by (C/3 -T)* (C^ -?)"' cf-'-f Y. E m ^ M i [^1 - ^ l ] N N ,N : U £ 2 3 (Cfi-i)"' 2 2 M 2 [M + m ] 2 2 3 2 X 1 JV; = JV - (6.44) which satisfies the bound (6.37) of Lemma 6.4.4) with q = 0. By (5.45) we have Y M M r m i E , N » < M i - ™ i l n i S U ' 6 P E * 1 M (oi,6i,a;i,yi) < — m -3=5- [Mi-mi] *i By Remarks 5.2.1-5.4.5 and (6.17), the bounds sup ,_ / hi(v - v') < t lI 2 (6.45) and hi( — ') < K, together with bounds on the rest of the lines in F (L') imply that v 2~2v-v' v 3 m € ^ M 3 ^~*}2i- C ( E sup £ M ^ ( a , 6 3 , x , 0 ) < f 3 + m i , i V 3 a 3 ' b 3 *3 3 [M + 3 mi] (-) 6 46 2 Therefore by Lemma 6.1.4, to prove that (6.42) is bounded by (6.44) it is enough to show: sup YI ~ ' meH ,N n2<M *' * x v v a M2 ^ 2 2 d 7-2 M k ^pY (^b ,x ,0)(v-v') M N2),n2,9 m 2 2 b 2 (6.47) ~6 2 2 and E E (N ),g,n, E sup£M^(a ,6 ,x ,0)(,-,') r 3 v-v' me'HM ,N n <M ' 2 2 3 3 x a2 b2 2 2 2 3 (6.48) M 2 2 Again by Lemma 6.1.4, to show (6.47) it is sufficient to prove CB v', w) < -r^CB '^ 2 sup sup g (u -u',v- 2 ktl v—v' u—u' ™ 61/ CB 2 (6.49) ' and sup Y 9kA u ~ '^ ~ u v 117 v > M < ^T P ~^C 2 (6.50) Similarly, to show (6.48) it is sufficient to prove E sup g j(u - u,v , , u—u' / / \ o 61/ 2 1 —5-,- - v ,w) <CB Z k CB (6.51) * 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 is also the endvertex of some other bond (by definition of a lace, in the case we are considering here the endvertex of e strictly on branch 2 could only be the endvertex of e ) 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 and obtain the extra factor /3 ~~^, (see Figure 6.13). 2 2 3 2 2 Case (2). Finally suppose both endvertices of e 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 B ~ ~^. 2 2 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 M 2 1 m <M l 2 ~ 2] M 2 m 2 2 [M + m \ 3 2 2 For the q = 1 case of Lemma 6.4.4 we use \XJ\ < 2N'Y^, 2 2N _ 1 \ j,i\ u 2 ( tms gives the TV factor) where the Ujj denote the displacements along the backbone of 2 diagram Fj(L'). If the extra factor \UJJ\ occurs on a part of the diagram F'(L) 2 where F\(L') and -F (L') are joined by g ,i then we can use Lemma 6.1.4 to include 2 k a factor | u | on one of the lines in the 4-star lemmas, and proceed to get the extra 2 factor cr 1| Af || OQ. Otherwise the extra factor p- ||iVf ||oo comes by applying Lemma 2 2 6.1.4 to the diagram Fi(L') where the |uj,/,| is attached. 2 This proves that the contribution (up to permutation of branch labels) to Ylg\xj\ ir^(x) 2 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 - ^ ^ M £ 2 2 x L -—Z=E- L m <M [^2-m ] 2 - z . i [M +m ] 2 2 3 I (6.54) 2 2 This completes the proof of Lemma 6.4.4. • P r o o f of L e m m a 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 according to these restrictions gives rise to 4 terms. One such term is 2 E ]T X h m i m 2 < . ^M i m YI 2 <Mi (2) x 2 Ml (U2 ~ w)hM -m 2 ~ Ul)p (u mi(ui)h -mA l h 2 [x + W - U ) 2 2 (6.55) U 1 X S U p / i M - m ( « 2 +UJ - U ) ( h u ^ 2 2 2 m i * p ( 2 ) */j 2 <—i"—j M 2 x 2^ - Ul) d=4<°p 2^ ^ ^ ^ ( ' " i + m j ) 2 m 2 J (ttf) ' —T—r» d 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. B y 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 to leave us 2 119 with 4 terms. The most difficult to bound is Y £sup Xl x h X .Ml m 2 (u - w)h 2-m SUP m2 2 h {ui)h -mAxi X mi M (x + w- 2 2 -^"zC X X X Af, Xl Z2 ! X /i T O 2 J ^ -S M i ^, M i 2 CB < ^ (^2 - w)hM -m 2 2 u) 2 M -m {xi ~Ui)p {u h 1 {2) 1 2 - ^ l ) K i ,Uo (x 2 2 + W - U) 2 L P 2X ^ M -rn U ~ ui)pW(u - ui) Mx <^I«l.«2 (X h 2 2 2 +W 2 - U) ( E s u p E A/f,i i i , ^ M m2<Y C/3 < +W h 2 2 m2 2 (6.56) 2 XI X ( <Ml«l,«2 P 2 < ^ - s u p , £ '1 m <2 £p L - U) 3 : 1 ~ «l)P Xl ( 3 ) (n 2 ( 2 ) ( « 2 - «l)/lm '(«2 - ™) 2 - xi)h {u m2 2 - w) J 2 X ^ 2 sup 2 2 2 m C/3 2 x Xl x p( \u -ui)h (u -w) i.r. \ ,ito . A,f, \"ll> j SWp 2_^ M -m {X u M* M - ui) h ™ x sup < Mi-mAxi zC ^ ^ M r C/? ~T 2 m / < C/? M 2 -^KCB'-ir, 2 x 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 Xl,X2 E E m\<M\ 77X2 <M X h m2 ^ ( 2 ) ( u - Ui) 2 U\,U2 2 M 2 Y 2 E mi(ui)p^{u ~ Ul)h (u 2 m2 2 - w) 2 Y Y 2 m <Mim <M 2 2 h u\,u 2 mi<iW"i m <JVf 1 2 Y m i < M i m <M <C - ui)p (u - U))h 2-m2 {x + w - U ) Y C hmiMh-Mi-rmixi E hrnAuijiiMx-mAxi - u i ) p ( 2 ) ( u - ui)h (u '2 m2 2 w) ^' ^ 6 57 ui,U2 2 [mi + m ] 2 2 2 <C/3 -^". 2 • 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)-«)V(«-yV(2-«) ( 2 V v.\v-y\>^ <c E . v:\v-z\>^ p ( - ^ - y}p( - ) + {2) v u z < Cb(z - y) (p( ) (z-u)+ 3 z p v Y C p ( - ") ( - W {2) v u p v y z (y-u)), ( 3 ) (6.58) Therefore for the first bound, s u p X E Y ^ - ^ '~^YP^^~ )P( v)p( '~ HM m<M M X U u <Cb(z-y)swp <C^b{z-y) Mi HM v ) z v v E m<M u £ . m Y ^ HM ^ M 2 < HM ' )(pW(z-u) ^ - y ) ^ + ' pW(y-u)) (6.59) , Mi m<f where we have used Proposition 5.1.4 and the fact that either m > y O r M - m > 4f 121 For the second bound, we again use Proposition 5.1.4 to get, Y Y Y ^ ~ ^ hm x m<M hM m ~ )Y ^^ x U P u ~^^ ~^^ ~ ^ V u p v y p z v v <Cb{z-y)Y Y Y h ^ m h —m(x x m<M u <Cb(z-y) Y M U Y ^ ){p ( -u)+p^(y-u)) h m<M u (3) z ( 6 - 6 0 ) 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 definition of BN(M) from 6.2. Lemma (4.3.3). There is a constant C independent of L such that Y Y N B (M) < 3 N C N Y N'WMWooBNiM) 0 2 Hoc < , J 10-rf ( Y ' 2 VO , log||n||oo, M M<n and (6.61) tfd^io ( g 6 2 ) i/d=10. Proof. Summing over ./V first gives the factor CB *d ,forsmall enough 8. Summing 2 over each Mj separately we have, -ir<HLr> Y M: > Mj Mj nj M 2 (6-63) 2 and ( 3 Eiwioon4^< M<n j=iMj 2 I H I o 10-rfyo ° 2 [log||n||oo, ' (6-64) ifd=10, as required. This verifies the Lemma for the first component of 122 BN(M). For the second component of BJV(M) (ignoring the sum over j) we sum over M i separately to get V __J ^ - V M ^ ^ M , m 2 "l 2 ^ [«i] 2 2 2 ^ 3 2 3 2 1 1 [M -m } M ,M m <M 2 L < M [ M - m ] 2 [M + m ] 2 2 2 2 - { 6 6 5 ) 2 [M +m ] 2 2 3 2 We leave it as an exercise for the reader to show (by summing separately over m < ^2 d > ML) that 2 a n m 2 E 77; m <M [Afe E M M 2 l 3 2 2 T^TT, - 2 WI2] rer^- [M + m] 3 ( - ) 6 66 2 2 Furthermore E M:M >n 2 <c 2 2 2 M 2 2 2 M 2 2 ~ rn \ M Z\Z M ;> n 2 2 2 M 2 m 2 3 + 3 m m ]~ 2 1 2 2] _ „ r , d-4 [M + m ] 2 3 . 1 2 >^2. t 2 M - m M 2 2 3 2] 2 [M 77T , t m + EE E 3 M 1 fc.- •' /E [M ] [M 2 E E E 2 2 2 t 2 - 2 1 , d-6 Mz <K ^ M >n 3 1 mo< 3 — + c —^ [2 M EE E M >n 2 1 m <M 3 6 2 y y M >ra S C - m ]°'2 [ M + m ] V mt<M W 2 l y E i = ± M 2 2 d-4 T 2 2 d-6 I + m] ^ 2 + m] 3 C M 2 n >^ M \M >m 3 2 2 m > Q ^ M 3 [M2 - m ] 2 2 [Ms + m ] 2 2 y , [M , d-4 3 2 2 2 2 + m] ^ [ m >-g2 m 2 1 ] 2 ' J 2 "-2 (6.67) Similarly we get E M : M 3 > n 3 ~l=fi E 77; M i [^2 - a m <M 2 2 ,<j=fi m] 2 2 TT; [M 3 ,d=4 + m] 2 2 < ~ E F n ( 6 6 8 ) 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 M < E „ ll^ooll—1=!M n r m 2 1 1 2 < M lfcfiri [M 2 ~ 2 m] > 2 2 i|n||oo 2 ^ { log ||n||oo, 1 [Af + 3 , if d + 10 if d = 10. 2 m] 2 (6.69) If 11 Moo 11 oo = M\ this follows easily by summing first over M i and using (6.66). If llMHoo = M , as in (6.67) we get a bound of 2 c E [M ; ^]"^ bE E Ah 2 M 3 m 2 < M 2 _ 1 [M + m ] 3 V 2 (6.70) +C EE m 2 E M 3 m 2 \M >m 2 W2 2 ~ m] 2 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 . After permuting the labels 1,2,3, this verifies the 3 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 i x a'skeleton network jV(a,n), with a £ E and recall Definition 2.1.1, where b is the r branch point neighbouring the root of jV. Let M. C j\f(a,n). If U t € {—1,0} for s each st, then trivially for any A C E ^ , n n i + «], [1+< 1 u (6-7i) so that in particular for any finite collection of disjoint sets G j C E^vi, n [ +u,t]<Yi i ste^M i steGi n [i+^i- (- ) 6 72 We will 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 t (K) M < K*". 124 (6.73) Proof. Label the branches of M, 1,..., #M a d write rij for the length of branch n 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. will receive multiple labels. Using #M J[[l n + U ]<\{ st steM ' ll + Ust], (6-74) st e A^J = s < t < ni »=i 0 < we have that #M « M ( y ) < / » ( o ) ^ n & < W ' £ ( 6 7 5 *=1 2/i y-£.1&(#M> 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 n E (n E ( t +^) 1 1 - n t +^)> 1 ( e ?6 ) where U t is given by (4.15). In this section we prove that a E ^ ( ^ E T A y eeE n (- ) 6 77 2 e Let M denote the branch of M corresponding to edge e of a and let lZ ' = {st € e,e e TZ : s G A/" , * G A/'e'}- We claim that when U G { —1,0} for all st, e st E i-n[i+^i< n h - t i + c / ^ (6.78) e,e < e, e' e BJV" : Me n A/"E' = 0 m <n m/ <n/ e e 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 M r\N > =0). To verify (6.78), e e observe that each of the quantities 1-Y[[l steiz + Ust], 1- [I [1 + U ], 8t steii ' ' e e 125 -U i e > m e W t < ) , (6.79) are either zero or one. Suppose the left hand side of (6.78) is non-zero. Then there exists some st G 1Z with U t = — 1- B y definition of 1Z, st covers two branch points s of j\f so that st G TZ ' for some e, e' that do not have a common endvertex. For e e> this e and e', we have 1 — YlsteH ' ' [1 + U t] = 1 and the first inequality is verified. e e s Now for fixed e, e', if 1 — Ylsteii ' ^ ^*] * ee s sn o n ~ z e r the* there exists st G o 1 1Z ' e e> with U t = ~ 1 - But s = (e,m ), t = (e',m i) for some m < n , m i < n i so that e s e for this m and m ' , — f( ,m ),(e',m^) e e e e e e e 1- This proves the second inequality. = e Examining the second quantity in (2.4) when U t G {—1,0} for all st we have, s o< n [i+u }(i-H{i+u \\ st st£Etf\1l < * E m <n M n Mi = 0 m / < ni e,e'6%: m <n rrv < e ' i+c/ e E e e M E [-^:;<]n n e n [ ^ e x s,teM : /#e,e' e n [ ^ s,teM • m <s<t< n e e e n e [i+^t] mi 0 <s< t< (6.80) f 0<s<<<n^ i+c/ n s,t£jV •• 0 < s<t<m [i+^i st£E \Tl e e .A/e n A/" / = 0 ) n E e st st£Tl e.e'GBjv-: e x \ n [i+o*t], m < s < t< ni e e e where we have used (6.72) in the final step. Breaking up w (in 6.76) at every branch point and at (e,m ) and (e',m r) e e and applying inequality (6.80) we obtain E^(y)<P(o) 2 R - E E 2 V V e hm u Ve hn e X h , (v! me where ve(y) = V i{y))hn ,-m , Ylf^eVf e a n < ^ II M w ) m <n m / < ni e e e e /#e,e' e e e e : + y -u) Y ^ ~ ^ e~rn (v (y) x E e, e' e M n A/" / e t n e ^ ^ 6-81 (Ve' + Ve'iv) ~ u')p^{u - u ' ) , notation / - w e 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 nEM*/) E e , e ' £ Bjv = A/ n A/„, /^ > ' e e e SU X E P v,w * ( ~~ ) n -m E U hm v h e (v + y ~ u) e e (- ) 6 82 x h , (v! - w)h ,^ , (w + y > - u')p \u - u'). {2 me ne me e By translation invariance, the last two lines of (6.82) are equal to the sup over z °f Ylxi x2 9n ,n i(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 i n Lemma 6.4.2. Break up the sums over m i into the two terms m i < ^ and m i > ^ and similarly for m . Then we are left with 4 terms, one of which is e e 2 E E m {u\)h h e n -m e Xl,Z ,«l,«2 m 2 e "V > -f- x h , (u - w)h - (xi me = ne £ e > Se. £ - u ) p ( u - v!) (2) me h (ui)hrn ,(u me - w)p( \u - v!) 2 e r; 1 % X (6-83) E^ne-meO^l - «l)£^n ,-m ,(a:2 + e X\ < E m e ~ U) e 2 X2 > [m + m ¥ e 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' this is bounded by a e constant times m C(3 ~% ^ 2 e [m j 2 >^ e " [m /]V 2 C/3 -^ 2 d-8 2 e ^ e C/3 ~^ m iCfi 2~T- 2 1 d-8 ^ , 2 e' The other 3 terms give the same bounds by symmetry. 127 e (6.84) 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 - ^ £ « # ( * ) < £ y e£E d-8 (6.85) ' 2 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 b o u n d of L e m m a 4.2.1 Recall that <fij/(y) was defined in (4.23) as £ M(V)= E W(u)U u££lM(.y) W &) E seAf R eT{oj(s)) ^ (m))n^((AT\5 (m)) ), A A i meH s i = l nb (6.86) where Un = ({rh : 0 < m , < n i = 1,2, 3} fl {rh : 0 < m,j < n - 2, i € G}) \ %n u b { b (6-87) and Un = {rh : 0 < mj < -± i - 1,2,3} n { m : mj < m - 2, t G G } . o As in Lemma 4.3.4, |<%/-(y)| is bounded by b (6.88) 3 C E EIIE E men . fi ~ ')*AAU ) (6.89) a <^EE E^wEfiE^-^VGW, i=l Vi where j\ff = (j\f \ <S4), and y 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 Vi oo 3 <?E E E ^ n ^ fi ^ I m e ^ j «=i oo =c£ £ £4ro N=lm€Hri oo 3 b (6.90) fi 3 <E" iEm:mj>?jE B„W)<T, 3 =1 128 2-% rR 0 d-8 ' 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 (6.91) Xl^(y)l<c£4rj=i n . It follows immediately from the definition of <fi%-(y) that (6.92) ojefijviy) serf R er(ui(s)) s 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 £ ^ 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 b be the other branch point in jV incident to branch j\f . Let b e e Ep/j- = {r G £tf '• for every e G F, Ab{T) contains a nearest neighbour of b } e (6.93) Then, £n>= E res w E n>- E n^> ster (6.94) where some of these sums could be empty if G ^ {2,3}. Thus, E n>« < E (6.95) E G(Af) F C F^0 Note that if r = 4 then one of £ ^ ^ or £ y ^ is empty and £ may also be true for r > 4, depending on the shape a. b b 3 b 2 .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 is empty. This 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 6 for some e € F and b € Ab(F), or E e • s,t G Af for some e E F. e Let <SF(F) be the largest subnetwork of Af covered by Tp C T . Clearly r | s ( r ) = Tp is a connected graph on SF(T). For each e E F , <SF(F) by definition contains a nearest neighbour of 6 in Af, and may contain b itself. Since TF contains at most one bond that covers 6 , if b G SF(T) then it is not a branch point of SF(T). Moreover if F = {2} or F = {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. F E e E 1 e F i x Af,F. Write S ZZF following properties: if <S C M is a star-shaped network with the (a) for every e G F, S contains a vertex v that is adjacent to a branch point 6 of E Af, and (b) S contains no branch points of Af other than 6 and b , e G F. e Such star-shaped networks are exactly those for which there exists T G GJ/ such that S = 6V(r). Define C* to be the set of laces L on S such that 1 S 1. For each e in F, if 6 G <S then there is exactly one bond s t G L covering e e E branch point (of Af) b ^ b, and that bond has s or t strictly on branch Af e e 130 2. If F = {2} or F = {3} then there is exactly one bond in if JP = {2,3} there are at most 2 bonds in covering b. covering b, while 3. L contains no elements of 7c (i.e. no bonds which cover > 2 branch points of AOThen recalling the definition of L r from Definition 2.1.4 we have n^=E E r F M n>« E serf •S (V) e r6£ , ^er b -F,M £* F • E E »,: =S **er \JUst .steL F E r e g~ ' n con H". r =L (6.96) •. ster\L =L n «, u E r* F n E .steL h E s't'er\L = 5,Lr 5 (r) • E n ^ E re -V. egS,M\S " stev* 5 (Lur*) = s F where $SM\s = i 6 N N F f o r e v e f y s t > [sG5,t€^\5]or[i.e5,ae^\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 1 of { s \ t i , s t } , we have Jlren Uster st = Ti^eiA + st]- Let Cf* be the set of laces in C* consisting of exactly N bonds. Then (6.96) is equal to : G r u u s E ( - D " iV=l E E s\z M Lec ** LsteL 1 F n stec(L) n. st n s e s,t e J\f \ s •. S (LUst) = S,st£K F (6.98) 131 If U t € {—1,0} for each st, then each quantity involving U t i n (6.98) is s s nonnegative and we have E II"- < E E E n N=l -Ust i n .steL SCFJVLeCg* i + u ° t ] stec(L) (6.99) n n t + 't\ 1 u st £ E ^ i si £ TZ ( s ) 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 b and 63 if F = {2, 3}) each of the form 2 00 / EII N=l n +(n /-l)\ m E e \e€F m =n -l / e e mi=0 II £ / E e n -l\ n Eh e \e£{2,3}\F i 1 + m =0J e st} U (6.100) stec(L) n - n n Me((AA\S|)'0e : 0 < s < £ < n^m)") where e' denotes one of the two branches (other than e) incident to b , e star-shaped network defined by (2.12), and (j\f\S^)' % branch J\f i is being removed if m >n . e e e is the denotes the fact that part of In addition ne(rh)' is the length of branch 1 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 132 Y^rze^ l i s t e r Ust is bounded by a constant times oo £ / n +(n /-l)\ ni E I IE F C G(jV) N = 1 e £ e \ e e F m =n -l e j e / n -l II E i e x m i = 0 \ e e { 2 , 3 ) \ F m =0y c F#0 E n[ ^ i+c/ steL - n n <=1 (6.101) **ec(L) n M€((Jv-\ $£)")*: e6(JvV£)« 0 < s < t < rie(m)") 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 Ch _^yi(»), with displacements summed over. O n the latter on which we use (5.6) with I = 1 to bound | | / j „ ( , 7 j ) ' i | | i by a constant and we obtain an upper bound on (6.92) of a constant times n r oo E / n«+(n ,-l)\ E e £ 1 1 F C G(J\f) N=l \eeF £ ( n -l\ n E E * e J m i = 0 \ e e { 2 , 3 } \ F m -0/ m =n -l e m u e e (6.102) F^Q m - n n * • v By Proposition 4.3.2 this is bounded above by oo / n +(n /-l)\ \eeF m =n -l E EWlI E c F C G(J\f) N = l e e e e ( n -l\ II E e \ee{2,3}\Fm =0J e (6.103) 2—1 F C G(jV) E J mi=0 F ^ 0 <C m e€F • n d-8 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 M (D(M (R ))) (defined by (1.17)) to the canonical measure of super-Brownian motion ( C S B M ) . d F F 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 Theorem 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 moments 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 Branching Random Walk 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 ~ N t > ' " - > = { ( a , . . . , a ) : a, G N , m G Z+}. 0 =0 1 m 0 134 m (7.1) We start with a single particle, and set ao = I for all a. Let \(ao,... ,a )\ = m m be the generation of a and write a < a a — (ao,... ,a%) for some i < \a\, if a is an ancestor of a. If a — (ao,..., a _ i , a ) and a = (ao, • • •, « - i ) i.e. m m r a then we say that a is the parent of a and a is the a^ child of a. F i x 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}) ] = < o o . 2 7 Let : a G 1} be i.i.d ~ Y random variables. Let Go = {(1)} and {Y a for each m G N we define G m a = (ao, • • •, a -i) m recursively as follows. A t time m~, each particle £ G - i gives birth to Y children m a (ao,.-.,a -i,l),...,(ao,...,a -i,Y ), m m and immediately dies. We let G be the set of particles alive at time ra. Note m that each particle a G G m (7.2) a satisfies \a\ — m and has a unique parent a G G -i by m = 0 then G = 0 for every n > m. A well known result definition. Clearly if G m n 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 ^ { G ^ 0}) = 0 so that G = U ^ G 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 ' s is finite for each m. Therefore Q is a countable set. = 0 m = 0 m m 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 as follows. Let Cl = {B : G H - » Z ,B((l)) = 0} be the set of maps from G to Z 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 d d d G P'((G,B) = (G*,B*)) = P(G = G*) D(B*(a)-B*(a))I . , , (7.4) {B &G } (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 M (R ), d F iG N n by n,(G,B) x y C[ = ^ x:y/C nx£B(Gi) 2 We extend this to all t G M+ by X M (D(M (R ))) F »' (H) n ' j B ) = X ^' . n B) Finally we define ^ G by d F n , ( G ( G H}) , = nC' P' {(G, B) : {X?' ' } {G z B) tm+ 135 He B(D(M (R ))). d F (7.6) 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), M (R )) —> d F [0, 00] is defined by S {{X } t ) = inf{s > 0 : X = 0 } , teR+ s (7.7) M where 0M is the zero measure on R satisfying O M ( ^ ) = 0. Let d D*(M (R )) = {{X } e D([0,oo),M (R )) Co(M (R )) = {{X } € D* : X = 0 ,X. is continuous}, d : S({X }) > 0, X = 0 d F t d F F t 0 t t M V* > S}, M (7.8) with the topologies inherited from D([0,00), M {R )), C([0,00), M (R )) (the topology for C is the topology of uniform convergence on compact sets). Note that d d F ,fi' (S< e) > nC' P' (x ^ n n F = 0 ) = nC' P ( G B) 3 M > nC' P ( G i = 0), 3 = 0) [ n e J for n > ^ 3 ( 7 - 9 ) —> 00 as n -4- 00. Let us now also define the finite dimensional distributions of v € Let R > 1, and t = {ti,.,.,t } R M (D(M (R ))). d F £ [0,oo) . Let h : D(M {R )) R £ F denote the projection map satisfying h^{{X,}) = (X^,... Xt )• dimensional distributions of v are the measures vhZ e M {M {R ) ) d F vhZ (H)=v{{X.}:hA\{X.})eH), d R n u (S > e) < • n 00, S > e) ==> ^ o o ( ) S > e), - 136 n (7.10) M (D*),, a on D* if for every e > 0, Vn € N U 00, and as n ^ R given by {u : n G N U o o } C the set of a-finite measures on D*. We write v ==> n R F Definition 7.1.1 (Convergence in TJ*). Suppose v (», d F H £ B{M (R ) ). 1 w F M (R ) Then the finite R 1 F -> d 00, (7-11) where the weak convergence in the second condition is convergence in We write v ==> MF(D(MF on D* if for every e > 0, n 00, v (S > e) -> 1/00(5 > e) < n (7.12) and for every m G N and t G [0, o o ) , m u (S > e) < n «, Vooh- -1(•, f / i _-1 . (•, S > e) ==> 00, 00, and as n —> 00, Vn G N U 5 > e), n (- ) 7 13 where the weak convergence in the second condition is convergence in M i ? ( M / r ( l R ) ) . rf 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 by (7.5-7.6) C[,C', C' such that the measures p!n defined Z for the branching random walk model satisfy (a) for every s > 0 there exists R G MF(Mp(R ) \ {OM}) such that for every D s s>0, p! {X G . , X ? 0 ) =2* Rs{») on M (R ), and R (M (R ) D n s s M \ {0M}) = — • D F s F 7s (7.14) (b) There exists a a-finite measure No on CQ{MF(R^)) D*{M (R )), F 1. N ( X 0 such that u' ==> No on n and for every s > 0 D G ;S > s) = R (m), S 8 2. P' ( { X N , ( G T ' } G •\Xs' ' B ) {G ^0 ) B) M N ( « | 5 > s) on 0 £>(M (R*)). F Definition 7.1.3. The a-finite measure No on C*([0,00), Mjr(]R )) defined by part d (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?' G M (R ) T therefore {X?' } T D X t r = v k l G D(M (R )) D teR+ F X x:Vcr vnx£Ti 2 137 F ^ , a n d ^ defined by =^ . (7.15) F G Mi(7o) defined by F({T}) = • p G M (D(M (R ))) F P(0) (7.16) ' defined by d n W(T) R Pn(H) = nVAp(0)¥ ( { T : {X?> } G T teR }) . (7.17) 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 : X ^ = 0 n for all t > e) M n > nVAp(0)P(T : x g j (7.18) = 0 ) M n > nF74p(0)P(T = {0}) = nVAp{0), i.e. p (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. n T h e o r e m (1.3.1). There exists LQ 3> 1 such that for every L > Lo, with p defined n by (1.17) the following holds: For every s > 0, A > 0, m G N , t G [0, o o ) F : Mp<(R ) d m — > •C m and every bounded by a multinomial and such that N o / ^ ( ^ V ) = 0, 1. E llnh. 1 Xs{l)F{X) E^ X (l)F(X)] a , (7.19) and E Hnh 1 r F(X)I{x (i)>\} F{X)I (i)>\} s {Xs (7.20) We show in Section 7.4 that the convergence of the survival probability (together with our results) would be sufficient to prove the following conjecture. C o n j e c t u r e 7.2.1. Let p n 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, fin => N , 0 on D*. (7.21) 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 ISE 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 . Thus scaling space by n~ 2 should be equivalent (in terms 2 of the leading asymptotics) to scaling space by N~~i. For fixed N e Z+ and T G TN, define = N- E XN,T <-) 7 22 where D\ is a constant defined in [7]. Since T contains exactly N vertices, X ' is a probability measure on R . Keeping N fixed, choose a random tree according to N T d which is independent of p. Then X is a random probability measure described N,T by/x^. Define X G M i ( M i ( M ) ) by d N TN{A) = p ({T N :X G A}), N,T • A G B(Mi(R )). (7.24) d 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 ) ) is called integrated d super-Brownian excursion. This is a statement that for all / G Cft(Mi(E )) (i.e. / d bounded continuous on M n j fdl N -> j fdl. (7.25) Derbez and Slade [7] prove (7.25) for functions of the form 4 » - / e*M<&), 139 (7-26) 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 , and y € Z ( ~ \ d 2r the set T£{y) of trees of skeleton shape a with skeleton 3 r displacements yi and the generating functions G Tc y { 'ffc E ) = E neZ ;- ^! 2 v^)- (- ) w ? 27 T€7V(*,a)(0,iO 3 They then write 2r-3 G '"(/?) = V ~ r r 2 ^ TT + E ' JH), r a f\ + C7 (l - % ) \ + C ( 1 - 0 ) Cn 2 = 2 P S ' (7.28) C for specific constants C\, C 2 , C 3 . They show that .&'-(«;) is an error term when: • r = 2, 3 for all p < p and c HCIloo < 1> and when • r > 2 for all p < p and £ = 1c 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 = p , (7.27) implies that the coefficient of Hfj^ c £ y (T) W = a ) (j=). (" J in ^ ' " - ( ^ j ) is (7.29) TeT (n, ){Q,y) N a Using the fact that for x < a, = ^ YlnLoia ^ i n g that E is an error term, (7.28) implies that this same coefficient is approximately an< T/r-2 2 £-3 1 yr-2 J_^ a s s u m 2 _ 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 {X } G C*([0, o o ) , Mp(R )). B y definition of the survival time, S, we have that X — OM for every s > S, and Xt / OM for all t G (0, £ ) . t d t If 5 is finite (note that under No, S is indeed finite almost everywhere) then by continuity on [0,5], s u p X ( I R ) < K for some 0 < K < oo. Define a measure n*) = on R by d f t d poo Y(.) = / Jo X (»)dt. (7.31) Then by the above discussion, Y ( M ) < f bility measure V on R by t d S Q Kdt < o o , and we may define a proba- d where for A G tf(R ) and C > 0, £ = {x E R : Cx G A}. Now if we choose X randomly according to No ( • | Y ( l R ) = l ) (which is a probability measure on D(M (R ))) then V has law 1. d d fl! t d F 7.3 Finite dimensional distributions 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 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 finitedimensional distributions. n D e f i n i t i o n 7.3.1 (Tightness for finite measures). A set offinitemeasures 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(K ) < n. A set F C Mp(E) is tight if it is spatially tight and sup^p fi(E) < oo. c 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) 141 (weak convergence). Proof. Let {/i } C n we have \i ni F. n {F) If there exists a subsequence /_*„, such that ni — > 0 then —> OM by definition (for every bounded continuous /....) and we are done. So without loss of generality there exists no > 0 such that mi fi (E) n = no- n Therefore Mn(*) (7.33) are probability measures. Let n > 0. Since the \i are (spatially) tight there exists n K C E compact such that sup p (K ) < nrjo- Therefore c n n su p in = s u p ^ < ™ (7.34) V, n so {P } is tight as a set of probability measures. Therefore there exists a subsequence P n( n n Hn{E) 0 n Pn k P & Since {p } is tight, {fj, (E)} is a bounded, real-valued sequence, and there nk nk t 1 fore has a convergent subsequence /J„*(J5) —»• C > no- So P\ ~ F 'oo and k • fi *(E) —> C > 0 and therefore / j * —> 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(R ) convergence determining class of bounded continuous functions (j) : R C (i.e. u -» v in Mp{R^) if and only if v {4>) -> v{4>) for every (f> € J ), that contains a constant function, (f)(x) = C? / 0. Hereafter when H G M (D(M (R ))), we interpret nhZ G M ( ( M F ( M ) ) ) as the measure on the space consisting of only one point x, that satisfies phZ (x) = /z (D(M (R ))). The main result of this section is the following proposition. n n d d n n 7 d F l o! F 0 F l d F P r o p o s i t i o n 7.3.3. Let a > 0 and fj, ,fj, G Mp (D(Mp Suppose that for n every I G Z + and every t G [a, oo)', m G Z+ we /lave 1. there exists a S = S(i) > 0 suc/i £/iai for all 9i < 5, E -i [e^<=i h < oo, 2. /or every 0 = O n , . ' . . , <fo } G F^lLi> m( nil < oo, (7.35) in Mp ({Mp(R )) ). i=ij=i i=ij=i where an empty product is 1 by definition. Then for every m G N and every t G [a,oo) , p KZ m l n -> / x / i l 1 d The importance of the I = 0 case in the Proposition is evident from the following example. 142 m Example 7.3.4. Let u' n G Mp (D(Mp(W ))) be the measure that puts all its mass i (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, = fx' + H i.e. n n Mn(*) = Then E, nUn^i /i = ^ i - (7.36) =n!=in^ *i(^) = n U n S i ^ ( ) » 1 a n d 1 m, I "iin^+nn w - o+n n *,-<D- nn^^) E, n8sq_ + St!, i=ij=i i=lj=l i=lj=l i=lj=l (7.37) Thus we have I nn tin mi (7.38) i=lj=l i=lj=l for every I > 1, rn ^ 0, t, and E^| e £ ***<(*) = T,8i < e 0 0 - However / i n (5 > e) n + 1 (Vesp. p {D{M {R ))) =n + 1) and (j, (S > e) = 1 (resp. n (D(M (R ))) = 1), so that none of thefinite-dimensionaldistributions can converge, and no subsequence of \i can converge in Mp (D(M (R ))). Note that {p, }nef$ * spatially tight but not tight. d n d F F d n s F n We prove Proposition 7.3.3 in the form of 5 lemmas. 7.3.5 establishes tightness of the {a hZ l n The first, Lemma : n G N } for each fixed l,t. Thus every subsequence of the jj, h? has a further subsequence that converges. The second, n * , Lemma 7.3.6 states that any limit point of the {n h~ n _ i /x/i- t 7.3.7 : n G N } must have the same moments (7.35) as . The third, Lemma 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 £ * = ' ' ' ^ ] , fa > 0 - 1 x 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 all (7.35), the limit points coincide, and thus the whole sequence converges to that limit point. Lemma £74 [1] < 0 7.3.5. 0 d an Let p ,/x n G Mp (L>(Mj?(IR ))), and a > d 0. Suppose that £ ^ [ 1 ] -» that! for every t G [a, 00), and every <j> G T, , (7.39) 143 Then for each m G Z the set of measures {/J, h- and every t G [a, oo) , m + 1 n : n G N} is tight. Proof. The m = 0 case is trivial since j E ^ f l ] -> -E>[1] < oo. This also gives ->• /J (7J(Mi?(E ))) < oo so it remains to prove spatial tightness fj, (D(MF(R ))) d d n for m > 1. We first prove the m — 1 case. Let e > 0, i > a. Define v n and v = E -i[X]. — E -i[X], h Then u(R ) = Lo < oo and applying Fubini to (7.39) we have d h / fa(x)u (dx) —>• J fa(x)u(dx) for every fa E T hence f n n —> IA Therefore there exists no such that for every n > no, u (R ) < Lo + 1. Since the u are finite, there d n n exists L\ such that u (R ) < L\ for all n < no. d n Let L = (LQ + 1) A Li and choose M such that L > s u p ^ /,-1[X(M )] > BupE < | . Then -i[MI d =MsupUnhT^XiW ) h ] X(RD)>M (7.40) > M). 1 n Dividing through by M , we get that upu hr (X(R )>M)<^< -. l S d (7.41) e n F i x n > 0. There exists K^i C K. compact such that K ^ - i ) < ^. Furd thermore there exists KQ C K compact such that U(KQ) < u(Kti) (e.g. the set . d KQ = {x : d(x,K-i) < 1}). Since u ^ u m M/?(M ) and K% is closed, d n limsupi/ (iv" ) < v(K§) < c n V (7.42) 0 Therefore there exists no > 0 such that for all n > no, U (KQ) < rf. Also since u u\,...,f -i a r no e finite measures there exist Ki C R compact such that Ui(Kf) < n d Then K = U"=o ^ 1 i s compact and supz/„(iv' ) < n. (7.43) c n Now s u p r y i ^ V 1 ( X ( K ) > tj*) < sup£ C n * ' -x [ x ( K ) J c h ' n t X{Kc)>r)-i n 1 = supu (K ) < T]. c n (7.44) Dividing through by r/4 we get that sup/i„/i _1 t ( X ( X ) > r/i) < n C 144 (7.45) Choose 774 = X. Then there exists Kj C M compact such that d s u p / i n ^ .(x(Kj) >±^<±j. (7.46) 1 Choose m > + 1 so that .< § and let pfrr K = p | I.X : * ( * ? ) < 1} f|{X : X(M ) < M}. (7.47) rf Now K is (sequentially) compact (see for example in the proof of Theorem II.4.1 of [27]), and K= (J tx-.X(K<)>±j\\j{x-.X(R )>M}. c (7.48) d Thus, sup/in/ir^K ) < s u p 6 : X{KD > 1) ( M tx 1 + sup/inV ({X : X ( R ) > Af}) d 1 (7.49) <su P f; p hi : X(K<) > • 1]) (tx l n + I 00 - 2^ ^37 2 + j=m - 8 ^ 1 2 + < 6 ' which verifies that the j^nh^ are spatially tight, for m = 1. For m > 1, and f € [a, o o ) , We have from (7.49) that for each i € { 1 , . . . , m} there exists K; C Mp(W ) compact such that sup n h^. {K.\°) < ^ . Let K = Ki x K x • • • x K . Then K C (M {R )) is compact and 1 m 1 x n d 2 m n m F ( TO \ jJlXiXiGK^} i=i / m < supE/irA- ( { X : X j G K i } ) 1 = supYV^. i=i n 1 c (K; ) < c VsupMn/it. <=i n 1 (Ki >< c e, (7.50) which gives the result. 145 Lemma 7.3.6. Fix I > 0 and t [0,oo)'. G Suppose that the second hypothesis of Proposition 7.3.3 holds for n ,fj, G MF{D(MF(W ))), for this t and for every 1 n v in Mp ({Mp(R )) ), then for each rh G Z d l rrii i=i j=i 1 and faj G T, l I + mi =E (7.51) i=i j=i Proof. The I = 0 case is trivial, so .we may assume that / > 0. Let p h- ==> v. 1 nk Then in particular we have /z„ /tr (1) -> v{\). Assume v{\) / 0. Then there exists fc k-o such that for every k > ko, < n h~ {l) l nk < 2/v(l) and we define for k > k 0 the probability measures, P(.) = (7.52) 1/(1)' P as probability measures. Let X ~ P and X Then we have that P X and since (Mp is separable, we may assume that X and Then X, X are defined on the same probability space ( f i , . F , P ) . B y Corollary 1 of Theorem NK 7 NK NK 5.1 of [3], F(X ) N F ( X ) for every F such that P ( X G V ) = 0, i.e. such that F P(T>F) = 0, where Vp denotes the set of discontinuities of F. We apply this to the continuous function F^ : X —> Yli=i ITj=i -^t(^u')We now show that E Ef i ^ P O j . [F$'X ) F nk < oo. But, [10], it is enough to show that sup .Ef» fc = sup£ fc supEfc fc B y Example 7.10(15) of P E (7.53) (ni=inSi^(^-)) sup fc < CO, A*nA- (1) since (Y[\ i YYJ^i Xi(4>i3)) is also a polynomial of the form appearing in (7.35). Thus we have (7.54) Ef 2 = which implies that ^ ?[F}m\->E [F£X) hh Since we also have E,M n , ^ 1 v [ F $ ( X \ E /j.h in the case v(l) ^ 0. 146 (7.55) we have verified the claim Consider now the case that u(l) = 0. Then u is the zero measure and we have E v n ; = i n ^ i ^ ( ^ i ) J = 0. B y Cauchy-Schwarz, ( 2 E , - i ^k t h < E ^ k [ ] h f E ^ - i l2 i=ij=i l 2-| mt i=ij=i (7.56) Since 1 is a bounded continuous function and u / i - —» O M we have that the first ex1 Uk pectation on the right converges to 0. Since s u p fcJ l~Ij=i Xi(faj)) 1 < E ^ ^ - I (n!=i oo we obtain I mi niiw E, 0. (7.57) i=lj=l Since also I E mi ,-x -> E h (7.58) - i i=ij=i i=ij=i nL n^i we have that E -i h = . o = [nu nri ifies the result. which ver• r Lemma 7.3.7. Suppose I > 0, p, p,' € M ((M (R )) ). d F l mi l E„ = Etf i=i j=i If 1 F mi nn^(^) (7.59) i=l .7=1 holds (and both quantities are finite) for every (j) S JF^ then (7.59) holds for every mi 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 . 5 9 ) , using the facts that oo (by choosing fa — ^[JlUi lTj=i Xii )} 1 ^ 0, the constant function in J ) and the fa are bounded 7 we have /nn^ / /nn^ mi » unxiidxij) i=i j=i / j / l mt mi IJlUXiidxij) i=l j=l -=ij=i (7.60) Since T is a determining class for M (R ) one can verify that T^ d mi F class for M (R ^ ) d F mi is a determining (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 I rrii (Hmi (7.61) i i=ij=i i k=i so the products of <f>ij in (7.60) uniquely determine the measure defined by u(dx) = 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. • ^tnUi L e m m a 7.3.8. Suppose ft,n' G M ({M (R )) ) d F and assume D C m 0 F (B (R ,R+)) d b -bp satisfies A f = (B (R ,R )) . d b m + If for all <f> G D 0 (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 (C {R ,R+)) . We verify the stronger result that the class £ of $ for which (7.62) holds contains (B (R , R+ )) . d m b d m b Let if> G C be such that (f % fc Now by dominated convergence (using the fact that /J is a finite measure and dominating by e° = 1), n n = Eu, \ l i m E„ -T.T=\ Mi,n) x e Ln—>oo lim E u = 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 (B (R ,R+)) C C as required. Define e r : ( M ( M ) ) -> 1+ by eAv) = ~ £7=i ^ (<fe). Now let d m b d m F e U = {# G # 6 ( ( M F ( R ) ) , M ) : E^{X)] d M = ^[*(X)]} (7.64) and ^ o = {ex:<?e(C (M ,R )r}. d 6 148 f (7.65) m (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 $ G M,$n finite measures. Then $ G % by dominated convergence since U,\J! are n 4) Let fi, f € Uo- Then f = e^. and 2 { = -Er«^(*i.i) -sr-i^(*w) = e - s r . i ^ ( ^ + ^ » ) / l / 2 e e = e 0 1 + 0 2 e (7.66) 5) Mo CH was verified i n part (a). (c) There exists a countable convergence determining set for (M (R )) . d m F We use the construction of Proposition 3.4.4 of [8] to obtain a countable set V C (C (R ,! )) d 6 such that v m + -> V i n (M {R )) d n i f and only i f ff ($) ->• £(<£) for m F n every ^ G V). Let { ^ i , 92, • • • } be an enumeration of Q ^ , a dense subset of R . For d GN each 2 define fi,j( ) = 2 (1 — j\x — qi\) V 0, x and for A C N 2 (7.67) define / \ A1. AW = (7.68) i,j < m It is an exercise left for the reader to verify that Vo = {g :meN,Ac{l,...,my}cC (R ), m (7.69) a b is a countable convergence determining set for M (R ). d F {(4>i,..., It follows that V — 4> ) : <pi G Vo U {0}} is a countable convergence determining set for m (M (R )) . d m F Define G = a(e :$£V). $ 149 (7.70) (d) B({M (R )) ) d C g C cr(^o), where Q = a(e^ :$eV). m F is trivial since V C (C[,(R , R)) . d (M (R )) d m F The second inclusion We claim that Q contains all the open sets in m and hence contains B((M (R )) ). d Define the metric m F m oo (7.71) 2 n j = l 71 = 1 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 ((M (R )) , g'). Now M (R ) is separable so every open set is a countable union of balls B i(v, r) and therefore to show that U G Q, it is enough to show that Bff(v,r) G g. But d m d F F Q B ,(v,r) = { p' e E E ^^ ~ ^^ NDLJ,J 271 NDVJ ^ (7.72) G^ < R j= l n=l since an infinite series of measurable functions is measurable. We have now verified that g contains all the open sets of (M (R )) and therefore contains B({M {R )) ). (e) Conclusion. We have now verified that B({M {R )) ) C g C a(H ). Therefore every bounded continuous function is measurable with respect to a(%o). Furthermore 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 / : (M (R )) —> R, we have proved the result. d m d F m F d m F d 0 m F Lemma 7.3.9. Let /J, G M ((M (R )) ). d F Suppose there exists a 6 > 0 such that m F for all 9i < 6, E„[e^eiX (^) 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. Fix £ = . . . , 4> ) and let x = X(fc). Then for all z G C"*-, m oX{<t>)-AE„ E .1=0 150 (z-x) (7.75) 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 ... I=O oo ' E 1 z=o t[(ziXi) (7.76) ni ne Y.Tli=l which is a multivariable power series i n zi,...z m with coefficients that are linear combinations of quantities of the form n E„ Li=l Li=l Now let (7.77) = E„ 6 and note that 0 < Xi = Xi(fc) < Xi{R ) for 0 < fc < 1. Then d Halloo < / lim / ( z + A 2 i ) - x _ gi -: - e A^ Xie ' dn z x gAzj-Xi _ 1 = lim -xAd/j, Az AZJ->0 { (Azj)'- a| /! 2 = lim AziY Az;-»0 < 2 e:2:i 1=2 l i m \Azi\ [ eZ?=ite('j)zj 2 \A*i\xi A*;->0 y x e •• d/Ji l i m \Azi\ / " e C ^ J + ^ + l ^ l ^ + ^ i ^ ^ ^ e - ^ d u . = Now £ e ~ (7.78) < C so the integral converges for all z s u c h that Re(zi) + e + \Az{\ < 6 e and Re(zj) < d for all j / i. Thus the limit i n the above is zero. Choosing i — 1 we get that for fixed z-\ = (z2,---z ) with HiLiHoo < m 5, ip {z) l = f e d/j, is analytic i n z\ such that zx Re(z\) + e < 8 for every e > 0, and thus i n particular for z\ such that Re{z\) < 0. In particular, <p {z) is the l analytic continuation (in z\) of J e dp for ||z||oo < 8 and as such zx tp {z) 1 is uniquely determined by the moments (7.77). For 1 < j > i i < m, and fixed Zj such that Re(zj) suppose we have <p ~ (z) is analytic in each l l < 0 for j Zj < i and in the regions Re(zj) < S for < 0 for j < i and \ZJ\ < 5 for j > i. Then we define f (z) as follows. l As in the last line of (7.78), and using the fact that 151 ZjXj < 0 for j < i, we have lim (z+Azt)-x__z-x z-£ (z+Azi)-x e e e A5 Az»->-0 / < r e (7.79) l i m [AzA / ( ^ ) + + l ^ l ) ^ + E r > i ^ ) ^ e e ~ A ->o y 2i a ; 2 -^ e < i * 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 dn (7.80) £S 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,ip ~ \z), as a function of Z{. As such, ip ~ is uniquely determined by the moments (7.77). Therefore we have (p (z) — / e d \ i is analytic i n each i n 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 l l m l l zx j e^da =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 (X {l) = A) = 0. 0 b 152 2. For every t e [0, o o ) there exists a S(t, b) > 0 such that for all m E n o \x (l)e^ b 6 i X t (7.83) i^<oo. Proof. B y Theorem II.7.2(iii) of [27] we have for b > 0, N (X (l)&A)=Q (7.84) J^dx. 2 0 b 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,...,t ) G (0, o o ) and b > 0, and we set to = b. Then Theorem 11.7.3(c) of [27] implies that for 0* > 0, m m E So [ £r=o^(i)] = e x p | y S r = o ^ i ( i ) - i d e e N o ( )|, (7.85) l / where {Yj}t>o is a super-Brownian motion starting at Jo (i-e. with initial law S$ ). 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^ \b where co is 0 C some constant depending on m. Therefore for ||0||oo < |^| standard application of the Dominated Convergence Theorem allows us to take differentiation through the integral on the left side of (7.85) and obtain CQ v E S O = £ £ o * * i ( i ) a b ^ [ ^ ( 1 ) ^ ^ ( 1 ) (7.86) 0 3=0 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 (7.87) exp o=0 Therefore we have d Ei* [n(i)<.££i«i*ii(a) | £r=o^(i)-idNoH e D = 0 E s o [ £r=o^(D e = H{b,t,e) < oo. 153 (7.88) B y Fatou's Lemma we have f HlbXO) = lim 0oMV - i e e ^i^W E dNoW «o^(i) _ i (7.89) e 9 \0 0 = y E£i*i^(i),^(i) dNo(u). e Thus for ° ^ m^b lo we £ h a v e [x (l)e ^i^ E N o 6 ( 1 ) (7.90) l < H(b,t,(f) < oo, 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 fi defined by (1-17) the following holds: For every s > 0, A > 0, m G N , t G [0, o o ) and every F : Mp(R ) -» C bounded by a multinomial and such that N Q / I ^ ( P F ) = 0, n m d m 1 Eilnh £ ^ - 1 [x.(l)FLY)]'-, X (1)F(X)] S 1 (7.91) v and 2. HnhZ E l [E(X)I{x (i)>\} [^(^K{X.(l)>A}J 3 • (7-92) Proof. Define j««,Ng G M (D(M (R ))) by d F F < ( A ) = f X (M )d/x , d s JA n Ng(A)= [ . X (R )dN . d s 0 (7.93) JA That these measures are finite (in fact bounded uniformly in n) follows from the fact that p4 (D(M (R ))) d F We take T = {e ikx = E„ [X (l)) n s EMO [ X ( l ) ] < oo. s (7.94) : k G R } which is a convergence determining class of d bounded C-valued functions containing the constant function C?'— 1. Now for all 154 I > 0, rh e Z , l + l mi / E — rrii I t=i j=i mi ^(l)IIII'.^) E,,s i=l j=l 1 i=lj=l / rrii 1^(1)1111^^) E No i=ij=i l rrii i=l j=l (7.95) where even i n the I = 0 case, the presence of the factor X (\) ensures that the s convergence i n (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 [Xo(l)] = 0, while 0 E [X (l)} lin = nCE 0 B y Lemma 7.3.5 the measures {p h s n also uniformly bounded by (7.94), r a_ 0. n (7.96) } are spatially tight. Since they are } are in fact tight. B y 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 ,No to get n (7.98) Thus jj, h-. (l) s l n NQ/I- (1). In particular, there exists rio(s,i) such that for 1 n > no, (7.99) < x^ (i)<2N^r (l) 1 i / Therefore for n > n we may define P * G M i ( ( M ( M ) ) ) by d 0 f m F (7.100) 155 Now n hZ ==> W hZ implies that P » ==> P i as probability measures, where s l l n P'(*) a f = ?, J-ul- Let X H n since ( M p ( R ) ) d ~ i " and X ~ P° r is separable, we may assume that X m n n -2+ X , and and X are defined on the n same probability space (SI, T, P). F(X ) Then we have X B y Corollary 1 of Theorem 5.1 of Billingsley -2+ F(X) for every F such that f(X € V) F = 0, i.e. such that P?(V ) = 0. F We now show that if F is also bounded by a multinomial, then Ef |^P(A ")j —> ? r Ef - . 1 P ( X ) . B y Example 7.10(15) of Grimmett and Stirzaker, it is enough to show J L that s u p P n [ | F ( X ) |121 < oo. But, n P |P(X)| Q{xf 2 = Eps _ \F(X)\ ' Ef \\F(X )\ n 2 2 Mi n,t where Q is a polynomial such that | P | < Q. Since sup by definition of p n -i h Q(X) 2 (i) (7.101) < oo holds and the convergence of the r-point functions we have the result. s n Thus we have that Ef F(X ) (7.102) Ef F(X)] , n which implies that F(X) E n° hz l F(X) Em n (7.103) Therefore for every function F that is bounded by a polynomial and that satisfies W (D ) = 0, 0 F (7.104) Define , if X (l) s Pi = { x. (i) ( Then P i is continuous except at X (l) s <A (7.105) , otherwise. A> = A, and is bounded above by j. Thus, Lemma 7.4.1 and (7.104) show that » f E nh [Xs(mF] -> E ^ - i [X.imF], i.e. (7.106) • 156 7.5 A note on convergence of finite dimensional distributions Recall the definitions of p, (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. n C o n j e c t u r e 7.5.1. There exists an L\ 3> 1 such that for every L > L\, and for every e > 0, fi (S>e)^-N (S>e). n (7.107) 0 C o r o l l a r y 7.5.2. / / Conjecture 7.5.1 holds then there exists L such that for every 2 L>L ,e>0,leZ+,rhe Z , te 2 + mi I E, [0, oo)' and <f e l mi / nn^(^) {5> ) 7 , nn '.^) )^} N E 0 E m i 1 7 (7.108) »=ij=i i=ij=i where T = {e ' : k 6 lk x 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 i x L > L where L = 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 X (<pij). B y convergence (and finiteness) of the Fourier transforms of the r-point functions, we have E^ F {XAl)) -»• E [F (XA[T))] < C and therefore there exists C = 2 2 ti 2 2 NO 0 Co(t, rh, 4>) such that su ^ P n 2 \F {XA\\)) 2 n L <C . (7.109) JL (7.110) 0 J Choose Ao = Ao(n, Co) sufficiently small so that No(X (l)€(0,A ])< e 0 6C 0 B y (7.107) and (7.92) with F = 1 we have . ii ( X ( l ) > 0) -> N (X (l) n e 0 > 0), and e Mn P Q ( 1 ) > A ) ^ No ( X ( l ) > A ) . 0 e Therefore fi (X (l) e (0, A ]) -»• N (X (l) there exists no such that for all n > no, n e 0 0 £ (0, A ]). It follows from (7.110) that e 0 ^(x (i)e(o,A ])< e (7.111) 0 0 157 (jl-J (7.112) Using I = I (i)>x } + {s>£} Vn < E, n + {Xe F(X0)I ] ] e h a v e [F(X0))I No {s> -En {XeW>Xo} w 0 - E {s>c} [F(X0)I J{A%(i)e(o,A ]} 0 [F(X i$))I 0 t E,Vn F(X?(<£))I{x (i)e(o,\o}}\ + ^N e ] (7.113) {Xe(1)>Xo} [F(Xti$)) {x (i)e(o,x ]} I 0 e We bound the right hand side of (7.113) as follows. 0 B y (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\. O n the second term we use Cauchy-Schwarz to get Vn \F(X )\I ( ( r {Xe 1)e <E, Vn F (Xf{l)) 2 0M]} <C 0 ,1 2 2 E^ n [/{x«(i)€(o,Ao]}] JL =1 3C : (7.114) 3' 0 The third term is handled in exactly the same way. Therefore we have shown that for n > n i , E,Vn F(X0))I ] - E {s>e} [F(X^))I No {s>e} < V, (7.115) • which proves the lemma. F i x e > 0 and define n ,W G M E n 0 (D(M (R ))) d F F /z„(»,S>e), by (7.116) N§(.)=No(.,5>e). 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 h 'o 1 < F 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- hZ ^%hz\ £ l n which is the statement that fj, =$• No (Conjecture 7.2.1). n 158 (7.118) • Appendix A Extending the inductive approach A.l Motivation We have already noted in Chapter 1 why we expect a Gaussian scaling limit for our lattice trees model i n dimensions d > 8. We have also discussed results of Derbez and Slade [7] i n Chapter 7, and i n 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, sup fc \f (k;z)\ < K with /o = 1, n n and n+l fn+i(k;z) = £ g (k;z)f i- (k;z) m n+ (n > 0 ) . +e i(k;z), m n+ (A.l) m=l For the following nonrigorous argument we also suppose that pi (A:; 1) = D(k) « D(0) — ^23~, h e r e D is defined i n 1.2.4, and that e ,g l w m Then we have / n + 1 « gf x m+ 1.2 2 \ / and so f (k) « gi{k) « (1 - . Thus . n n « 0 for m > 1. n n 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 i n the recursion equation ( A . l ) , that ensures that there exists a critical z ss 1 at which / „ ^ - ^ = ; z j - —>• e~%s. The c c 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 d — 4. In each case the lace expansion is used to derive a c recursion relation of the form ( A . l ) and the required bounds on the quantities i n the recursion equation are shown to hold (provided d > 4) by estimating Feynman diagrams. The required bounds are typically of the form \h (k,z)\ < m some functions h m c , for 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 d / 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 d . In such cases the lace expansion for d > d suggests setting 9 = 2 + ^Y^. In particular for percolation (d = 6) we would expect 9 = 2 + Note that in the case d = 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 f 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. c c c c c n A.2 Assumptions on the Recursion Relation Suppose that for z > 0 and k G [—7r, ir] , we have fo(k; z) = 1 and • d n+l fn+i(k;z) = E 9m(k;z)f i- (k;z) n+ m + e i(k;z), n+ . (n > 0), . (A.3) m=l where the functions g and e are to be regarded as given. The goal is to understand the behaviour of the solution f (k; z) of (A.3). m m n 160 A.2.1 Assumptions 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 f remains bounded in a weak sense. This assumption remains unchanged from [19]. n Assumption S. For every n 6 N and z > 0, the mapping k H-> f (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 e (-;z) and g (-;z). In addition, for each n, \f (k; z)\ is bounded uniformly in k £ [—TT, ir] and z in a neighbourhood of 1 (which may depend on n). n n n d 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 (A.4) i=i 3 x x k=0 Let (A.5) a(k) = 1-D(k). Assumption D. We assume that fi(k\z) = zD(k), (A.6) ei(k;z ) = 0. 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 xez d (ii) There is a constant C such that, for all L > 1, Halloo < CL -d a = a< 2 161 2 CL , 2 (A.S) (iii) There exist constants rj, c\,c > 0 such that 2 ciL k 2 < a{k) < c L k 2 2 (||fc||oo < L ) , 2 (A.9) - 1 2 a(k) > r) (pHoo > L' ), (A.10) a{k)<2-n (k£[-ir,Tr] ). (A.ll) 1 d Assumptions E and G of [19] are now adapted to general 6 > 2 as follows. The relevant bounds on f , which m 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, \\D f (-,z)\\ < 2 m K p d p d A m 2 a , \fm(0;z)\<K, \V f (0;z)\<Ka m, 2 (A.13) 2 m 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 || • || 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. p 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 - » C (K), such that if (A.13) holds for some K > 1, L > LQ, e z £ I and for all 1 < m < n, then for that L and z, and for all k £ [—TT, Tr] and d 2 < m < n + 1, the following bounds hold: |e (A:;z)| < C {K)Bm- , \e {k;z) - e {0;z)\ < C {K)a{k)Bm- . e m 9+l e m Assumption Gg. There is an and a function K H-> C (K), g LQ, m e (A.14) an interval / C [1 - a, 1 +.a] with a £ (0,1), 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\ and d 2 < m < n + 1, the following bounds hold: \9m(k\z)\ < C {K)Bm-\ g \V g (0;z)\ 2 m 162 < C (K)a Bm- , 2 g e+1 (A.15) \d g (0;z)\ z (A.16) <C (K)Pm- , 9+1 m g \9m(k; z) - g (0; z) - a(k)a- W g (0; z)\ < C (K)8a(k) <' - ^ '\ 2 2 l+ m m g e (A.17) l+e m 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 > d and 6(d) > 2, and assume that Assumptions S, D, c EQ and GQ all hold. There exist positive LQ = Lg(d,e), z = z (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 c f(-^- )=Ae-&[l c + 0(k n- ) + 0(n- )], 2 Zc 5 (A.18) 6+2 with the error estimate uniform in {k G R : a(k/Vvo n) < ^n~ logn}. 2 d l (b) V /„(0;z ) = _^,_ „ri , n(t>~-S\ va n[l + 0(8n)}. fn(0;z ) 2 22 c (AJ19) 6 c (c) For all p > 1, |£> / (-;z )|| <^^. (A.20) 2 n c P Lpn p 2 (d) The constants z , A and v obey c 00 1 = E 9m(0;z ), c m=l A _ 1 (A.21) 1- 2^m=l VmKV^c) 22m=i gm(0;zc) m °~ 2 Em=l ffm(0;^ )' m c It follows immediately from Theorem A.2.1(d) and the bounds of Assumptions E and G that z = l + 0(/3), A = 1 + 0(8), c A.3 v = l + 0(B). (A.22) Induction hypotheses The recursion relation (A.3) is analysed using induction on n, as done in [19]. The induction hypotheses involve a sequence v , which is defined exactly as in [19] as follows. We set VQ = bo = I, and for n > 1 we define n " = —2E b 1 n m=l n v 2 5m(0; z), c = Y(mn m=l 163 l)<? (0; m z), b v = ——. n (A.23) 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) < A < e. 0- 7 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.{C {cKi),Cg{cKA,Ki}, (A.25) e where c is a constant determined in Lemma A.3.6 below. To advance the induction, we will need to assume that K > K > K' > K > 1, 3 x 4 K> 4 KuZK'i, 2 K > K. 5 (A.26) 4 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 z recursively by n n+1 z = 1-^2 n+1 9m{0;z ), n > 1. n (A.27) For n > 1, we define intervals I = n [ z - K n 1 p n - e + 1 , z „ + K Pn- ]. 0+1 l n (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 I and all 1 < j < n. n (HI) \zj-Zj-t\KKrfj- . , (H2) K-^-i| (H3) For k such that a(k) < 7 J 0 : <K 0j- + . d l 2 - 1 log j , fj(k; z) can be written in the form i f (k;z)=Y[[l-v a(k)+r (k)], i=l j i i with r{(k) = ri(k;z) obeying \n(0)\,< K 3 r z e + \ \n(k) - (0)\ < ri 164 K 0a(k)r . s 3 (H4) For k such that' a(k) > yj \fj(k;z)\ < 1 log j , fj(k; z) obeys the bounds* K a(k)- r , X 6 4 \fj(k;z) - fj-i(k;z)\ < K a{k)- j~ • x+l 6 b Note that these four statements are those of [19] with the replacement P+ 24 A ; (A.29) in (H4) and the global replacement (A.30) By global replacement we also mean that such quantities appear in exponents. A.3.1 4 0 — 2, etc. whenever 49-1, 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 I D 2 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)\ < K (2 + C(K 3 + K )8)8a(k)r , s 2 165 3 (A.31) 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, \ (k)\ < [l + 2K 0] [(l + \vi - i\)a(k)n(0) + \n(k) - n(0)\) 3 Si < [1 + 2K 0] (1 + CK 0)a{k) 2 3 < + 7rT ^ [1 + 2^ /?][2 + C^ /3] < 3 (A.32) [2 + C(K 2 2 + tf )0. 3 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 a(k) < 7 J logj, G I n and assume (H2-H3) for 1 < j < n. Then for k with _ 1 (A.33) \fj(k;z)\ < CK3P -(i-c{K K )p)jaW. e a+ e a The middle bound of (A.13) follows, for 1 < m < n and z G I , directly from Lemma A.3.3. We next state two lemmas which provide the other two bounds of (A.13). The first concerns the || • || norms and contains the most significant changes from [19]. As such we present the full proof of this lemma. m p Lemma A . 3 . 4 . Let z G I n n, and p > and assume (H2), (H3) and (H4). Then for all 1<j < 1, d _d_ jP LP 2 where the constant C may depend on p, d. Proof. We show that I I * ' * ' ) ^ 5 ^ - ' < - > A 35 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 I and 1 < j < n, and define n Ri = R2 = {ke ["*, {k€ Rs = {k G i?4 = {k G ["*, *]' _1 a{k) < 7 J " logj, Halloo < L~ 1}, ll*lloo > 7T] a(fe) < 7J~ iog;', a{k) > 7 J " logj, 7T] a(k) > -yj~ iogi, IWloo > _1 -1 _1 166 Halloo L~ lh < L~ h l L~ The set R is empty if j is sufficiently large. Then 2 \\D m = i:f 2 {D{kf\ {k)\) ^- . (A.36) P fj d 1=1 i ' JR 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 D < 1, to obtain for all p > 0, 2 <n/_* -^^,<_^ 0e 1=1 - (A.37, ~L c < ldjd/2Here we have used the substitution k[ = Lki^/pJ. On R , we use Lemma A.3.3 and (A. 10) to conclude that for all p > 0, there is an a(p) > 1 such that 2 dk f d ,p dk d _,• L<R^ N w i ) ') ^^ (2*)*- L c = ~^ ca j - (A 38) where \R \ denotes the volume of R . This volume is maximal when j = 3, so that 2 2 \R2\ < \{k : a(k) < < ( S L 1 7 * < \{k : D(k) > 1 l iin n, < I \2rT-<i 2 los3J 3 1- ^ } | (A.39) 2 111 S \W (, 7 1 lo 3-) 3 K ^ ) L using (A.8) in the last step. Therefore ori\R \ < CL~ j~ l J , and d since ^- < C for every d 2 2 < CL- r l . d d 2 (/3(A;) |/ (fc)|) ^ 2 P j (A.40) On P and R4, we use (H4). As a result, the contribution from these two regions is bounded above by 3 ^yf D(k) P r* D 4 dk 2 d (k) P i=3 (A141) (27r) ' x d On # 3 , we use D(k) < 1 and (A.9). From (A.9), k <E R3 implies that L \k\ > Cj~ log j so that 2 2 2 x l CK f -J—d k d < * CK 167 f X r - - dr d l 2Xp (A 42) 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 / \ L p r ~^dr ^j9<2X °[ )« L ( ° \ j6p l2Xp Jo d 2 X p C d 2Xp dr PL P L CKl ^<j9 d- (A 43) (A.4o) PL 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 CK P 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 ^,.^d k d _ CKl f A / f X , A „ Ctf? where we have used the fact that p > 1 and |D| < 1. Since i f f < (1 + Ki) , this p completes the proof. • Lemma A.3.5. Let z £ I and assume (H2) and (H3). Then, for n |V /,(0; *)| < (1 + C{K + K )6)o j. - 2 2 2 1 < j < n, 3 (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* '|. : ( A * ) ( (0 f 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 g +in 168 Lemma A.3.6. Let z € /„, and assume (H2), (H3) and (R~4). For k € [—IT, 2 < j < n + 1, and e' G [0, e], £/ie following hold: (i) (ii) (iii) (iv) \ {k-z)\<K' Bj- , \V g,(O;z)\<KyBj- +\ \d (0;z)\<K'8j- \ \g (k-,z)-g (0-z)-a(k)a-^g (0;z)\<K> ^ e gj A 2 0 6+ zgj j (v) j j 4 \ (k;z)\<K' Br , e ej 4 (vi) \e {k;z) - ej{Q\z)\ < K' a(k)Bj- . A.4 advanced d+1 3 4 T h e induction 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 ^ 771=2 m e - i - e ' < °°' 771=2 H ^ ( + 2-ro)*- " 2 ( A > 4 9 ) n 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, i n (H3) we require that there exists a q > 1 but sufficiently close to 1 so that ( n + l)-Mog(n l ) x ( ^ ) ° ' \log(n + l), + + 1 V M V** (9 = 3), • (A.50) is bounded by (n + 1)~ . This holds since <5 + 7 < l A ( 0 - 2 ) by (A.24). This corresponds 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 < 1 A (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 log n, C n> C " 5 _ 1 e q'y+\-e n n - e )x- n . ai<k ^ - 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) ~ < C (recall that a(k) < 2 from ( A . l l ) ) to get ^ < _ c _ x 2 The proof of Theorem A.2.1 now proceeds as i n [19] with the global replacement (A.30). 169 Tr] , d Bibliography [1] R. Adler. Superprocess local and intersection local times and their corresponding 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 polymer correlation functions. Journal. Stat. Phys., 110:503-518, 2003. [5] D . Brydges and T . Spencer. Self-avoiding walk in 5 or more dimensions. Commun. 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. NorthHolland, 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. O n 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 i n 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 networks of self-avoiding walks. Canadian Journal Math., 56:77-114, 2004. E . Janse van Rensburg. O n the number of trees in Z . J. Phys. A: Math. Gen., d 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 Probabilites 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 Lectures on Probability Theory and Statistics, no. 1840, Ecole d'Ete de Probability's de Saint Flour 2002. Springer, Berlin, 2004. 172
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Convergence of lattice trees to super-brownian motion...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Convergence of lattice trees to super-brownian motion above the critical dimension Holmes, Mark 2005
pdf
Page Metadata
Item Metadata
Title | Convergence of lattice trees to super-brownian motion above the critical dimension |
Creator |
Holmes, Mark |
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 |
Graduation Date | 2005-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-ubc_2005-10502X.pdf [ 7.06MB ]
- Metadata
- JSON: 831-1.0079636.json
- JSON-LD: 831-1.0079636-ld.json
- RDF/XML (Pretty): 831-1.0079636-rdf.xml
- RDF/JSON: 831-1.0079636-rdf.json
- Turtle: 831-1.0079636-turtle.txt
- N-Triples: 831-1.0079636-rdf-ntriples.txt
- Original Record: 831-1.0079636-source.json
- Full Text
- 831-1.0079636-fulltext.txt
- Citation
- 831-1.0079636.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0079636/manifest