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Decomposition of free fields and structural stability of dynamical systems for renormalization group.. Bauerschmidt, Roland 2013

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Decomposition of free fields and structuralstability of dynamical systems forrenormalization group analysisbyRoland BauerschmidtBachelor of Science in Physics, Eidgen?ssische Technische Hochschule Z?rich, 2007Master of Science in Physics, Eidgen?ssische Technische Hochschule Z?rich, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2013c? Roland Bauerschmidt 2013AbstractThe main results of this thesis concern the spatial decomposition of Gaussian fieldsand the structural stability of a class of dynamical systems near a non-hyperbolicfixed point. These are two problems that arise in a renormalization group methodfor random fields and self-avoiding walks developed by Brydges and Slade. Thisrenormalization group program is outlined in the introduction of this thesis withemphasis on the relevance of the problems studied subsequently.The first original result is a new and simple method to decompose the Greenfunctions corresponding to a large class of interesting symmetric Dirichlet formsinto integrals over symmetric positive semi-definite and finite range (properly sup-ported) forms that are smoother than the original Green function. This result givesrise to multiscale decompositions of the associated free fields into sums of inde-pendent smoother Gaussian fields with spatially localized correlations. Such de-compositions are the point of departure for renormalization group analysis. Thenovelty of the result is the use of the finite propagation speed of the wave equa-tion and a related property of Chebyshev polynomials. The result improves severalexisting results and also gives simpler proofs.The second result concerns structural stability, with respect to contractive third-order perturbations, of a certain class of dynamical systems near a non-hyperbolicfixed point. We reformulate the stability problem in terms of the well-posedness ofan infinite-dimensional nonlinear ordinary differential equation in a Banach spaceof carefully weighted sequences. Using this, we prove the existence and regularityof flows of the dynamical system which obey mixed initial and final boundaryconditions. This result can be applied to the renormalization group map of Brydgesand Slade, and is an ingredient in the analysis of the long-distance behavior of fourdimensional weakly self-avoiding walks using this approach.iiPrefaceChapter 1 is an introduction and motivation for the problems studied in the remain-der of the thesis. No originality is claimed and, to give an informative exposition,we explain a number of ideas from a number of references mentioned, but withoutexplicit reference to the origin of each single idea.Chapter 2, in slightly modified form, has been accepted for publication in thejournal Probability Theory and Related Fields; see reference [8].Chapter 3 is based on joint work with David Brydges and Gordon Slade; a ver-sion of it has been accepted for publication in the journal Annales Henri Poincar?;see reference [11].Chapter 4 discusses ideas developed together with David Brydges and GordonSlade.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Outline and preliminaries . . . . . . . . . . . . . . . . . . . . . 11.2 Random polymers . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Random fields and local time . . . . . . . . . . . . . . . . . . . 101.4 The renormalization group . . . . . . . . . . . . . . . . . . . . . 202 Decomposition of free fields . . . . . . . . . . . . . . . . . . . . . . 392.1 Introduction and main result . . . . . . . . . . . . . . . . . . . . 392.2 Proof of main result . . . . . . . . . . . . . . . . . . . . . . . . 522.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 Structural stability of a class of dynamical systems . . . . . . . . . 653.1 Introduction and main result . . . . . . . . . . . . . . . . . . . . 653.2 The quadratic flow . . . . . . . . . . . . . . . . . . . . . . . . . 733.3 Proof of main result . . . . . . . . . . . . . . . . . . . . . . . . 824 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.1 The weakly self-avoiding walk with contact attraction . . . . . . 1014.2 Logarithmic corrections to scaling behavior . . . . . . . . . . . . 102Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105ivTable of ContentsAppendicesA Perturbation theory and coordinates of the renormalization group . 114A.1 Flow of coupling constants . . . . . . . . . . . . . . . . . . . . . 114A.2 Bounds on the coefficients . . . . . . . . . . . . . . . . . . . . . 115A.3 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 119vList of SymbolsChapter 1(X, E) graph with vertices X and edges Ewt position of a walk on a graph at time tLx local time of a walk, see (1.7)?x field on a graph, for example, ? : X ? RHt (L) energy (or Hamilton function) of a walk of length t as a functionof its local time, see (1.9)Chapter 2? a smooth function on Rwith rapid decay, ? ? 0, and supp(??) ?[?1, 1] where ?? is the Fourier transform, see Lemma 2.2.5E Dirichlet formL generator of Dirichlet form, see (2.20)? Green form corresponding to a Dirichlet form, see (2.24)a coefficients of a generator, see Examples 2.1.3?2.1.4?f (?) Fourier transform of a function f(P?,?) finite propagation speed condition of wave equation associatedto generator of Dirichlet form L(P??,B) discrete finite propagation speed condition associated to LChapter 3j discrete time parameterX j = K j ? R state space of dynamical system at time jx j = (K j ,Vj ) position of dynamical system at time j? j evolution map ? j : X j ? X j+1 of dynamical system at time j?? j quadratic part of evolution map, see (3.1)g?j solution to the recursion relation g?j+1 = g?j ? ? j g?2jXw space of sequences with weight w, see Definition 3.3.1w, r specific choices of weights for sequences, see (3.95)viAcknowledgmentsI thank my research advisers, David Brydges and Gordon Slade, without whom thisthesis would not have been possible. I feel fortunate that I was introduced to theirarea of research, and their support throughout the course of this thesis, in academicand non-academic matters, was indispensable for its success.I also thank Joel Feldman for serving on my thesis committee, Tyler Helmuthfor proofreading the introduction, and the members of the Probability Group at theUniversity of British Columbia for having provided an inspiring research environ-ment. Part of the research that led to this thesis was carried out during stays at theInstitute Henri Poincar? in Paris and the Department of Mathematics and Statisticsat McGill University in Montreal, and I thank these institutions for their hospitalityduring my stays.I also gratefully acknowledge financial support from several sources, includingthe University of British Columbia, the Government of British Columbia, and theFondation Math?matique Science de Paris.Finally, I thank my family and friends for their emotional and financial supportduring the course of my studies, without which I could not have carried these out.viiChapter 1Introduction1.1 Outline and preliminaries1.1.1 OutlineThe main results of this thesis concern the spatial decomposition of Gaussian fieldsand the structural stability of a class of dynamical systems near a non-hyperbolicfixed point, and are given in Chapters 2 and Chapter 3, respectively. The primarymotivation for the study of the these problems is an application to a renormaliza-tion group method for the analysis of four-dimensional weakly self-avoiding walksdeveloped by Brydges and Slade. However, the results of Chapter 2?3 are not spe-cific to the application to self-avoiding walks, and we expect that they may also beuseful for renormalization group analysis of other models.The aim of the present chapter is to sketch the background of the problemsstudied in Chapter 2?3, in particular their advent in the renormalization group con-text. In Section 1.2, some aspects of random polymer models are introduced; thesemodels of phenomena from polymer chemistry are our primary motivation. Theirrelation to the problems studied in this thesis is indirect, however, via random fieldswhich are introduced in Section 1.3. Random fields are related to a broad range ofmodels of statistical mechanics, for example the description of interfaces describ-ing phase separation and models for ferromagnetism. In the description of randompolymers, they appear as the local time of a perturbed Markov process. The mainresults of this thesis are discussed in Section 1.4.In statistical mechanics, the behavior at large distances of a model is of maininterest. For random polymer and random field models, the large distance behavioris notoriously difficult to study, however, because both classes of models involve alarge number of strongly coupled degrees of freedom. The renormalization group,which is discussed in Section 1.4, is a program to study the large-distance behaviorof random fields, pioneered in this sense by the theoretical physicist Wilson. Themathematical realization of Wilson?s ideas has been a major challenge since theirseminal proposal. We discuss some of the difficulties involved in it, and then sketchimportant aspects of one of several approaches to resolve these difficulties, initi-ated by Brydges and Yau, and much generalized and improved in recent work of11.1. Outline and preliminariesBrydges and Slade, based on work of many others. The emphasis of this discussionis on how, specifically, the problems studied in the main part of this thesis pertainto this program, but we also aim to give an introduction to the general ideas.1.1.2 PreliminariesGeneral notation. We use the usual Landau notation:f (t) = o(g(t)) as t ? T if limt?Tf (t)/g(t) ? 0; (1.1)f (t) = O(g(t)) as t ? T if lim supt?T| f (t)/g(t) | < ?; (1.2)and also the usual asymptotic notation:f ? g as t ? T if f (t) = g(t)(1 + o(1)) as t ? T (1.3)where T is often 0 or ?.Limits are abbreviated by f (t?) = lims?t? f (s). The indicator function 1z isgiven by 1z = 1 if condition z is satisfied and 1z = 0 otherwise. The symbols C andc will mostly denote constants whose values are allowed to change between twooccurances. The dependence of a constant on a parameter is sometimes emphasizedby a subscript. The letter d is reserved for the dimension of the relevant physicalspace, i.e., of Zd or Rd , and for metrics (which of the two should be clear from thecontext). The expectation value of a random variable, ?, is denoted by E(?).Graphs. It will be convenient at various places to use the language of graphs, butwe do not use any non-trivial results from graph theory. We say ? = (X, E) is a(simple) graph if X is a finite or countable set of vertices and E ? P2(X ) is a set of(undirected) edges, where P2(X ) denotes the set of subsets of X with exactly twoelements. The words simple and undirected will be implicit from now on. Verticeswill typically be denoted by the letters x and y and edges by the letter e. The edgeconnecting two vertices x and y is written as xy = yx = {x , y}. The graph distanced(x , y) between two vertices x , y ? X is the number of edges of the shortest pathbetween two vertices x and y, if there is one, and ? otherwise. That x and y areneighbors, xy ? E, is denoted by x ? y. All graphs will be assumed to be locallyfinite, i.e., for any x ? X there are only finitely many y ? X with x ? y.The graphs of primary relevance for this thesis are lattice graphs which can beembedded in Rd or in the torus Td = Rd/Zd . The most important examples are theEuclidean (or hypercubic) lattice Zd with nearest-neighbor edges, i.e., xy ? E(Zd )if |xi ? yi | = 1 for exactly one i ? {1, . . . , d} and otherwise |xi ? yi | = 0, and thediscrete torus of side length n, denoted Zdn , for which xy ? E(Zdn ) if |xi ? yi | = 1mod n for exactly one i ? {1, . . . , d} and otherwise |xi ? yi | = 0 mod n.21.2. Random polymers1.2 Random polymers1.2.1 The simple random walkLet ? = (X, E) be a graph. A walk on ? of length t ? (0,?] is a right-continuouspath w : [0, t) ? X with finitely many jumps in finite intervals, i.e., d(ws ,ws?) ?1 for all s ? (0, t) with equality for only finitely many s in each bounded subinter-val of (0, t). Let Wt denote the set of all walks of length t. An important subclassof walks are discrete walks, denoted W ?t ? Wt , for which a jump happens at s ifand only if s is an integer and 0 < s < t. For t < ?, each walk w ? Wt can bespecified uniquely by an integer n ? 0, a finite sequence t0 = 0 < t1 < t2 < ? ? ? <tn < tn+1 = t, and an element w? ? W ?n asws = w?n if s ? [tn , tn+1). (1.4)The walk w? is called the skeleton walk of w and (t1 , . . . , tn ) are the jump times.Let Wx ,t = {w ? Wt : w0 = x} and W ?x ,t = W ?t ? Wx ,t be the sets of (contin-uous and discrete) walks starting at x. There are several natural probability mea-sures on Wx ,t that arise as restrictions of measures on Wx ,? and W ?x ,? (stochasticprocesses). The simple random walk is the discrete Markov process which choosesuniformly from its neighbors at each step. The constant-speed simple random walkis a continuous Markov process with skeleton walk given by the simple randomwalk and the times between two jumps distributed independently with exponentialdistribution with parameter 1. The variable-speed simple random walk likewisehas the simple random walk as skeleton walk, but the waiting times between twojumps now have exponential distribution with parameter given by the degree of thevertex before the jump. This can also be interpreted as that each edge has a supplyof exponential clocks with parameter 1 and that the next jump is along the edgewhose clock rings first. The two continuous processes only differ by rescaling ofthe time when the graph is regular, i.e., when all vertices have the same degree asis in particular the case for the graph of main interest, ?= Zd .To illustrate what is understood, consider (any of) the simple random walks onZd . Then, for any t ? 0,??1/2w?t ? Nt (? ? ?) (1.5)where the convergence is in distribution and Nt is a vector of independent Gaussianrandom variables with mean 0 and variance t (or 2dt for the variable-speed walk).This result is essentially the classical central limit theorem. It shows that wt growstypically like?t as t ? ?. A less precise way of measuring this is the statementthat E|wt |2 ? t as t ? ?. But much more is understood. It is also a well-known31.2. Random polymersresult that if (Bt )t is the Wiener process, a continuous random path [0,?) ? Rd(thus not in W?) with Gaussian distribution defined by B0 = 0, E(Bt ) = 0, andE(Bit B js ) = ?i j min{s, t} for i, j ? {1, . . . , d}, that then the convergence of (1.5)holds on a space of paths [0,?) ? Rd , i.e., for all t simultaneously in a sense.Proofs of the latter result, Donsker?s invariance principle, can be found in manytextbooks on advanced probability theory; see e.g. reference [105]. It describes thebehavior at large distances of the paths of (wt )t .1.2.2 Polymers and local timeIn polymer science, a linear polymer is a long chain of molecules (monomers).The simplest mathematical model for a linear polymer is the uniform ensemble,the uniform probability measures on W ?x ,t , for some graph (in particular for Zd),but it is not a well motivated approximation. For example, polymers should not beable to intersect themselves due to the finite extent of each molecule. A model thattakes this into consideration is the strictly self-avoiding walk, the uniform measureon (discrete) walks conditioned on the event that walks do not intersect themselves.For regular graphs, the uniform ensemble and the simple random walk are thesame. This has turned out to be an important observation for the study of randompolymers. Moreover, in some aspects, the continuous-time random walks have fa-vorable analytic properties over the discrete-time random walk. For regular graphs,the constant- and variable-speed walks are identical, up to rescaling of time by theconstant vertex degree, their jump sequences are Poisson processes, and the skele-ton walks are simple random walks, thus uniform when conditioned on the numberof jumps. In view of the last aspect, the continuous-time random walks are naturalvariants of the uniform ensemble.In reference [43], den Hollander gives a broad overview of mathematical mod-els for random polymers. Like the strictly self-avoiding walk, these polymer mod-els for example suppress self-intersections by giving smaller weight to intersectingpaths with respect to a reference measure. Natural choices for the reference mea-sure P0x ,t are any of the simple random walk models on the regular graphs Zd whered = 1, 2, . . . . These models are then defined by an energy or Hamilton function,Ht : Wt ? R, assigning an energy cost to every path, as a probability measure PHx ,ton Wx ,t byPHx ,t (dw) =1Ze?Ht (w) P0x ,t (dw) (1.6)where Z = ZHx ,t is a normalizing constant, called the partition function. The mea-sure PHx ,t can be viewed as a kind of Gibbs measure on walks. For a number ofinteresting models, the energy function is a functional of the local time. The local41.2. Random polymersFigure 1.1: Polymer with one self-intersection and several self-contacts.time of a walk w is given by:Ltx (w) =? t01ws=x ds. (1.7)where we recall the indicator function: 1a=b = 1 if a = b and 1a=b = 0 otherwise.To say that H is a functional of the local time means that there is H : M+(X ) ? R,where M+(X ) = {m : X ? R+ : ?x?X mx < ?}, such that1Ht (w) = H (Lt (w)) (w ? Wt ). (1.8)For example, an interesting class of Hamilton functions is given byH?,? (L) = ??x?XL2x ? ??x?X?y?X :y?xLxLy (?, ? ? 0). (1.9)This model is known under a number of names. If ? = 0, it is called the weakly self-avoiding walk, soft polymer, discrete Edwards model, and Domb-Joyce model [15,43, 88], and with self-attraction is added to the name if ? > 0 [43]. The repulsiveforce (? > 0) models the effect that polymers should not intersect themselves bysuppressing self-intersections of walks, as can be seen from the elementary identity?x?XLtx (w)2 =? t0? t01ws1=ws2 ds1 ds2. (1.10)The (optional) attractive force (? > 0) models the effect of a solution in which thepolymer is immersed, by making it energetically beneficial for a polymer to be incontact with itself (rather than the solution). This can be understood from?x?X?y?X :y?xLtx (w)Lty (w) =? t0? t01ws1?ws2 ds1 ds2. (1.11)1Observe that ?x?X Ltx (w) = t < ? for w ? Wt and thus Lt (Wt ) ? M+(X ).51.2. Random polymersNote that the strictly self-avoiding walk is obtained in the limit ? = 0, ? ? ?of the discrete-parameter version of (1.6); see e.g. references [82,88]. It can also berelated to the continuous-parameter model, but then the relation is more subtle [26].Figure 1.2: A trapped self-avoiding walk.Unlike the simple random walks, random polymer models like (1.6) are almostnever stochastic processes. For example, it is easy to see that strictly self-avoidingwalks can get trapped as shown in Figure 1.2. The parameter t of the measures PHx ,tcan therefore not be interpreted as time, but it is rather a measure of the lengths ofthe polymers described by the measures. In analogy to the classical theory of gasesin statistical mechanics, the measures PHx ,t describe ensembles of walks (whichtake the role of particle configurations of a gas) with fixed length (taking the roleof a fixed number of particles in the gas).As a consequence, the standard tools for the analysis of stochastic process arenot available to study the measures (1.6), making their analysis decidedly moredifficult than that of simple random walks. It turns out that random polymer modelsdepend sensitively on the presence of an interaction given as in (1.6). For example,it is believed (but only proved in dimension d = 1 so far; but see Section 4.2) thateven arbitrarily small values of ? > 0 can change the asymptotic behavior of thewalks drastically compared to the case ? = 0. On the other hand, the behavior forall ? > 0 is believed to be similar.1.2.3 Asymptotic behavior and universalityFrom now on, the discussion will be restricted to polymer models on the Euclideanlattice Zd ; we also consider only spatially homogeneous interactions, i.e., interac-tions that are invariant under translations like (1.9). To simplify the notation, wethen set the starting point to 0 and drop it from the notation, for example in (1.6).The perhaps most interesting mathematical problem about random polymers isto determine the typical growth of the distance between the starting and endpointwith its length, t. For the simple random walk, this, and almost any other question,are very well understood, for example by (1.5). However, for self-interacting ran-61.2. Random polymersdom polymers (H , 0), it is in general a difficult (open) problem to determine thegrowth of the end-to-end distance Et |wt |2. It is a general conjecture that the end-to-end distance is asymptotically described by a power law, i.e., that for ?, ? ? 0,there are constants c > 0 and ? ? 0 such thatEHt |wt |2 ? ct2? as t ? ? (1.12)where EHt (F) is the expectation value of a random variable F = F (w) under PHt .For the simple random walk, the exponent is ? = 12 , in any dimension. It is believedthat, for general polymers, the constant c > 0 depends on all of d, ?, and ?, butthat the exponent ? is universal, i.e., constant for appropriate ranges of ? and ? andalso independent of the lattice of a given dimension d. It does in general depend ond. In Figure 1.3, the conjectured phase diagram for the weakly self-avoiding walkwith self-attraction is shown; it was conjectured by v.d. Hofstad and Klenke [110].??? = 1/d? = 1/(1 + d)? = 0? = ??? = ?SAWFigure 1.3: The phase diagram conjectured (for discrete-time) in d ? 2, from [110]The kind of universality described in the last paragraph is one of the paradigmsin equilibrium statistical mechanics, yet in general only understood in few specificexamples mathematically. For the self-avoiding walk, without self-attraction, sem-inal results by Brydges and Spencer [20] and by Hara and Slade [73?75] providean essentially complete picture in dimensions five and higher. In particular, theseresults include the result EHt |wt |2 ? ct which is the same behavior as for the simplerandom walk, except for the constant. In dimension two, there is strong evidence71.2. Random polymersthat the long-distance behavior of (strictly) self-avoiding walks is described by theso-called Schramm-Loewner-Evolution [83]. This is a subject of intense research,but proofs are not known at the time this thesis is written. The weakly self-avoidingwalk, even without self-attraction, seems even more difficult to understand in twodimensions, but it is believed to be in the same universality class as the strictlyself-avoiding walk in any dimension. The term universality class refers to the classof models that share the same scaling limit (or at least the same critical exponents).The validity of the former conjecture is known only for dimension one [111] and,as discussed, in dimensions five and above without self-attraction. For the physi-cally most interesting dimension three, only numerical estimates of the values forthe critical exponents are known [41]. Dimension four is expected to be critical, inthe sense that the behavior of self-avoiding walks changes from behavior similar tothat of the simple random walk to complex behavior as d gets smaller through 4.The critical dimension. Many models of discrete equilibrium statistical mechan-ics can be defined, by the ?same? specification like (1.9), on an (essentially) arbi-trary graph. It is a paradigm of statistical mechanics that when such models aredefined on Zd , there is a critical dimension, dc , such that for d < dc , the behavioris complex, meaning for self-avoiding walks, for example, that it is different fromthat of the simple random walk, while for d > dc , the model has the so-calledmean-field behavior, meaning for self-avoiding walks that the behavior is the sameas that of the simple random walk. The term mean-field stems from analogy withmodels of ferromagnetism, but it is standard terminology for more general models.For self-avoiding walk models, there is overwhelming evidence that the critical di-mension is dc = 4. In the critical dimension, the behavior is expected to be thatof the mean-field model with universal logarithmic corrections. For example, forself-avoiding walks (with additional small self-attraction allowed), it is conjecturedthat, in d = 4, in the phase where ? > 0, ? ? ?,EHt |wt |2 ? ct(log t)14 (t ? ?); (1.13)see e.g. references [17,28,29,50,88]. The exponent 14 is expected not to depend onthe ?details? of the model. Brydges and Slade have developed methods by whicha proof of (1.13) seems within reach (but not reached, see also Section 4.2). Thework of this thesis is a contribution to this program which we will therefore discussin some detail.81.2. Random polymers1.2.4 The two-point function as a Laplace transformLet Ea (F) be the expectation value of a random variable F = F (w) with respectto the simple random walk probability distribution P0a on walks starting at a, letcHt (a, b) = Ea (e?H (Lt )1wt=b ) (a, b ? X ) (1.14)be the probability weight function of the endpoint b for the ensemble of walks oflength t that start at a, and set cHt (x) = cHt (0, x) on Zd . The main goal in the studyof random polymers is to understand this function ?very well,? in the limit t ? ?.For example, this would enable one to understandEHt |wt |2 =?x cHt (x) |x |2?x cHt (x). (1.15)An approach to understanding cHt (x) is via its Laplace transform in t,GH? (x) =? ?0E(e?H (Lt )1wt=x )e??t dt (? ? R), (1.16)which is called the two-point function for the random polymer described by Hamil-tonian H . To recover information about ct (x), as t ? ?, from G? (x), it is partic-ularly important to understand G? (x) as the minimal value, ? = ?c , above whichthe Laplace transform converges is approached.To illustrate this, it is instructive to consider the simple random walk with, say,variable-speed. In this case, the two-point function is the Green function of ??+ ?where ? is the graph Laplace operator given by? f (x) =?y:y?x( f (y) ? f (x)). (1.17)By use of the Fourier transform, it is straightforward to establish the exact relations?xG? (x) = 1? ,?x|x |2G? (x) = 2d?2 (? > 0). (1.18)In particular, ?c = 0 and the Laplace transforms of the numerator and the denomi-nator in (1.15) can be inverted explicitly to obtainEt |wt |2 = 2d ? t (1.19)as explained in [28, p. 526]. Even though it may not be the most efficient way tocompute Et |wt |2 for the simple random walk by analysis of the two-point function,91.3. Random fields and local timeas the result is elementary there, this approach has proven fruitful for the analysisof interacting models as we will explain (see also [28, 29]).The two-point function is however also of independent interest. For the simplerandom walk, it is possible to determine the asymptotic behavior of the two-pointfunction for fixed value of ?. For reference, we record from [64] that if2 d > 2,G? (x) ????????????c|x |d?2(? = 0)c?|x |(d?1)/2e?M (?)b(x/ |x |)?x (? > 0)(|x | ? ?). (1.20)where b : Sd?1 ? Rd and the rate of exponential decay satisfies M = M (?) ? ??as ? ? 0. It is related to the divergence of (1.18); see e.g. [88, Appendix A]. Theparameter ? ? 0 is also called the killing rate of the simple random walk because ithas an interpretation in terms of random walks that die (stop) after a finite randomtime, if ? > 0. In the context of the next section, ? is also called the square of themass and we write ? = m2 also in the context of the simple random walks.It turns out that questions about random polymers are related to questions aboutrandom fields.1.3 Random fields and local time1.3.1 GeneralitiesLet X be a countable set. It should be thought of as a spatial configuration of points;in the main examples, it is the vertex set of a graph, ?= (X, E). Let us call any map? : X ? R a real-valued field on X . It is also of interest to consider vector-valuedfields or more generally maps ? : X ? M that take values in a manifold M , butmost of the discussion will be restricted to the simplest case of real-valued fields,M = R. The space of fields is MX = {? : X ? M }.Random fields, or probability measures on MX , are one of the main structuresof interest in equilibrium statistical mechanics, in particular with X an infinite set orin the limit when X tends to an infinite set. Examples of random fields in statisticalmechanics include spin models, i.e., models of ferromagnetism in which a randomfield describes spins of particles located at the vertices of a graph, the descriptionof dislocations of particles from a crystal, the modelling of phase interfaces, heightfunctions of some configuration models (e.g. dimers), the local time of Markov (ormore general random) processes, and more.2The formula for ? > 0 also holds for d = 2, but for ? = 0, the homogeneous function 1/|x |d?2is replaced by ? log |x |. For simplicity, we restrict to d > 2.101.3. Random fields and local timeIn general, it is non-trivial to define random fields on an infinite set X so thattheir definition often proceeds through an approximation by finite sets.1.3.2 Gaussian fieldsA class of random fields of fundamental importance are Gaussian fields. These arespecial in many ways: they can be defined essentially directly on infinite sets (andalso in the continuum), many properties are accessible by elementary calculations,and they play an important role in the study of a number of non-Gaussian fields.Let X be a finite set and C = (Cxy )x ,y?X be a symmetric positive semi-definitematrix with real entries indexed by X , i.e., Cxy = Cyx for all x , y ? X and?x ,y?Xf xCxy fy ? 0 for all f ? RX . (1.21)The Gaussian measure PC on RX with mean 0 and covariance C is uniquely de-fined by the Fourier transform:?ei? f PC (d?) = e? 12 f C f for all f ? RX (1.22)where? f =?x?Xf x?x , f C f =?x ,y?Xf xCxy fy . (1.23)In particular, when C is a strictly positive definite matrix, i.e., if equality in (1.21)holds only if f x = 0 for all x ? X , then the inverse matrix L = C?1 exists and theGaussian measure PC is equivalently given by the densityPC (d?) = e? 12?L??det(2piC) ?X (d?) (1.24)where ?X denotes the |X |-dimensional Lebesgue measure on RX . We then say thatPC is a non-degenerate Gaussian measure. The matrix C is the covariance matrixor two-point function of PC in the sense thatEC (?x?y ) :=??x?y PC (d?) = Cxy (1.25)where we have introduced the notation EC (F) for the integral or expectation of arandom variable F with respect to the Gaussian measure PC .111.3. Random fields and local timeWick?s formulaThe moments of PC are given explicitly in terms of C by:EC????????2p?i=1?xi????????=?Pp?j=1Cxn j xmj (1.26)where the sum ranges over all pairings P of 1, . . . , 2p into p distinct unorderedpairs {n1 ,m1}, . . . , {np ,mp }; the odd moments vanish [104, Proposition 1.2].ConsistencyGaussian fields are consistent, in the sense that if ? is a Gaussian field on X withcovariance matrix C = (Cxy )x ,y?X , then for any subset Y ? X , the restriction of ?to Y is also a Gaussian field with covariance (Cxy )x ,y?Y ; this follows from (1.22).The consistency implies the existence of Gaussian fields on infinite index sets.A matrix C on an infinite index set X is positive definite if, for every finite subsetY ? X , the restriction of C to Y is positive definite. For any positive definite ma-trix C indexed by a set X , Kolmogorov?s extension theorem [61, Theorem 10.18]implies that there exists a random field ? on X such that, for each finite Y ? X , therestriction of ? to Y is a Gaussian field with covariance the restriction of C to Y .Free fieldsNow suppose that X is the vertex set of a graph ? = (X, E). A random field ? onX is called a Markov field on ? if, for any A ? X , {?x : x ? A} is independentof {?x : d(A, x) > 1} conditionally on {?x : d(x , A) = 1}. Markov randomfields play an important role in statistical mechanics because the Markov propertydescribes local interactions. It is not difficult to see that a non-degenerate Gaussianfield on a finite graph is Markovian if and only if the matrix L = C?1 is local in thesense that Lxy = 0 if d(x , y) > 1; see e.g. [98, Theorems 2.1?2.2].Let E(?, ?) = ?L? denote the quadratic form associated to such an L. Note thatevery quadratic form compatible with the locality requirement is given by functions? : E ? R and ? : X ? R asE(?, ?) = ?L? =?e?E?e (??)2e +?x?X?x?2x (1.27)(= 2?xy?E?xy?x?y +?x?X(?x + 2?y:y?x?xy)?2x)where(??)2xy = (?x ? ?y )2. (1.28)121.3. Random fields and local timeA quadratic form of this form is called a Dirichlet form on X when ?, ? ? 0, butthere are also interesting situations in which the last requirement is relaxed and onlypositive definiteness of E is required [3, 4]. The inverse matrix C = L?1 is calledthe Green function of E. A Gaussian field whose covariance is the Green functionof a Dirichlet form E is called the free field associated to E. Much interest is alreadyin the simplest case where ? and ? are both constant, say, ?e = 1 for all e ? E and?x = m2 ? 0 for all x ? X . Then such a field is called the discrete free field on ?with mass m. This terminology has its roots in quantum field theory [104].Local perturbations of Gaussian fieldsIt turns out that a number of interesting problems can be studied through (approx-imately) local perturbations of Gaussian fields, in particular local perturbations offree fields. By a local perturbation, we shall understand a random field given on afinite graph by a measure of the formPC ,Z0 (d?) =1ZZ0(?) PC (d?) (1.29)where local means that Z0 is a product of local field functionals3,Z0(?) =?x?XZ0,x (?), (1.30)i.e., Z0,x depends on {?y : d(x , y) ? 1} only. The most interesting examples aregiven by homogeneous perturbations for which Z0,x is the same functional for allx which is analogous to the requirement that ? and ? are constant in (1.27).The term ?perturbation? might suggests that fields described by such measuresare very similar to free fields, in particular when ?Z0,x ? 1,? but it turns out that thelarge distance behavior can be drastically different, in a way very much analogousto the behavior of polymer models discussed in the last paragraph of Section 1.2.2.This is no coincidence. In Section 1.3.3, we will sketch how, in terms of a general-ized notion of Gaussian field, random polymers are models that can be described interms of such local perturbations. This description is closely related to spin models,subsequently discussed briefly in Section use the term field functional rather than random variable for several reasons. It emphasizesthe point of view that the former are defined on the fields themselves rather than a probability space.For example, it will become useful to evaluate field functionals on deterministic fields. The secondreason is that, in a generalized context involving differential forms (Fermions) introduced later, thenotion of random variable does not exist while the notion of field functional still does.131.3. Random fields and local time1.3.3 Local time of Markov processes and free fieldsThe local time of a Markov process on a graph ?= (X, E) is a random field on X ,given (for every t ? 0) by (1.7). It is of considerable interest for random polymers.For example, the ratio of weight functions cHt (a, b)/c0t (a, b) of Section 1.2.4 is theexpectation of a functional of the random field Lt under the conditional probabilitydistribution Pa ( ? |wt = b) of the simple random walk.The distribution of the local time of a Markov process is difficult to study di-rectly, but it is known that, for continuous-time Markov processes, the local time4 isclosely related to the free field associated to the Dirichlet form of the Markov pro-cess. (The connection of Dirichlet forms and Markov processes is discussed in thenext subsection.) These relations go back to Symanzik [107], Brydges, Fr?hlich,and Spencer [18], and Dynkin [51?54], and there are also a number of more recentresults [108]. For example, Dynkin?s so-called isomorphism theorem states [108]EC (?a?bF ( 12?2)) =? ?0(EC ? Ea )(F ( 12?2 + Lt )1wt=b )e??t dt (1.31)where C is the covariance of the free field ? with mass m2 = ? > 0, i.e., theGreen function of the variable-speed simple random walk wt killed at rate ?, ECis the expectation functional of the field ?, and Ea is the expectation of the simplerandom walk wt started at w0 = a.Parisi and Sourlas [92,93] and McKane [89] discovered a more direct relation-ship involving supersymmetry; see also Luttinger [87]. In notation to be introducedbelow, the so-called ?-isomorphism [17, 30] can be stated asEC ( ??a?bF ( ??? + ???)) =? ?0Ea (F (Lt )1wt=b )e??t dt (1.32)where the pair (?, ?) a supersymmetric Gaussian field with the same covariance C.Thus, if the square of the free field is replaced by the square of the supersymmetricfield on the left-hand side, 12?2 + Lt is replaced by only Lt on the right-hand side.The supersymmetric partner ? of the complex free field ? decouples the two sides.Dirichlet forms, random walks, and free fieldsThe theory of Dirichlet forms is concerned with far-reaching generalizations of thequadratic form (1.27); see reference [63]. Dirichlet forms stand in close connectionto continuous-parameter Markov processes. For example, the Dirichlet form (1.27)with constant coefficients, ?e = 1, ?x = m2 ? 0, is associated to the variable-speed4For continuous-time Markov processes, the local time is also often called the occupation time todistinguish it from the local time of the skeleton Markov chain.141.3. Random fields and local timesimple random walk on the graph ?. Indeed, (?e )e?E with ?e = 1 can be viewedas the adjacency matrix (Axy )x ,y?X of ?, defined byAxy = 1xy?E =???????1 (xy ? E),0 (xy < E). (1.33)Let Dxx =?y?x Axy be the number of neighbors of the vertex x, and set Dxy = 0if x , y. The generator of the form (1.27) with m2 = 0 can then be written as thegraph Laplace operatorL = ?? = D ? A. (1.34)Standard theory of Markov process implies that there is a Markov process (wt )t?0on X with Ex (1wt=y ) = [e?Lt ]xy . The two-point function of this process isGm2 (x , y) =? ?0Ex (1wt=y )e?m2t dt=? ?0[e(??m2)t ]xydt =[(?? + m2)?1]xy. (1.35)Thus the two-point functions of the simple random walk and the two-point functionof the free field are the same. The connections between a Markov process and thecorresponding free field go much further, however.Complex and supersymmetric Gaussian fieldsA natural variant of (real) Gaussian fields are complex Gaussian fields. In general,a complex field is merely a two-component real field, but we restrict to symmetriccomplex Gaussian fields which means that the real and imaginary parts of the fieldare independent real Gaussian fields with the same covariance [79]. The symmetriccomplex field is then determined byE( ??x?y ) = Cxy , E(?x?y ) = E( ??x ??y ) = 0 (1.36)and C is called its covariance. (The real and imaginary components of ? both havecovariance 12C in the usual sense.)Let us consider the symmetric complex Gaussian measure on CX with strictlypositive definite covariance matrix C for a finite set X . Then, with L = C?1, theexpectation of a random variable F : CX ? C is given byEC (F (?)) = 1det(2piiC)?CXF (?) exp??????????x ,y?X?xLxy ??y????????d ?? d? (1.37)151.3. Random fields and local timewhich is interpreted as follows: in terms of two real fields, u and v, ? and ?? aregiven by ?x = ux + ivx and ??x = ux ? ivx , and the measure d ?? d? is a shorthandfor d ??x1 d?x1 ? ? ? d ??xn d?xn , if X = {x1 , . . . , xn }, whered ??i d?i = 2i dux dvx (1.38)with dux dvx the usual Lebesgue measures on C  R2.Now observe that the probability density of the complex Gaussian measure isthe top degree part of the differential form?C = exp??????????x ,y?X?xLxy ??y ?12pii?x ,y?Xd?xLxyd ??y????????. (1.39)Here differential forms are multiplied with the anticommuting wedge product (sup-pressed in the notation above), and the exponential function is defined by expansioninto a power series (which is unambiguous because the argument has even degree).An interesting property of this formula is that the normalization factor of the mea-sure does not appear explicitly. The expectation (1.37) can now be written asEC (F (?)) =?CXF?C (1.40)with the convention that the integral of a differential form is the integral of the topdegree part of the form only, in the usual sense of integrals of differential forms.Observe that, while equation (1.37) only has an interpretation for ordinary ran-dom variables F (?), i.e., differential forms of degree 0, equation (1.40) has a natu-ral interpretation when F is a more general differential form, namely as the integralof the top degree part of the differential form F?C . Differential forms can then beviewed as functionals of the field ?x and the differential form ?x = (2pii)?1/2d?x .5?x and ?x appear in a (formally) symmetric way in the formula for ?C . Inthe terminology of quantum mechanics, ? has the interpretation of a Boson field,while ? can be interpreted as a Fermion field. The formal symmetry between ?and ? is called a supersymmetry and has several fascinating implications whichwe will not discuss, but see references [17, 30]. We still call the pair (?x , ?x ) thesupersymmetric Gaussian field with covariance C. The identification of Fermionfields with differential forms in this context is due to Le Jan [85, 86].To exemplify in which ways supersymmetric Gaussian fields behave like ordi-nary Gaussian fields, let us mention that the sum of two supersymmetric Gaussianfields can again be interpreted as a supersymmetric Gaussian field whose covari-ance is the sum of the covariances [34, Proposition 2.6]. The covariance isE( ??x?y ) = E( ??x?y ) = ?E(?y ??x ) = Cxy . (1.41)5The complex square root function is fixed in an arbitrary way.161.3. Random fields and local timeThis can be generalized to a version of Wick?s formula for the moments:EC????????p?i=1??xi?yiq?j=1??u j?v j????????=????????????Spp?i=1Cxi ypi (i)?????????????????????Sq(?1) |? |q?j=1Cu j vpi ( j )?????????(1.42)where Sn is the symmetric group of order n, and (?1) |? | is the sign of a permutationpi ? Sn . More details are given in [29,30], but the upshot is that again, as in (1.26),all moments can be calculated in a simple way in terms of the covariance.Local time and supersymmetryFinally, we can discuss the connection between random walks and supersymmetry,discovered by Parisi and Sourlas [92, 93] and McKane [89], in the form stated inreference [33]. To explain it, define differential forms ?x , x ? X , on CX by?x = ??x?x +12pii d??xd?x = ??x?x + ??x?x . (1.43)For F : RX ? R smooth, it is natural to define a differential form F (?) as the finiteTaylor series around the degree 0 part of ? which is ??? = |?|2:F (?) =|X |?m=11m!?x1 ,... ,xm?XFx1 ???xm ( ???)m?j=112pii d??x j d?x j (1.44)where Fx1 ???xm (t) is the mth derivative of F (t) in direction (ex1 , . . . , exm ). TheTaylor series is finite because differential forms on a finite dimensional space havea maximal degree (the dimension of the space). It is unambiguous because thedifferential form ? is even.Theorem 1.3.1. Let X be a finite set, (wt )t?0 be a continuous-time Markov processon X, and C be the Green function of (wt )t?0 with killing rate m2 > 0:Cxy =? ?0Ex (1wt=y )e?m2t dt. (1.45)Then, for any smooth F : RX+ ? R that does not grow too rapidly,? ?0Ex (F (Lt )1wt=y )e?m2t dt = EC (F (?) ??x?y ). (1.46)Proof. See [30, Propositions 2.7 and 4.4]. 171.3. Random fields and local timeTheorem 1.3.1 with F = ?x e?g?2x?(??m2)?x for some m2 > 0 implies that thetwo-point function of the continuous-time weakly self-avoiding walk on a finitegraph is equal to the two-point function of a local perturbation of the supersym-metric free field on the same graph, in the sense of Section 1.3.2 with the Gaussianmeasure PC replaced by the supersymmetric Gaussian ?measure? ?C . If we writeg instead of ? and set ? = 0, the two-point function (1.16) is thus, more explicitly,G? (a, b) = EC ( ??a?bZ0) (1.47)where C = [?? + m2]?1 andZ0 =?x?Xe?g?2x?(??m2)?x (1.48)is a local perturbation. In fact, there is some flexibility in the split of perturbationand Gaussian measure, for example, by choice of m2. It turns out that this split canbe made use of in the context of the renormalization group, and that then, it is alsonecessary to consider a more general splitting, C = (1 + z)[?? + m2]?1 withZ0 =?x?Xe?g?2x?(??zm2)?x?z??x (1.49)and??,x =12[?x (? ??)x + (??x ) ??x + ?x (??)x + (??)x ??x]. (1.50)The study of the perturbation (1.48) is actually also very interesting when theFermionic (differential form) part of ? is dropped, and then such perturbations havebeen studied extensively, as spin models which are models of ferromagnetism.1.3.4 Spin modelsLet ? = (X, E) be a finite graph. A spin model on ? is real- or vector-valuedrandom field on ? with distribution given by [60]P(d?) = 1Ze?H (?)?x?X?(d?x ) (1.51)where Z is a normalizing constant, ? is a probability measure on RN called a priorimeasure of the spin model, ? : E ? [0,?) are pair interactions, andH (?) = ??xy?E?xy ?x ? ?y . (1.52)181.3. Random fields and local timeThe best-known case is when ? is constant, i.e., ?e = ? > 0, and the a priorimeasure is given by the uniform (surface) measure of the unit sphere SN?1 ? RN .These so-called N-vector models include the Ising model (N = 1), the rotor or XYmodel (N = 2), and the Heisenberg model (N = 3). Much attention has also beendevoted to the ?4 models, given by ?e = 1 and a priori measure?(d?x ) = e?g |?x |4?s |?x |2 . (1.53)The ?4 models include the N-vector models as limits with g ? ? and s ? ?g; seereferences [60, 100]. They can be written in exact analogy to (1.48) asdP = 1ZZ0 dPC , Z0 =?x?Xe?g |?x |4?? |?x |2 . (1.54)where PC is the Gaussian measure with covariance6 given by C = [?? + m2]?1and ? = s ? 1 ? m2.Spin models and walksThe relation (1.54) with N = 2 components is the same as (1.46) with ?x replacedby its 0-degree part, |?x |2. Thus the weakly self-avoiding walk model is a super-symmetric version of the two-component ?4-model. It is known that spin modelsalso have interpretations in terms of walks, but with additional loops [18, 60]. Infact, the discovery of the relations between walks and fields departed from thisdirection in the study of field theories in terms walks and loops [107].De Gennes [42] also argued that the self-avoiding walk is described by the limitN ? 0 of the N-vector model (also see [30, 88]), but this limit does not have ameaning at the level of probability measures. The supersymmetric version is a wayof giving rigorous meaning to it, in the context of the weakly self-avoiding walk.The essential idea is that the Fermion components of ? count, in a sense, negativelyto the number of components due to the minus sign in equation (1.42), in this sensegiving ?N = 2 ? 2 = 0.? For a more complete discussion, see reference [30].Behavior of spin modelsIn view of the connection between spin models and interacting walks (with loops),it is not surprising that many qualitative features of the weakly self-avoiding walkare shared by the spin models. In the context of spin models, the critical value ?chas an instructive interpretation. For example, consider the ?4 model with N = 16Each component is an independent Gaussian field with this covariance, in the vector-valued case.191.4. The renormalization groupand g > 0 fixed (or the Ising model). It is known, see e.g. reference [6], that thereis ?c > ?? such that its infinite volume limits on Zd , d ? 2, satisfy?(?) =?x|E(?0?x ) |???????< ? (? > ?c ),= ? (? < ?c ).(1.55)The field ?x can be interpreted as a kind of spin of a particle (an arrow) located atvertex x. For ? < ?c , (1.55) means that the spins are ordered. This corresponds tothe ferromagnetic phase of a magnet in which most spins point in the same direc-tion. On the other hand, the case ? > ?c corresponds to a disordered phase. Thevariation of ? corresponds to a variation in inverse temperature. The critical point? = ?c corresponds to the critical temperature of the phase transition between theordered and the disordered phase. For N > 1, the picture is similar, but much moredelicate due to the continuous O(N )-symmetry of the model on finite graphs. Thiscontinuous symmetry is ?spontaneously? broken in the ordered phase in the infinitevolume limit [62], giving a different magnitude of difficulty to the problem.1.4 The renormalization group1.4.1 The concept of renormalization in statistical mechanicsRandom polymer models on the Euclidean lattice are expected to have scaling lim-its. The fundamental example of this is the convergence of the simple random walkto the Wiener process (1.5). This is a statement about large distances and timesrelated by diffusive scaling. The basic idea of renormalization is to study the large-distance behavior of a model by reduction of the degrees of freedom of the modelby a version of coarse graining, i.e., disregarding information about the behaviorat small distances, say, smaller than 0 ? L ? ?. The fundamental hypothesisof the renormalization idea is that, after coarse graining and rescaling, the modelshould be similar to the original model with modified parameters. The combinationof the two operations of coarse graining and rescaling is called a renormalizationgroup transformation. However, concrete formulations of such transformations formodels of self-avoiding walks on the Euclidean lattices, in any dimension, defineddirectly in terms of walks and amenable to analysis, seem not to be understood.7The renormalization group concept is, however, much better understood in thecontext of (near) critical random fields, in particular if these are local perturbations7On hierarchical groups, the work of Brydges, Evans, and Imbrie [17,28,29] has an interpretationin terms of walks, and there is also work in preparation by Ohno in which renormalization of self-avoiding walks on hierarchical lattices is studied directly in terms of walks.201.4. The renormalization groupof a Gaussian field on Zd , i.e., described in finite volume by measuresPC ,Z0 (d?) =1ZZ0(?) PC (d?) (1.56)where PC is a Gaussian measure on Zd , Z0(?) is a local perturbation, in the sensediscussed in Section 1.3.4, and Z is the normalization constant Z = EC (Z0). In thiscontext (but not only in this), the renormalization group has been used successfullyto study the long-distance behavior of a number of such models. It also providesan approach to a renormalization group study of random polymers via (1.46). Theterm critical random field refers, for example, to a spin model at the critical point;see Section 1.3.4. In the context of models of walks, the near critical behavior isrelated to the behavior of long polymers as discussed in Section 1.2.4.Let us mention two historically important ideas for the renormalization groupstudy of random fields: Kadanoff [80] proposed the intuitively appealing idea toreplace a random field in an L ? L ? ? ? ? ? L block of points in Zd by an effectiveblock spin field, constructed for example by averaging the field in that block. Heclaimed that this block spin field should behave in a similar way as the originalfield, but did not provide arguments to justify such an approximation. Wilson laterargued, still non-rigorously but with deep insight, how a variant of this idea may bejustified. He was awarded the Nobel Prize in Physics in 1982 for his contributions[113]. Following the introduction of [1], let us sometimes refer to the mathematicalrealization of Wilson?s ideas as Wilson?s program. There has been quite remarkableprogress in the realization of approaches like Wilson?s renormalization group. Wedo not attempt to provide a comprehensive list of references, but let us only mentiona few relevant references: Benfatto et al. [13], Feldman et al. [58], Gawedzki andKupiainen [67], and Brydges and Yau [22]. Unfortunately, these works all involvenumerous technical challenges, and it seems unlikely that the full capacity of therenormalization group idea has been attained yet. Nonetheless, it is one of the mostpowerful tools available for the study of random fields.We will give a short heuristic account of our interpretation of the challengesof Wilson?s program and also sketch very briefly aspects of the approach initi-ated by Brydges and Yau [22], in a further developed form of Brydges and Slade[10, 34?38]. The latter authors conceptualized, simplified, and generalized the ap-proach in significant aspects to study weakly self-avoiding walks via (1.46). Themethod of Brydges and Yau has, however, also been applied to a number of othermodels, including the dipole and Coulomb gases [44?46, 48, 49, 55], gradient in-terface models [3], as well as problems from quantum field theory [1, 16, 32, 47].Introductions to concepts of the method are given in [12, 24, 25, 106]. Our dis-cussion is inspired by many of the references previously mentioned and by thegeneral expositions on the renormalization group [14, 67, 100, 113]. The focus of211.4. The renormalization groupour discussion is on the relation to the problems studied in this thesis.1.4.2 Progressive integration, dynamical systems, and coordinatesLet us consider a random field that is a local perturbation of the free field, (1.56),with covariance given by the Green function C = [??+m2]?1 of the graph Laplaceoperator on ? ? Zd . The perturbation Z0 makes sense only if it depends on a finiteset ? and then m2 > 0 may be required, but the goal is to analyze such measuresPC ,Z0 in the limit?? Zd and m2 ? 0; we will, however, not devote much attentionto the details of these limits.In principle, the measure PC ,Z0 can of course be studied in terms ofEC (F Z0), (1.57)for enough field functionals F which we call observables. For instance, with F = 1,(1.57) expresses the normalization factor in (1.56), and with F = ?a?b , it gives theunnormalized two-point function. It is well-known, however, that it can be useful tostudy a measure in terms of a transform, e.g., its Laplace or Fourier transformation.Let us denote the Laplace transform of the unnormalized measure Z0 dPC byZ f := EC (e?? f Z0(?)) =: EC (Z f0 (?)). (1.58)To study the large distance behavior of the field, the class of test functions f shouldbe insensitive to fluctuations at short distances. For example, a scaling limit wouldbe determined by increasingly smooth f = f ? given by f ?x = ?? ?f (?x), (x ? Zd ),for some exponent ? > 0 and ?f ? C?c (Rd ), in the limit ? ? 0. It is, however, alsointeresting to consider pointwise asymptotics of correlation functions, for examplewith f = f ab = ?a?a +?b?b as d(a, b) ? ?, where ?c are constants (c = a, b),and (?c )x = 1 if c = x and (?c )x = 0 otherwise. Then the normalized two-pointfunction is the derivative of log Z f with respect to ?a and ?b .The accurate analysis of expectations like (1.57)?(1.58) is however highly non-trivial because the free field is strongly correlated: for example, see (1.20), (1.18),EC ((?x ? EC (?x )(?y ? EC (?y ))) = EC (?x?y ) ? 0 (1.59)so slowly that?y |EC (?x?y ) | ? ? as ? ? Zd and m ? 0. The crucial propertythat will facilitate the analysis is that the perturbation Z0 is local, i.e., a productZ0 =?x??Z0,x (1.60)where each Z0,x is a local field functional; see Section 1.3.2. This factorizationproperty provides the important structure, as we will sketch, for the iterative anal-ysis of such expectations by a particular form of coarse graining.221.4. The renormalization groupThe fundamental idea is to decompose the free field ? into a sum of two in-dependent Gaussian fields, ? = ?s + ?l , corresponding to small and large dis-tances. The coarse graining step is then implemented by taking the expectation ofthe field ?s which is called the fluctuation field because it captures the small dis-tance fluctuations that are to be eliminated. Wilson?s renormalization group pro-gram involves iteration of this procedure and rescaling of the underlying physicalspace after each step. The motivation is that this renormalization group transfor-mation, the combination of coarse graining and rescaling, should bring a criticalmodel approximately back to its original form so that the transformation can beiterated to obtain an effective description for increasingly large distances.In practice, it can be convenient to omit the rescaling step and instead consider?increasingly smooth? test functions, as discussed below (1.58). Furthermore, theiterated decomposition of the Gaussian field, or equivalently of its covariance, intosmall and large distance contributions can be implemented by a priori decomposi-tion of the initial covariance,C = C1 + C2 + ? ? ? (1.61)into a sum of covariances corresponding to geometrically increasing length scales.This idea goes back to Wilson, but was perhaps first explicitly formulated by Ben-fatto et al. [13]. The somewhat vague term length or distance scale means that eachCj should account for the fluctuations of the free field in an exponential range ofdistances L j?1 . |x | . L j for a fixed L > 1. This is discussed in the next section.From a pragmatic point of view, the covariance decomposition C = C1+C2+? ? ?allows to evaluate the expectation EC (Z f0 (?)) progressively, in terms of a sequenceof field functionals Z fj which are integrated with respect to the Gaussian fields withcovariance Cj+1 + ? ? ? , defined byZ fj+1(?) := E j+1Zfj (?) := E??C j+1(Z fj (? + ??)), (1.62)where the expectation on the right-hand side is that of the fluctuation field ??. E j isthus the convolution operator of the Gaussian measure with covariance Cj . It thenfollows that the expectation is given by8Z f = Z f? (0) := limj??Z fj (0). (1.63)The progressive integration (E j ) can be regarded as a time-dependent dynami-cal system, with the scale parameter j in the role of ?time:? if N is an appropriate8The limit requires some mild assumptions on the decomposition. Moreover, in practice, it can bemore convenient to stop the iteration after finitely many steps, when the decomposition has reachedthe size of the finite set ?; we will ignore such details.231.4. The renormalization groupspace of field functionals and N j ? N a subspace of field functionals which areintegrable with respect to the Gaussian measure with covariance Cj + Cj+1 + ? ? ? ,then E j+1 : N j ? N j+1 ? N. This picture, in itself, is not a simplification of theproblem since the dynamical system (E j ) j is enormously complicated and time-dependent, and it must be understood in the limit ? ? Zd . To analyze particularaspects of this dynamical system, one must find appropriate coordinates in whichan aspect of consideration becomes tractable, uniformly in ?.In particular, it is natural to consider the evolution of the perturbation Z0 only,without f . For example, by an elementary calculation for Gaussian measures,Z f = Z f? (0) = e f C f Z? (C f ) (1.64)where Z? on the right-hand side does not have a superscript f . Thus, in principle,i.e., given sufficient knowledge about Z?, the general case can be reduced to it.The goal of the next subsections is to outline how coordinates x j can be foundin which the action of the Gaussian convolution with covariance Cj+1 on Z j isexpressed in a much simpler form by a map ? j acting on x j :E j+1( ?Z j (x j )) = ?Z j+1(? j (x j )) (1.65)for some coordinate maps ?Z j that map an ?abstract? coordinate x j to a field func-tional ?Z j (x j ) = ?Z j (x j , ?). In his pioneering work, Wilson argued how this shouldbe possible and, with the previously mentioned rescaling step, his dynamical sys-tem is approximately autonomous. In the rigorous approach of Brydges and Slade[10,34?37], it has turned out useful to allow the coordinate spaces to depend on thescale j. Thus there is a sequence of spaces X j such that x j ? X j and the evolutionmaps are given as ? j : X j ? X j+1, but approximate invariance under rescalingmust, of course, still play a role. Finding such coordinates x j , rigorously, is at theheart of the difficulties of the renormalization group.Let us mention again, with the more specific context that has now been intro-duced, that the main results of this thesis are the following.? Chapter 2 provides a new method for decomposition of Green functions thatgive decompositions of free fields with particularly useful properties for theanalysis of the renormalization group transformations that they induce.? Chapter 3 is the analysis of a class of general dynamical systems ? = (? j )that arise as coordinates of the renormalization group map for four-dimen-sional weakly self-avoiding walks [10, 37].The outline of this subsection will be expanded with further details in the followingsubsections. In Appendix A, we provide some concrete details how the covariancedecomposition of Chapter 2 gives rise to the assumptions of Chapter 3.241.4. The renormalization group1.4.3 Decomposition of the free fieldThe starting point for the renormalization group, in the form discussed in the previ-ous section, is a decomposition of the free field, or equivalently the decompositionof its covariance C, into distance scales:C = C1 + C2 + ? ? ? . (1.66)The covariance should here be regarded as an infinite (in the limit ? ? Zd) sym-metric matrix (Cxy )x ,y?Zd that is positive definite in the sense that?x ,y?Zdf xCxy fy ? 0 for all finitely supported f : Zd ? R. (1.67)The decomposition (1.66) must be such that each term Cj satisfies (1.67), in orderfor the Cj to be the covariances associated to Gaussian fields, and, at the same time,the covariances Cj must ?capture? the distance scales L j?1 . |x | . L j for somefixed L > 1, where L j = L ? ? ? ? ? L. These are two competing constraints.In Chapter 2, in particular in Theorem 2.1.2 and Example 2.1.3, we prove that,if C is the Green function of a quadratic form in general class (containing Dirichletforms on a general graph, not necessarily Zd), then a strong form of the decompo-sition of the above kind is possible. There exists ?t (x , y), t > 0 such thatCxy =? ?0?t (x , y) dtt (1.68)where ?t is positive definite, for each t > 0. The use of the scale-invariant measuredt/t on [0,?) in (1.68), rather than the Lebesgue measure dt, is not important buta natural choice. The kernel ?t satisfies the finite range property?t (x , y) = 0 if d(x , y) > t, (1.69)and natural estimates. For example, if C is the Green function associated to thelattice Laplace operator, then, for all multi-indices lx , ly ? N{?1,... ,?d}0 ,????lxx ?lyy ?t (x , y)??? ? Ct?(d?2)?|lx |1?|ly |1 (1.70)where negative components of l denote discrete gradients in the negative coordinatedirections and |l |1 =?di=1(li + l?i ). Moreover, ?t is then also translation-invariant,i.e., ?t (x , y) = ?t (0, y ? x), and symmetric, i.e., ?t (0, x) = ?t (0, ?x).251.4. The renormalization groupTo obtain a discrete decomposition, as in (1.66), the integral (1.68) can be splitinto integrals over finite intervals. For example, for any L > 1, set[Cj ]xy =?????????????????? 12 L0?t (x , y) dtt ( j = 1),? 12 Lj12 Lj?1?t (x , y) dtt ( j > 1).(1.71)The properties of ?t immediately imply???????????????????Cj is positive definite;Cj has the finite range property: [Cj ]xy = 0 if d(x , y) > 12 L j ;Cj is translation-invariant: [Cj ]x+a ,y+a = [Cj ]xy ;Cj satisfies |[?lxx ?lyy Cj ]xy | ? O(L?(d?2+|lx |1+|ly |1)( j?1)).(1.72)In addition, we show that the ? of the Euclidean lattice has a scaling limit. Forthe discrete decomposition, this means that there exists c ? C?c (B 12 (0)) such that[Cj ]xy = L?(d?2) jc(L? j (x ? y)) + O(L?(d?2+1) j ). (1.73)An analogous result also holds for all discrete gradients of Cj . The existence of thescaling limit implies that certain functions of Cj can be computed very precisely inthe limit j ? ?, as illustrated in Appendix A.As hinted at, the two constraints that Cj is positive definite and finite range arenon-trivial to satisfy simultaneously. It is a natural question if covariance decom-positions in which Cj is localized exponentially, e.g., for some c > 0,|[Cj ]xy | ? O(L?(d?2)( j?1)e?cL? ( j?1) |x?y | ), (1.74)would be equally useful. It is much easier to find decompositions with this relaxedlocalization property. The answer is that such decompositions are almost as useful,and, in fact, they have been used in earlier results on the renormalization group, seein particular [13, 65]. The use of the finite range property, originally proposed byBrydges [90], leads to simplifications of the method and, in some aspects, slightlybetter results.1.4.4 Formal perturbation theoryPhysicists have long understood that the evolution Z j ? Z j+1 = E j+1Z j becomesformally simple when expressed as an exponential function. Let ?Vj = ? log Z j be261.4. The renormalization groupthe effective potential. In particular, for the weakly self-avoiding walk model, by(1.49),?V0 =?x??(g0?2x + ?0?x + z0??,x ) (1.75)is parametrized by the three coupling constants (g0 , ?0 , z0). Formally, by whichwe mean by expanding the exponential function into a power series without payingattention to its convergence,?Vj+1 = ? log(E j+1(exp(? ?Vj )))? E j+1( ?Vj ) + 12 (E j+1( ?Vj )2 ? E j+1( ?V 2j )) + ? ? ? (1.76)where ? means in the sense of a formal power series in ?Vj . This relation is calledthe cumulant expansion and also perturbation expansion in the physics literature.If ?Vj was a polynomial (or formal power series) of the field, as for ?V0 in (1.75),then the terms of each order of on the right-hand side of (1.76) could be calculatedexplicitly in terms of the covariance by Wick?s formula (1.42). Ignoring a numberof problems with (1.76), Wilson observed that in this formal series of monomials ofthe field, a few terms seem to be much more important than the others. He arguedthat, in dimensions four and above9, ?Vj can be approximated by a polynomial of thesame form as ?V0. Effectively, this reduces the complexity from an infinite numberof variables to three variables, (gj , ? j , z j ), parametrizing ?Vj as in (1.75).First-order perturbation theory and local field monomialsTo explain Wilson?s argument, some terminology is convenient. A field functionalM is a local field monomial, localized at x ? ?, if M can be expressed as a mono-mial in ?x and ??x , and corresponding terms in other fields (such as the Fermionicfield ?). Moreover, P is a local field polynomial if there is X ? ? and local fieldmonomials Mx for x ? X such that P =?x?X Mx . For example, ?2x is a local fieldmonomial and ?x?? ?2x is a local field polynomial. In particular, ?V0 is a local fieldpolynomial.Then, to explain a fundamental idea, suppose that ?Vj+1 is given by the first termof the right-hand side of (1.76) only, i.e., ?Vj+1 = E j+1 ?Vj . Observe thatE j (?x ) = ?x , (1.77)E j (?2x ) = ?2x + 2[Cj ]xx?x = ?2x + 2[Cj ]00?x , (1.78)E j (??,x ) = ??,x , (1.79)9In fact, he also considers dimension ?4 ? ?,? but we will not be concerned with this case. Belowwe outline the considerations in general dimension d, but for field theories with ?quadric interac-tions,? these will only be useful in d ? 4 which is our main interest.271.4. The renormalization groupby the definitions of ? and ??, (1.43) and (1.50), Wick?s formula (1.42), and transla-tion-invariance of Cj , i.e., [Cj ]xx = [Cj ]00. The exact expressions of the right-handsides in (1.77) rely on the differential form parts of ?x and ??,x , but for many otherpurposes one can think of ?x and ??,x simply as their degree 0 parts, ??x?x and12 ( ??x (??)x + (? ??)x?x ), and we will then do so, and also replace the complexfield by a real field if the distinction is not important.10 It follows that, in thelinear approximation, all ?Vj are local field monomials of the same form as ?V0 with(g0 , ?0 , z0) replaced by (g?j , ?? j , z? j ) determined by the recursion relation(g?j+1, ?? j+1 , z? j+1) = (g?j , ?? j + 2[Cj+1]00g?j , z? j ). (1.80)Now observe that, according to the discussion about the decomposition of theGreen function in the previous section,Var(?lx? j+1,x ) = [?lx?lyCj+1]xy???x=y ? cL?(d?2+2|l |1) j = cL?2([?]+|l |1) j (1.81)for any multi-index l. The constant [?] := 12 (d ? 2) on the right-hand side is calledthe dimension of ?. A measure of the typical magnitude of a field is the square rootof its variance and, in this sense,|? j+1,x | ? L?[?] j . (1.82)Moreover, by (1.81), each discrete derivative ? of ? j+1 decreases this typical mag-nitude by an additional factor of L? j (up to an absolute constant). The dimension[M] of a local field monomial M is defined so that |M (? j+1) | ? L?[M ] j accordingto this heuristic, i.e., by adding a summand [?] for each factor of ? and a summandof 1 for each discrete gradient ?. For example,[?4] = 4[?] = 2(d ? 2), [(??)2] = 2[?] + 2 = d. (1.83)In dimensions d > 2, the typical magnitude of a fluctuation field decreases as jincreases, by (1.82), but at the same time its range increases like L j . For a scalinglimit, the natural ?effective size? of a field monomial is that of its sum over a blockB of approximate diameter L j , i.e.,?x?B|M (? j+1,x ) | ? L(d?[M ]) j . (1.84)This gives rise to the following classification of local field monomials:10The expressions with differential forms are simpler than those of their degree 0 parts alone. Thisis because of cancellations due to supersymmetry. It corresponds to the cancellation of ?loops? in therandom walk representation; see Section The renormalization group? if [M] > d, the heuristic magnitude of M contracts (M is irrelevant);? if [M] < d, the heuristic magnitude of M expands (M is relevant);? if [M] = d, the heuristic magnitude of M remains the same (M is marginal).It is therefore natural to consider the coupling constants (gj , ? j , z j ) with respectto the ?normalized? field monomials11 L(d?4) j?2x , L?2 j?x , and ??,x . In the formalfirst order approximation, the evolution of these is given by:(gj+1 , ? j+1, z j+1) = (L?(d?4)gj , L2(? j ? 2L2 j [Cj+1]00gj ), z j ). (1.85)These heuristic considerations lead to the following predictions for the largedistance behavior of the perturbed field. In dimension five and higher, the only non-contracting local field monomials compatible with the symmetries12 of the modelare ? and ??; in particular gj ? 0, and the large distance behavior is expected tobe that of the free field. In dimension three and lower, there are several relevantlocal field monomials, finitely many in dimension three, for example ? and ?2, andinfinitely many in dimension two, and in both cases the large distance behavior isexpected to be non-trivial (different from the free field). In dimension four, there isonly one relevant field monomial, ?, and only two marginal field monomials, ?2 and??, and the first-order approximation is not sufficient to (heuristically) determinethe long-distance behavior. A second-order analysis reveals that the long distancebehavior should be like that of the free field, but in a much more subtle way thanin dimensions above four.Higher-order perturbation theory and approximation by local polynomialsThe immediate difficulty encountered when trying to formally include higher-orderterms of (1.76) in the previously described heuristic procedure is that such termsare not local field monomials. For example, an (important, as it will turn out) termarising at second-order is?g2j?x ,y[Cj+1]2xy ??x?x ??y?y . (1.86)This term involves ?x and ?y with d(x , y) ? L j+1 and is therefore not a local fieldpolynomial. However, such terms which arise in (1.76) can always be replaced by11For simplicity, we refer to, e.g., ?x = ??x?x + ??x?x as a ?monomial,? even though it is actuallya sum of two monomials in the fields in the previously introduced terminology.12 The model is symmetric under Euclidean transformation that preserve the lattice and under aso-called supersymmetry [10, 35].291.4. The renormalization groupa local field polynomial and a contracting non-local remainder term. For example,for the above term, one can make the replacement?x ,y[Cj+1]2xy ??x?x ??y?y  C (2)j+1?x( ??x?x )2 (1.87)where we have introduced the abbreviationC (2)j =?y[Cj ]2xy (1.88)which is independent of x, by translation-invariance of Cj . The right-hand side of(1.87) is again a local field monomial and, as such, it can be included as a second-order correction to the flow of coupling constants (gj , ? j , z j ) 7? (gj+1 , ? j+1, z j+1).The above term results in a contribution to gj+1 like gj+1 = gj ? ? jg2j + ? ? ? with? j > 0. More details of the resulting equations are given in Appendix A.The difference between the right- and left-hand sides of (1.87) is?x ,y[Cj+1]2xy ( ??x?x )( ??y?y ? ??x?x ). (1.89)This term ?contracts? in dimensions d ? 4, roughly, since the difference between alocal field monomial at two points decays faster than the individual monomials, by(1.72), if the distance between the points remains fixed. To illustrate this, consider(1.89) with y = x + re where e is a unit lattice vector and r an integer with |r | ?O(L j+1); the latter restriction on r is because of the finite range condition thatCj (x , y) = 0 if d(x , y) ? cL j . Then??x?x ( ??y?y ? ??x?x ) =r?1?k=0??x?x (?e ( ???))x+ke . (1.90)This term, at scale l > j, i.e., if tested with fluctuation covariance Cl , has the effec-tive size O(rL(d?4[?]?1)l ) = O(L?(l? j ) L(d?4[?])l ) which decreases exponentiallyin l (because r and therefore j remain fixed). This argument can be made for eachterm appearing in (1.76). Brydges and Slade developed a systematic treatment [35].1.4.5 Dynamical systemsThat the space of relevant and marginal spatially homogeneous local field polyno-mials has finite dimension (in dimension four and above), and that every term in(1.76) can be approximated by such a local field polynomial with a ?contracting er-ror,? is the principal idea of Wilson?s renormalization group. Wilson argues [113]301.4. The renormalization groupthat the contractive terms should not influence the critical behavior of the model,determined by the evolution of the relevant and marginal terms, in our example,the three dimensional system (gj , z j , ? j ) 7? (gj+1, z j+1, ? j+1).There are numerous mathematical difficulties encountered when trying to jus-tify this picture given by formal perturbation theory; these are discussed (in part) inSection 1.4.6. Formal perturbation theory suggests that there should be coordinatesx j = (K j ,Vj ) determining Z j where Vj = (gj , z j , ? j ) is the three dimensional vec-tor describing the marginal and relevant monomials of formal perturbation theoryand K j is an infinite-dimensional vector capturing all of the irrelevant directions.The evolution of Vj should approximately be given by a ?localized? version of(1.76) as illustrated in (1.87), while K j should be contractive in some sense.In Chapter 3, the following abstract version of this set-up is considered. Weassume that there is a sequence of Banach spaces K j such that K j ? K j , that thejoint evolution of (K j ,Vj ) is described by an evolution map? j : K j ? R3 7? K j+1 ? R3 (1.91)of the form? j (K j ,Vj ) = (? j (K j ,Vj ), ?? j (Vj ) + ? j (K j ,Vj )) (1.92)with ? j and ? j contractive in K j and third-order in Vj , and ?? j a quadratic poly-nomial of Vj . The quadratic polynomials ?? j describe the formal second-order per-turbation theory of the relevant and marginal directions and therefore depend on Vjonly; ? j describes higher-order contributions which can either be due to the rele-vant and marginal coordinates or due to the contracting directions. The maps ? jare allowed to have a weak scale-dependence. In addition, we assume that the ?? donot have constant parts which allows for the interpretation ? j (0, 0) = (0, 0) suchthat 0 = (0, 0) can be considered a kind of fixed point13 of the dynamical system? = (? j ) j . This corresponds to the fact that the evolution of Z j is trivial if Z0 = 1.The main interest is in the long-time behavior of this dynamical system, asthis is related to the large distance behavior of the fields. For a dynamical systemnear a hyperbolic fixed point, the structure of the flows near the fixed point are wellunderstood. A dynamical system? : X ? X on a Banach space X has a hyperbolicfixed point 0 if the spectrum of D?(0) is bounded away from 1. Informally stated,the stable manifold theorem [99, Theorem 6.1] asserts that if ? is a hyperbolic Crmap (for some integer r > 0), then there exists a decomposition X = Xs ? Xu suchthat, near 0, the domain of attraction M ? X is the graph of a Cr map Xs ? Xu ,and that the convergence under iteration of ? of points on M to 0 is exponentially130 is a different element in every space X j = K j ?R3, but we will neglect this point in the presentdiscussion.311.4. The renormalization groupfast. This gives rise to a schematic phase portrait as shown in Figure 1.4. In thecontext of the renormalization group, the choice of V0 on the stable manifold of afixed point corresponds to a critical model, whose scaling limit is the same as thatof the perturbed Gaussian measure. (This is known as infrared asymptotic freedomin the physics literature. It will be discussed again in Section 1.4.8.)stable manifold fixed pointunstable manifoldFigure 1.4: Schematic phase portrait of the renormalization group.The ?fixed point? of the dynamical system arising in the renormalization groupanalysis of the four dimensional weakly self-avoiding walk, outlined above, is nothyperbolic; the reason is that ?2 is marginal. The analysis of the (local) long-timebehavior of non-hyperbolic fixed points is more subtle than that of hyperbolic onesand depends on specific properties of the dynamical system. For example, a smallchange of the value of a single coefficient of the quadratic term ?? above can changethe long-time behavior in an important way; see e.g. Example 3.1.6.In Chapter 3, we study dynamical systems of the form (1.92), and prove that,for the class of dynamical systems considered, an analog of a stable manifold the-orem holds. The exponentially fast convergence along the stable trajectory of thestable manifold theorem is replaced in our result by a polynomial bound with log-arithmic correction (which is likely optimal). Informally said, we show that, forsufficiently small V0 and K0, there is a codimension two manifold of (K0 ,V0) suchthat the solution to (K j+1 ,Vj+1) = ? j (K j ,Vj ) exists for all j ? N and is a pertur-bation of the solution to the analogous two dimensional manifold for the recursion?Vj+1 = ?? j ( ?Vj ) which can be studied by elementary means.In Appendix A, we provide the explicit expression of the quadratic part, ?pt, of321.4. The renormalization groupthe dynamical system that arises in the renormalization group map for the weaklyself-avoiding walk [10]. It is expressed in terms of the covariance decomposition ofChapter 2. It turns out that ?pt is not exactly of the form of ?? studied in Chapter 3.We provide an explicit transformation which expresses ?pt in terms of a map ?? towhich the result of Chapter 3 can be applied.1.4.6 The error coordinate and polymer gasesFinally, we provide some indication how the error coordinate K j can be found. Thisis, of course, the major mathematical difficulty in implementing Wilson?s program.In essence, this amounts to obtaining an approximate version of (1.76) with a usefulremainder estimate.This was first achieved for the ?4 model, in a somewhat different formulation,by Gawedzki and Kupiainen [65?67]; this model has also been studied by differentapproaches, see e.g. [58]. An infamous difficulty, known as the large field problem,is that (1.76) can only be a good approximation when V and, thus ?, are small. Thisproblem, in its simplest form, is already present in perturbations of the standardone-dimensional Gaussian measure. For example,I (g) =?Re?gt4e??t2 dt (1.93)is a singular function of g at g = 0 because e?gt4 is not integrable for g < 0. Largefields turn out to cause difficulties for the applicability of certain expansion meth-ods, but their probability is very small (in a large deviation sense). The solution ofGawedzki and Kupiainen to the large field problem involves a separate treatmentof small and large fields, in which the small field contribution gives rise to similareffective action as the formal analysis of Section 1.4.4, while the large field contri-bution is very small. Brydges and Yau [22] developed a different solution in whichno distinction between small and large fields has to be made, by use of well-chosenweights on the space of field functionals.The main issue, however, is that the perturbations Z j involve an unboundednumber of variables (as ? ? Zd) and that it is difficult to estimate the error to aformal approximation like (1.76) in a uniform way. This difficulty has historicallybeen handled by cluster expansions [22,65?67]. There, an important role is playedby a polymer gas14 which, informally said, can describe the irrelevant directions ofthe formal analysis. The previously mentioned references use covariance decom-positions C = C1+ ? ? ? in which the Cj are only exponentially localized, rather than14As a warning, we emphasize that the polymers that appear in the polymer gas are not the samekind of polymers as in Section 1.2.331.4. The renormalization groupfinite range, discussed in Section 1.4.3. The use of the finite range property allowsa simplified treatment without cluster expansion [25, 27, 90].Polymer gasesThe simplest version of a polymer gas is defined as follows; see [23,68]. Let P0(?)be the set of finite subsets of ?; for later convenience, we include the empty set ?in P0(?) although, at the moment, it would be more natural not to do so. Supposethat, for each polymer Y ? P0(?), there is a weight K (Y ), called polymer activity.The partition function of the polymer gas with activity K is given byZ =??N=01N!?Y1 ,... ,YN ?P0disjoint,Yi,?K (Y1) ? ? ? K (YN )=??N=01N!?Y1 ,... ,YN ?P0K (Y1) ? ? ? K (YN )?i, je?v (Yi ,Yj ) , (1.94)with the hard core interactionv(Yi ,Yj ) =???????0 (Yi ? Yj = ?),? (otherwise). (1.95)K (Y ) appears in analogy to the activity of the ?particle? at Y in the grand canonicalpartition function of a gas, which is why it is called polymer activity.A simplification is the connected polymer gas with configuration space givenby connected polymers CP0 ? P0. This requires a notion of connected polymerwith the property that each Y ? P0 has a unique disjoint decomposition Y = Y1 ?? ? ? ? YN into connected polymers in CP0. The activities K can then naturally beextended from CP0 to P0 byK (Y ) = K (Y1) ? ? ? K (YN ) (1.96)with the convention K (?) = 1. We identify polymer activities defined on CP0 withsuch defined on P0 satisfying (1.96) and call them connected polymer activities.The partition function (1.94) with P0 replaced by CP0 then has the simple formZ = 1 +?Y ?P0 ,Y,?K (Y ) =?Y ?P0K (Y ). (1.97)The first expression shows that ?Z ? 1? if K is ?small,? but observe that P0 has 2|?|elements, so that, for the sum to be small, K (X ) must be very small for most X .341.4. The renormalization groupFor the further development, it turns out convenient to introduce an algebraicstructure on polymer activities, introduced in [22, 25]. Define a commutative andassociative product on polymer activities by(F ? G)(X ) =?Y ?P0 (X )F (Y )G(X \ Y ) (1.98)where P0(X ) denotes the polymers contained in X . Let 1 denote the constantpolymer activity given by 1(Y ) = 1 for all Y ? P0. Then the partition function of aconnected polymer gas is given byZ = (K ? 1)(?). (1.99)1.4.7 Polymer representationIn the use of polymer gases to control the renormalization group, the polymer activ-ities K (X ) are local field functionals. More precisely, the space of field functionalsN is considered a commutative algebra with subalgebras N0(Y ) ? N of field func-tionals that only depend on the field in Y ? P0 and a ?small neighborhood? of Y .The polymer activities are then local in the sense that K (Y ) ? N0(Y ).The simplest example is the trivial polymer activity, denoted K = 1?, anddefined by 1? (X ) = 1 if X = ? and 1? (X ) = 0 else. 1? is the unit of the product ?.The initial partition function can then be written asZ0 = I0(?) = (1? ? I0)(?) (1.100)where I0 : P0 ? N is given by I (X ) = ?x e?V0,x .If the covariance decomposition has the finite range property, see Section 1.4.3,it turns out that all Z j can be expressed in a similar way, but to obtain a usefulrepresentation, the class of polymers must be restricted to reflect the increasinglylong range nature of the remaining fluctuation fields. More specifically, if ? is afinite torus or cube of side length LN for some integers L and N , let B j (?) be aset of mutually disjoint blocks of side length L j with the property that their unionequals ?. Let P j (?) be the set of finite unions of blocks in B j (?); these are calledscale j polymers. Then everything discussed in the previous section about polymergases has a straightforward scale j generalization, given by replacing P0 with P j ,and N0(Y ) with N j (Y ) which are field functionals that are allowed to depend on Yand a small neighborhood of blocks in B j near Y . In particular, the circle product? then depends on j, although we will not emphasize this in the notation.Brydges and Slade [25,37] show that, if the finite range decomposition is givenin terms of the same parameter L > 1, then Z j can be written asZ j = (K j ? I j )(?) = I j (?) +?X?P j (?) ,X,?K j (X )I j (? \ X ) (1.101)351.4. The renormalization groupwhere I j and K j and ? are defined on scale j polymers, I j is to second order es-sentially given by (1.76), and K j represents all of the higher order terms of formalperturbation theory in a rigorous fashion. To understand the significance that poly-mers must be at the correct length scale, observe that nth order terms of the formalapproximation (1.76) have range O(nL j ). This is easily understood by the exampleof the the second-order term (1.86). The polymer gas description becomes usefulif it can be arranged in such a way that nth order terms correspond, approximately,to polymer activities K (X ) on polymers with O(n) blocks so that K (X ) can beexpected to be smaller and smaller when X is large. This compensates ?loss oflocality? by smallness.Finite range propertyTo illustrate how the finite range property is helpful in obtaining the representation(1.101), we recall that the finite range property [Cj ]xy = 0 if d(x , y) ? cL j has theconsequence that, if ? j = (? j ,x )x is a Gaussian field with such a covariance, then? j ,x and ? j ,y are independent if d(x , y) ? cL j . In particular, if Y1 , . . . ,YN ? P jdo not touch each other, thenE jN?i=1K (Yi ) =N?i=1E jK (Yi ). (1.102)Now suppose that a local field functional I1,x = I1,x (?2+?3+? ? ? ), independentof the first fluctuation field, ?1, is given in some way, and let ?I0,x = I0,x ? I1,xwhere I0,x = I0,x (?1 + ?2 + ? ? ? ) does depend on ?1. ThenZ0 = I0(?) =?x??I0,x =?x??(I1,x + ?I0,x ) = (?I0 ? I1)(?). (1.103)The expectation of Z0 with respect to C1 can be written asZ1 = E1Z0 =?X?P0 (?)?????????x??\XI1,x????????E1????????x?X?I0,x???????= ( ?K1 ? I1)(?), (1.104)whereI1(X ) =?x?XI1,x , ?K1(X ) = E1????????x?X?I0,x???????(1.105)and the product ? on scale 0. However, ?K1(X ) depends on the field in a neighbor-hood of X of range O(L1) (or more generally O(L j ) at scale j). To write (1.104)361.4. The renormalization groupin terms of scale 1 polymers, one can restrict I1 to B1 and ?coarsen? ?K1 by settingK1(U) =?X?P0 (U ):X=U?????????x?U\XI1,x????????E1????????x?X?I0,x???????(1.106)for U ? P1, where the closure X ? P j+1 of a polymer X ? P j is the smallest scalej +1 polymer such that X ? X . The finite range property of E1 implies that K1(U)only depends on the field in U and in B1-blocks touching U; the appropriate choiceof N j is such that K1(U) ? N1(U).The representation Z j = K j ? I j is far from unique. There are many choices ofK j and I j that satisfy Z j = K j ? I j . It is crucial to choose the I j correctly to capturethe important directions, and the K j such that K j+1 contracts compared to K j in anappropriate norm. The details of this are quite delicate [25,37]. The representationZ j = K j ? I j bridges between the representations as an effective action, i.e., as anexponential and as a polymer gas. It resembles the expression e?Vj+K j sufficientlywell to serve as a replacement, but gives at the same time the flexibility to measurethe non-locality of the error.1.4.8 ConclusionThe renormalization group, in the sense sketched in the previous subsections, canprovide a complete description of the evolution of a local perturbation of a Gaus-sian field, Z j+1 = E j+1Z j , induced by a finite range decomposition of its covarianceC = C1 + C2 + ? ? ? , (1.107)in terms of tractable coordinates x j = (K j ,Vj ) defining field functionals ?Z j (K j ,Vj )such that, with Vj = (gj , ? j , z j ),E j ? ? ? E1 Z0(V0) = ?Z j (K j ,Vj ) ??x??e?g j?2x?? j?x?z j??,x . (1.108)The coordinates x j lie on the trajectory of a dynamical system ?,? j (K j ,Vj ) = (K j+1 ,Vj+1). (1.109)The long-time properties of the dynamical system ? can be used to establishproperties of the large distance behavior of the fields. For example, if V0 is chosencarefully, the flow Vj converges to the fixed point 0; this choice describes criticalmodels. The phenomenon Vj ? 0 is called infrared asymptotic freedom. Theterm infrared means that it concerns the large distance (short ?wavelength?) limit,371.4. The renormalization groupwhile freedom refers to the fact that V = 0 describes a free field. Together withdetailed estimates on K j , guaranteeing that its contribution is sufficiently small, theconvergence Vj ? 0 can, for example, be used to prove that the critical model hasthe same scaling limit as the perturbed Gaussian field (in an appropriate sense). Inaddition, the trajectories of ? close to the critical V0 reveal information about theapproach of the critical point, again with appropriate (non-trivial) estimates on theremainder part K j .In the next two chapters, two aspects of this program are studied in detail, thedecomposition of Gaussian fields and the analysis of a class of dynamical systemsthat arises in the renormalization group study of the weakly self-avoiding walk. Asmentioned, we provide some explicit details of the connection between the resultsof Chapter 2 and Chapter 3 in Appendix A.38Chapter 2Decomposition of free fields2.1 Introduction and main result2.1.1 The Newtonian potentialLet us place the result of this chapter into context via an example. Consider theNewtonian potential, the Green function of the Laplace operator on Rd given by?(x) = Cd???????|x |?(d?2) (d ? 3)log 1/|x | (d = 2) for all x ? Rd, x , 0. (2.1)For d ? 3 and any measurable function ? : [0,?) ? R such that td?3?(t) isintegrable, the Newtonian potential can be written, up to a constant, as|x |?(d?2) =? ?0t?(d?2) ?(|x |/t) dttfor all x ? Rd , x , 0. (2.2)This is true because both sides are radially symmetric and homogeneous of degree?(d?2), where homogeneity of the right-hand side simply follows from the changeof variables formula. In particular, ? can be chosen smooth with compact supportand such that ?(|x |) is a positive semi-definite function on Rd . The last conditionmeans that ?(|x |) is positive as a quadratic form: for any f ? C?c (Rd ), that is,f : Rd ? R smooth with compact support,?t ( f , f ) :=?Rd ?Rd?(|x ? y |/t) f (x) f (y) dx dy ? 0. (2.3)Similarly, if d = 2, and ? : [0,?) ? R is any absolutely continuous functionwith ?(0) = 1 and such that ??(t) is integrable, thenlog 1/|x | =? ?0(?(|x |/t) ? ?(1/t)) dttfor all x ? R2, x , 0. (2.4)Indeed, for x , 0,log 1/|x | = ?(0) log 1/|x | = ?? ?0??(s) log 1/|x | ds=? ?0??(s)? ss/ |x |dttds, (2.5)392.1. Introduction and main resultand thus, since ?? is integrable, by Fubini?s theorem,log 1/|x | =? ?0? t |x |t??(s) ds dtt=? ?0(?(t |x |) ? ?(t)) dtt, (2.6)showing (2.4) after the change of variables t 7? 1/t. Now suppose again that ?is chosen such that ?(|x |) is a positive semi-definite function on R2. Then thefunction R2 ? x 7? ?(|x |/t) ? ?(1/t) is positive as a quadratic form on the domainof smooth and compactly supported functions with vanishing integral:?t ( f , f ) :=?R2 ?R2(?(|x ? y |/t) ? ?(1/t)) f (x) f (y) dx dy (2.7)=?R2 ?R2?(|x ? y |/t) f (x) f (y) dx dy ? 0for all f ? C?c (R2) with?f dx = 0.The above shows that the Newtonian potentials (2.1) can be decomposed intointegrals of compactly supported and positive semi-definite functions, with the ap-propriate restriction of the domain for d = 2.Let us recall at this point that the positivity of a quadratic form has the impor-tant implication that it entails the existence of a corresponding Gaussian process,discussed briefly in Section 2.1.4. However, it is also of interest in mathematicalphysics for different reasons [71].2.1.2 Finite range decompositions of quadratic formsIt is an open problem to characterize the class of positive quadratic forms, S :D(S) ? D(S) ? R, that admit decompositions into integrals (or sums) of positivequadratic forms of finite range: for all f , g ? D(S), t > 0,?????????????????????????S( f , g) =? ?0St ( f , g) dtt ,St : D(S) ? D(S) ? R,St ( f , f ) ? 0,St ( f , g) = 0 if d(supp( f ), supp(g)) > ?(t),(2.8)where ? : (0,?) ? (0,?) is increasing and d is a distance function. The conditionof finite range, the last condition in (2.8), generalizes the property of compactsupport of the function ? in (2.3) to quadratic forms that are not defined by aconvolution kernel. The difficulty in decomposing quadratic forms in such a wayis to achieve the two conditions of positivity and finite range simultaneously. Note402.1. Introduction and main resultthat by splitting up the integral, one can obtain a decomposition into a sum from(2.8), and conversely, a decomposition into a sum can be written as an integral(without regularity in t).For applications, not only the existence, but also the regularity of the decom-position (2.8) is important. Let (X, ?) be a metric measure space, i.e., a locallycompact complete separable metric space X with a Radon measure ? on X withfull support (i.e., ? is strictly positive), Cc (X ) the space of continuous functionson X with compact support, and Cb (X ) the space of bounded and continuous func-tions on X . Let us say that the decomposition (2.8) is regular if Cc (X ) ? D(S) isS-dense in D(S) and if every St has a bounded continuous kernel st ? Cb (X ? X ):St ( f , g) =?st (x , y) f (x)g(y) d?(x) d?(y) for all f , g ? Cc (X ) ? D(S).(2.9)For the decompositions (2.2), (2.4), the kernels are of course given in terms ofthe smooth function ? by the explicit formula?t (x , y) = t?(d?2)?(|x ? y |/t) for all x , y ? Rd , t > 0. (2.10)Note that for d = 2 the second term in (2.4) could be omitted by (2.7), with theunderstanding that the quadratic form is restricted to functions with vanishing in-tegral. It follows in particular that|?t (x , y) | ? Ct?(d?2) uniformly in all x , y ? Rd . (2.11)This reflects the decay of the Newtonian potential. Moreover, for all integerslx , ly ? 0, the derivatives of the kernel ?t decay according to|Dlxx Dlyy ?t (x , y) | ? Cl t?(d?2)t?lx?ly , (2.12)reflecting that |Dl?(x) | ? Cl |x |?(d?2?l ) for all x ? Rd , x , 0.The main result of this chapter is a rather simple construction of decomposi-tions (2.8) with estimates like (2.11) for quadratic forms that arise by duality withDirichlet forms in a large class. We call such forms Green forms, motivated by theNewtonian potential, or Green function, that is a special case; this is explained inSection 2.1.3.The main idea of our method is that (2.8) can be achieved by applying formulaelike (2.2) to the spectral representation of the Green form, and then exploiting finitepropagation speed properties of appropriate wave flows. These are generalizationsof the fact that if u(t , x) is a solution to?2t u ? ?u = 0, u(0, x) = u0(x), ?tu(0, x) = 0 (2.13)412.1. Introduction and main resultwith compactly supported initial data u0 that thensupp(u(t , ?)) ? Nt (supp(u0)) (2.14)where Nt (U) = {x ? X : d(x ,U) ? t} for any U ? X .The idea of exploiting properties of the wave equation in the context of proba-bility theory is not new. For example, Varopoulos [112] used the finite propagationspeed of the wave equation to obtain Gaussian bounds on the heat kernel of gen-eral Markov chains, by decomposing it into an integral over compactly supportedparts. Our objective is slightly different in that we are interested in the constraintof positive definite decompositions.Decompositions of singular functions into sums or integrals of smooth andcompactly supported functions have a history in analysis. For example, Feffer-man?s celebrated proof of pointwise almost everywhere convergence of the Fourierseries [56] uses a decomposition of 1/x onR like (2.2), albeit without using positivesemi-definiteness. Hainzl and Seiringer [71], motivated by applications to quantummechanics such as [57], decompose general radially symmetric functions, withoutassuming a priori that they are positive definite, into weighted integrals over tentfunctions. These, like ?(|x |) in (2.2), are positive semi-definite. They state suffi-cient conditions for the weight to be non-negative, and thus obtain decompositionslike (2.2) for a class of radially symmetric potentials including e?m |x |/|x | on R3.Special cases and similar results have also appeared in earlier works of P?lya [94]and of Gneiting [69, 70].These results, like (2.2), make essential use of radial symmetry. One exampleof particular interest for probability theory?where radial symmetry is not given?is the Green function of the discrete Laplace operator:?Zdu(x) =?e?Zd :|e |1=1(u(x + e) ? u(x)) for any u : Zd ? R, x ? Zd . (2.15)Brydges, Guadagni, and Mitter [27] showed that also in this discrete case, thecorresponding Green function, or more generally the resolvent, admits a decompo-sition like (2.8) into a sum (instead of an integral) of positive semi-definite latticefunctions with estimates analogous to (2.12). Brydges and Talarczyck [21] gave arelated construction which applies to quite general elliptic operators on domains inRd , but estimates on the kernels of this decomposition are only known when thecoefficients are constant. Their construction was adapted by Adams, Koteck?, andM?ller [4] to show that the Green functions of constant coefficient discrete ellip-tic systems on Zd admit decompositions with estimates analogous to (2.12) andthat the decomposition obtained this way is analytic as a function of the (constant)coefficients. These results are based on constructions that average Poisson kernels.422.1. Introduction and main resultOur method, sketched earlier, is different from that of [4, 21, 27, 31] and yieldssimpler proofs of their results about constant coefficient elliptic operators?both indiscrete and continuous context. It furthermore naturally yields a decompositioninto an integral instead of a sum (with integrand smooth in t), and gives effectiveestimates for decompositions of Green functions of variable coefficient operators.2.1.3 Duality and spectral representation of the Green formLet us now introduce the general set-up in which our result is framed more pre-cisely. For motivation, we first return to the quadratic forms defined by the Newto-nian potentials (2.1):?( f , g) :=?Rd ?Rd?(x ? y) f (x)g(y) dx dy, f , g ? D(?) (2.16)where???????D(?) = C?c (Rd ) (d ? 3)D(?) = { f ? C?c (R2) :?R2 f dx = 0} (d = 2).(2.17)These quadratic forms are not bounded on L2(Rd ), as is most apparent when d = 2.They are closely related to the Dirichlet forms given byE(u, v) :=?Rd?u ? ?v dx , u, v ? C?c (Rd ). (2.18)The correspondence between the two is duality: for all f ? D(?),??( f , f ) = sup{?Rdf u dx : u ? C?c (Rd ), E(u, u) ? 1}. (2.19)This set-up admits the following natural generalization: Let (X, ?) be a metricmeasure space and L2(X ) be the Hilbert space of equivalence classes of real-valuedsquare ?-integrable functions on X with inner product (u, v) = (u, v)L2 . Let E :D(E) ? D(E) ? R be a closed positive quadratic form on L2(X ) with D(E) ?L2(X ) a dense linear subspace. It is sometimes convenient to assume that E isregular, i.e., that Cc (X ) ? D(E) is E-dense in D(E). That E is closed means thatD(E) is a Hilbert space with inner product E(u, v) + m2(u, v)L2 for any m2 > 0.For the example (2.18), the domain of the form closure D(E) of C?c (Rd ) is theusual Sobolev space H1(Rd ) and (u, v)H1 = E(u, v)+ (u, v)L2 is the usual Sobolevinner product.It follows [96] from closedness that E is the quadratic form associated to aunique self-adjoint operator L : D(L) ? L2(X ),E(u, v) = (u, Lv) for u ? D(E), v ? D(L), (2.20)432.1. Introduction and main resultwhere D(L) ? D(E) is a dense linear subspace in L2(X ). The self-adjointness ofL gives rise to a spectral family and functional calculus. This means in particularthat for any Borel measurable F : [0,?) ? R, there is a self-adjoint operator,denoted F (L) : D(F (L)) ? L2(X ), whereF (L) :=? ?0F (?) dP? , (2.21)D(F (L)) :={u ? L2(X ) :? ?0F (?)2 d(u, P?u) < ?}(2.22)with P? the spectral family associated to L, and (u, P?u) is the spectral measureassociated to L and u ? L2(X ). In these terms, E has the representationE(u, u) = ?L 12 u?L2 (X ) =?spec(L)? d(u, P?u), u ? D(E) = D(L 12 ), (2.23)where E(u, v) is defined by the polarization identity, if u , v. Similarly, the corre-sponding Green form can be defined by polarization and?( f , f ) = ?L? 12 f ?L2(X ) =?spec(L)??1 d(u, P?u), f ? D(?) = D(L? 12 ).(2.24)This representation will be our starting point for the decomposition of the Greenform. Before stating the result and its proof, let us sketch how the decompositionproblem arises in probability theory.2.1.4 Gaussian fields and statistical mechanicsThe linear space D(E) is complete under the metric induced by the inner productE(u, v) + m2(u, v)L2 for any m2 > 0, but it is generally not complete for m2 = 0.It may however be completed to a Hilbert space abstractly; we denote this Hilbertspace by (HE , (?, ?)E). Similarly, we can complete the domain D(?) to a Hilbertspace under the quadratic form ?; this Hilbert space is denoted by (H? , (?, ?)?).HE and H? are dual in the following sense: The L2 inner product can be restrictedto??, ?? : D(?) ? D(E) ? R, ? f , u? = ( f , u) = (L? 12 f , L 12 u) (2.25)which extends to a bounded bilinear form on H? ? HE. L acts by definition iso-metric from D(E) to D(?), with respect to the norms of HE and H?, and it extendsto an isometric isometry from HE to H?. Thus H? is identified with the dual spaceof HE naturally, via the extension of the L2 pairing ??, ??.442.1. Introduction and main resultRemark 2.1.1. To give some insight into the interpretation of the spaces HE andH?, let us mention how HE can be characterized in the case of the Newtonianpotential [40]:HE  { f : Rd ? R measurable :there exists an E-Cauchy sequence fn ? D(E) with fn ? f a.e.}/ ?d (2.26)where ?d is the usual identification of functions that are equal almost everywherewhen d ? 3. For d = 2, ?d in contrast identifies functions that may differ by aconstant almost everywhere. (It is therefore sometimes said that the massless freefield does not exist in two dimensions, but that its gradient does. The masslessfree field is the free field corresponding to ? in the terminology explained below.)To understand this distinction, take a smooth cut-off function ?1 on R2, e.g. with?1 ? 1 on B1(0) and ?1 ? 0 on B2(0)c , set ?n (x) = ?1(x/n), and note thatE(?n , ?n ) = nd?2E(?1 , ?1). Thus, (?n ) is bounded in HE whenever d ? 2, andthen (by the Banach-Alaoglu theorem) there is ? ? HE such that ?n ? ? weaklyalong a subsequence in HE; however, ?n ? 1 pointwise, so that ? ? 1 ? HE. NowE(1, 1) = 0 implies that the constant functions must be in the same equivalenceclass as the zero function.It is well-known that any separable real Hilbert space (H, (?, ?)H ) defines aGaussian process indexed by H [105]. This is a probability space (?, P) and aunitary map f ? H ? ? f , ?? ? L2(P) such that the random variables ? f , ?? areGaussian with variance ( f , f )H . Note that ? f , ?? is merely a symbolic notationfor the random variable on L2(P) that corresponds to f ? H . It cannot in gen-eral be interpreted as the pairing of f ? H with a random element ?(?) ? Hdefined for ? ? ?; see e.g. [101]. In particular, if (H, (?, ?)H ) is the Hilbert space(H? , (?, ?)H? ), this process is called the free field or the Gaussian free field (corre-sponding to Dirichlet form E or Green function ?).This is a generalization of the context introduced in Section 1.3.2 where X is acountable set and ?x ? H , (x ? X ) so that the field ?x = ??x , ?? has a pointwiseinterpretation.2.1.5 Main resultLet (X, ?) be a metric measure space. In addition, suppose that d : X ?X ? [0,?]is an extended pseudometric on X . (Extended means that d(x , y) may be infiniteand pseudo that d(x , y) = 0 for x , y is allowed. Example 2.1.4 below gives anexample of interest where d is not the metric of X .)Let E : D(E) ? D(E) ? R be a regular closed symmetric form on L2(X )as in Section 2.1.3 and denote by L : D(L) ? L2(X ) its self-adjoint generator.452.1. Introduction and main resultTheorem 2.1.2 assumes that (X, ?, d , E) satisfies one of the following two finitepropagation speed conditions that we now introduce: For ? > 0, B > 0, and anincreasing function ? : (0,?) ? (0,?), let us say that (X, ?, d , E) satisfies (P?,?)respectively (P??,B) if:supp(cos(L 12?t)u) ? N? (t ) (supp(u)) for all u ? Cc (X ), t > 0, (P?,?)respectivelyE(u, u) ? B?u?L2(X ) for all u ? L2(X ),supp(Lnu) ? N? (n) (supp(u)) for all u ? Cc (X ), n ? N,(P??,B)where as before Nt (U) = {x ? X : d(x ,U) ? t} for any U ? X . The left-hand sideof (P?,?) is defined in terms of functional calculus for the self-adjoint operator L.Note that if L = ??Rd = ??di=1 ?2xi is the standard Laplace operator of Rd,then u(t , x) = [cos(L 12 t)u0](x) is a solution to the standard wave equation (2.13),and the condition (P?,?) with ? = 1 and ?(t) = t is the finite propagation speedproperty (2.14). The property holds for more general elliptic operators and ellip-tic systems (not necessarily of second order), however; see Example 2.1.4 below.Similarly, if L = ??Zd is the discrete Laplace operator (2.15), then (P??,B) holdswith B = 2d and ?(n) = n, since Lu(x) only depends on u(y) when x and y arenearest neighbors. As for the property (P?,?), the condition (P??,B) remains true formore general discrete Dirichlet forms; see Examples 2.1.4?2.1.5.Let us introduce a further condition: The heat kernel bound (H?,?) holds whenthe heat semigroup (e?t L )t>0 has continuous kernels pt for all t > 0 and there is? > 0 and a bounded function ? : X ? R+ such thatpt (x , x) ? ?(x)t??/2 for all x ? X . (H?,?)Criteria for (H?,?) are classic; see e.g. [91] for second-order elliptic operators andalso the discussion in the examples below.Theorem 2.1.2. Suppose (X, ?, d , E) satisfies (P?,?) or (P??,B). Then the corre-sponding Green form (2.24) admits a finite range decomposition (2.8) with S = ?and St = ?t such that the ?t are bounded quadratic forms with|?t ( f , g) | ? C?,Bt2/? ? f ?L2(X ) ?g?L2(X ) for all f , g ? L2(X ). (2.27)Moreover, (H?,?) implies that the ?t have continuous kernels ?t that satisfy|?t (x , y) | ? C?,?,B??(x)?(y)t?(??2)/? . (2.28)462.1. Introduction and main result2.1.6 ExamplesExample 2.1.3 (Elliptic operators with constant coefficients). Let a = (ai j )1?i , j?dbe a strictly positive definite matrix in Rd?d andEa (u, v) =d?i , j=1?Rd(Diu(x))ai j (D jv(x)) dx , u, v ? C?c (Rd ), (2.29)E?a (u, v) =d?i , j=1?x?Zd(?iu(x))ai j (? jv(x)), u, v ? Cc (Zd ), (2.30)where Diu(x) is the partial derivative of u(x) in direction i = 1, . . . , d,?iu(x) = u(x + ei ) ? u(x) (2.31)with ei the unit vector in the positive ith direction, and Cc (Zd ) is the space offunctions u : Zd ? R with finite support. For m2 ? 0, further setEa ,m2 (u, v) = Ea (u, v) + m2?Rdu(x)v(x) dx (2.32)and define E?a ,m2 analogously. Assume that the eigenvalues of a are contained inthe interval [B2? , B2+], and in the discrete case also that m2 ? [0, M2+] for someB2? , B2+ , M2+ > 0; these assumptions are only important for uniformity in the con-stants below.In the continuous context, let d be the Euclidean distance on X = Rd and ?be the Lebesgue measure. It follows that (X, ?, d , E) satisfies (P?,?) with ? = 1,?(t) = B+t; see Example 2.1.4 for more details. In the discrete context, let d bethe infinity distance on X = Zd , i.e., d(x , y) = maxi=1,...d |xi ? yi |, and ? be thecounting measure. Then (P??,B) holds with B = B+ + M2+ and ?(n) = n.Theorem 2.1.2 implies that the Green functions associated to Ea ,m2 and E?a ,m2admit finite range decompositions. We denote their kernels by ?t (x , y; a,m2) and??t (x , y; a,m2). In addition to (2.28), it is not difficult to obtain estimates on thedecay of the derivatives of ?t and ??t , like (2.12), in this situation of constantcoefficients. Since these estimates are of interest for applications, we provide thedetails in Section 2.3.2 (in a slightly more general context). We show that there areconstants Cl ,k > 0 depending only on B? and B+, and in the discrete case also onM+, such that|Dlaa Dlm2m2 Dlyy Dlxx ?t (x , y; a,m2) | ? Cl ,k t?(d?2)?lx?ly+2lm2 (1 + m2t2)?k (2.33)and|Dlaa Dlm2m2 ?lyy ?lxx ??t (x , y, t; a,m2) | ? Cl ,k t?(d?2)?lx?ly+2lm2 (1+m2t2)?k (2.34)472.1. Introduction and main resultfor all integers la , lm2 , lx , ly , and k such thatlm2 < 12 (d + lx + ly ), (2.35)and that the following approximation result holds: There is c > 0 such that?lxx ?lyy ??t (x , y; a,m2) = cd?2Dlxx Dlyy ?t (cx , cy; a, c?2m2)+ O(t?(d?2)?lx?ly?1(1 + m2t2)?k ). (2.36)This reproduces and generalizes many results of [4, 27]. More precisely, weverify that there exists a smooth function ?? : Rd?[B2? , B2+]?[0,?) ? R supportedin |x | ? B+ such that?t (x , y; a,m2) = t?(d?2) ??(x ? yt; a,m2t2)(2.37)which has the same structure as (2.10) when m2 = 0; this is scale invariance.Moreover, by (2.36), the discrete Green function has a scaling limit and the erroris of the order of the rescaled lattice spacing O(t?1). This result improves [31].Example 2.1.4 (Elliptic operators and systems with variable coefficients). Let M ?N and ai j : Rd ? RM ?M , i, j = 1, . . . , d, be the smooth coefficients of a uni-formly elliptic system (or in particular, if M = 1, of a uniformly elliptic operator):B2? |? |2 ?M?k ,l=1d?i , j=1akli j (x)?ki ? lj ? B2+ |? |2 for all ? ? RdM , x ? Rd , (2.38)with B? , B+ > 0. Let us write u = (u1 , . . . , uM ) ? RdM with ui ? Rd , i =1, . . . , M . LetE(u, v) =d?i , j=1?Rd(Diuk (x))akli j (x)(D jul (x)) dx , u, v ? C?c (Rd ,RM )(2.39)and analogously in the discrete case (as in (2.29), (2.30)).To apply Theorem 2.1.2, (X, ?, d) is defined by X = Rd ? {1, . . . , M }, ? is theproduct of the Lebesgue measure on Rd and the counting measure on {1, . . . , M },and the distance is given by d((x , i), (y, j)) = d(x , y). In particular, d is only apseudometric on X . We may use the identification of u : Rd ? RM and u : X ? Rby u(x , i) = ui (x).It suffices to verify the condition (P1,B+t ) for smooth, compactly supportedu0 : Rd ? RM . For such a u0, set, by using spectral theory for self-adjointoperators:u(t) := cos((L + m2) 12 t)u0. (2.40)482.1. Introduction and main resultThen, since u0 is smooth, u(t , x) : R ? Rd ? RM is smooth jointly in (t , x), and?2t u + Lu + m2u = 0, ?tu(0) = 0, u(0) = u0 (2.41)holds in the classical sense. If M = 1, m2 = 0, and a is the d ? d identity matrix,(P1,t ) is the finite propagation speed of the wave equation.Similarly, in the general situation, the property (P1,B+t ) can be deduced fromthe finite propagation speed of first order hyperbolic systems. This is well-known,but the explicit reduction for the case of (2.41) with (2.39) is difficult is to find in theliterature. Let us therefore sketch how to convert (2.41) to a hyperbolic system forreaders interested in this case. For example, one can define v : R ? Rd ? R(d+2)Mby:vk0 = ?tuk , vki =d?j=1M?l=1akli j ?x j ul , vkd+1 = muk , (2.42)where i = {1, . . . , d} and k ? {1, . . . , M }. It follows that v satisfiesS?tv +d?j=1A j?x j v + Bv = 0, v(0) = (0, (aDu0)1 , . . . , (aDu0)d ,mu0) (2.43)where S,A j ,B : Rd ? R(d+2)M ? (d+2)M are defined as the block matricesS =?????????1M ?M 0dM ?M 0M ?M0M ?dM a?1 0M ?dM0M ?M 0dM ?M 1M ?M?????????, B =?????????01?1 0d?1 m01?d 0d?d 01?d?m 0d?1 01?1?????????? 1M ?M ,(2.44)andAi =?????????????????????0 ??1i ? ? ? ??di 0??1i 0 . . . 0 0...... . . .... 0??di 0 . . . 0 00 0 ? ? ? 0 0?????????????????????? 1M ?M , i = 1, . . . , d. (2.45)It is immediate that this system is symmetric uniformly hyperbolic, by the sym-metry and uniform ellipticity of the matrix a. The property (P1,B+t ) now followsfrom the finite propagation speed of linear hyperbolic systems; see e.g. [7, 84].Nash showed [91] that (Hd ,? ) holds when M = 1. In [77, 81], conditions aregiven for (Hd ,? ) to hold when M > 1. In particular, this includes the constantcoefficient case. The latter case can be treated by using the Fourier transform; seeSection Introduction and main resultExample 2.1.5 (Random walk on graphs). Let (X, E) be a (locally finite) graph,with vertex set X and edge set E ? P2(X ), where X is a countable (or finite) setand P2(X ) are the subsets of X with two elements. Let d : X ? X ? [0,?] bethe graph distance on (X, E), i.e., d(x , y) is the (unweighted) length of the shortestpath from x to y.Suppose that edge weights ?xy = ?yx ? 0, x , y ? X are given. These induce anatural measure, also denoted ?, on X by:?x =?y?X?xy , ?(A) =?x?A?x for all A ? X . (2.46)The associated Dirichlet form isE(u, u) = 12?xy?E?xy (u(x) ? u(y))2 for all u ? D(E) = L2(?) (2.47)and its generator is given byLu(x) = ??1x?y?X?xy (u(x)?u(y)) for all finitely supported u : X ? R. (2.48)L is called the probabilistic Laplace operator associated to the simple random walkon the weighted graph (X, ?) with transition probabilities ?xy/?x . Let us remarkthat a probabilistic interpretation (or a maximum principle) does not hold in generalfor Examples 2.1.3?2.1.4 (when a is non-diagonal or vector-valued).The Dirichlet form (2.47) is bounded on L2(?) with operator norm 2 so that theproperty (P??,B) holds with ?(n) = n and B = 2, and Theorem 2.1.2 is applicable.For applications, it is often useful to add a killing rate to the random walk: Theprobabilistic Green density with killing rate ? ? (0, 1) is defined by:G? (x , y) =?n?0pn (x , y)?n = (?L + (1 ? ?))?1(x , y) = (L? )?1(x , y) (2.49)where pn (x , y) is the kernel of the operator Pn on L2(?). Note that (2.49) onlyconverges for ? = 0 when the random walk is transient, but that L?1 still makessense as a quadratic form on its appropriate domain when the random walk is re-current, as in (2.16), (2.17) for d = 2. Note further that spec(L? ) ? [0, 2] for all? ? [0, 1], so that Theorem 2.1.2 is applicable uniformly in ? ? [0, 1].Closely related to the killed Green function G? is the resolvent kernel of L. Theresolvent of L is defined on L2(?) by Gm2 = (L + m2)?1 for m2 > 0. It is relatedto the killed Green density by:G? = ??1G(1??)/? . (2.50)502.1. Introduction and main resultOne difference compared with the killed Green function is that L+m2 is not bound-ed uniformly in m2 ? 0. To achieve the condition (P??,B) for fixed B > 0, it istherefore necessary to restrict to m2 ? M2+ with M2+ = B ? 2.Remark 2.1.6. Other examples which Theorem 2.1.2 is applicable to include Dir-ichlet spaces that satisfy a Davies-Gaffney estimate [103] such as weighted mani-folds and quadratic forms corresponding to powers of elliptic operators like ? RemarksRemark 2.1.7. Theorem 2.1.2 also gives the decomposition into sums as in [4, 21,27]: Suppose that the assumptions of Theorem 2.1.2 are satisfied and, for notationalsimplicity, that the resulting decomposition has a kernel. Then, for any L > 1,?(x , y) =?j?ZCj (x , y) for all x , y ? X ? X (2.51)where the functions Cj : X ? X ? [0,?), j ? Z are given byCj (x , y) :=? L jL j?1?t (x , y) dtt for all x , y ? X . (2.52)They satisfy the following properties:Cj is the kernel of a positive semi-definite form, (2.53)Cj (x , y) = 0 for all x , y ? X with d(x , y) ? L j , (2.54)and, if (H?,?) holds,|Cj (x , y) | ? c? (x , y)???????????L?(??2)( j?1) (? > 2)L(2??) j (? < 2)log(L) (? = 2)(2.55)with c? (x , y) is independent of L. Thus, (Cj ) j?Z is a finite range decompositioninto discrete scales of the Green function ?. Similarly, gradient estimates such as(2.33), (2.34), (2.36) in Example 2.1.3 have obvious discrete versions.Remark 2.1.8. More generally than in Theorem 2.1.2, we may consider a family ofsymmetric forms, (Es )s?Y , where Y is a domain in a Banach space, with generatorsLs . Let us assume that Es is smooth in s, in the following sense: There exists aprojection-valued measure P on a measurable space M and a function V : M ?Y ?(0,?), smooth in Y , such thatF (Ls ) =?spec(Ls )F (?) dPs? =?MF (V (s, ?)) dP? . (2.56)512.2. Proof of main resultAn example of this condition is Es ( f , f ) = E( f , f ) + s( f , f ) so that V (s, ?) =? + s and (Ls )?1 is the resolvent of L; similarly, the killed Green function ofExample 2.1.5 can be expressed in this way. Then the family of kernels ?s iscontinuous in s, and if (H?,?) holds for s = 0, and V (?, s) ? z2(s)V (?, 0)+m2(s),then|?st (x , y) | ? C?,?,l??(x)?(y)(z(s)t)?(??2)/? (1 + tm(s))?l . (2.57)This can be verified by a straightforward adaption of the proof of Theorem Proof of main result2.2.1 Spectral decompositionThe starting point for the proof is the spectral representation of the Green form(2.24):?( f , f ) =?spec(L)??1 d( f , P? f ) for all f ? D(?), (2.58)where f ? D(?) implies that the integral can be restricted to spec(L) \ 0. Themain result follows by decomposition of the function ??1 : spec(L) \ 0 ? R+.Different decompositions are needed under the two conditions (P?,?), (P??,B). Themain idea of the proof is that decompositions with good properties exist. The resultthat we prove after using it to deduce Theorem 2.1.2 is summarized in the followinglemma.Lemma 2.2.1 (Spectral decomposition). Suppose that L satisfies (P?,?) or (P??,B);in the second case, we assume that ? = 1. Then there exists a smooth family offunctions Wt ? C? (R), t > 0, such that for all ? ? spec(L) \ 0, t > 0, and allintegers l,??1 =? ?0t2? Wt (?) dtt , (2.59)Wt (?) ? 0, (2.60)(1 + t 2? ?)lWt (?) ? Cl , (2.61)and that for all u ? Cc (X ),supp(Wt (L)u) ? N? (t ) (supp(u)). (2.62)522.2. Proof of main resultRemark 2.2.2. More precisely, we will give explicit formulae for Wt that imply(1 + t2?)l?m??????m??m Wt (?)?????? Cl ,m (2.63)for all m and l, improving (2.61). This improvement is used in Section 2.3.2.Proof of Theorem 2.1.2. It follows from (2.59) that, for any f ? D(?),?( f , f ) =?spec(L)(? ?0t2? Wt (?) dtt)d( f , P? f ) (2.64)=? ?0t2?(?spec(L)Wt (?) d( f , P? f ))dtt=? ?0t2? ( f ,Wt (L) f ) dtt .The exchange of the order of the two integrals in the equation above is justi-fied by non-negativity of the integrand, by (2.60). The latter also implies that( f ,Wt (L) f ) ? 0 for all f ? L2(X ). The polarization identity allows to recover?( f , g) for all f , g ? D(?). Finally, (2.62) completes the verification of (2.8) for?t defined by?t ( f , g) = t2? ( f ,Wt (L)g). (2.65)It remains to prove that (H?,?) implies (2.28). The semigroup property and thecontinuity of pt imply that pt ? Cb (X, L2(X )) with?pt (x , ?)?L2 (X ) =?Xpt (x , y)pt (y, x) d?(y) = p2t (x , x), (2.66)?pt (x , ?) ? pt (y, ?)?L2 (X ) = p2t (x , x) + p2t (y, y) ? 2p2t (x , y) ? 0 as x ? y.(2.67)This implies that e?t L : L2(X ) ? Cb (X ) is a bounded linear operator (e?t L f (x) =(pt (x , ?), f )). Duality then also implies continuity of e?t L : Cb (X )? ? L2(X )(with respect to the strong topology on Cb (X )?). Let M (X ) ? Cb (X )? be thespace of signed finite Radon measures on X equipped with the weak-* topology.Let mi ? M (X ) with mi ? 0. Then:?e?t Lmi ?L2 (X ) =???????X(?Xpt (x , y) dmi (y))2d?(x)??????12=(?X?X(pt (y, ?), pt (z, ?)) dmi (y) dmi (z))12? 0 (2.68)532.2. Proof of main resultwhich means that e?t L : M (X ) ? L2(X ) is continuous (because X is separableand therefore the weak-* topology of M (X ) is metrizable). This implies that (1 +t2/?L)?l : M (X ) ? L2(X ) is likewise continuous for all l > ?/4. To see this, weuse the relation(1 + t2/??)?l = ?(l)?1? ?0e?s sl?1e?st2/?? ds (2.69)which holds by the change of variables formula and the definition of Euler?s gammafunction. The spectral theorem thus implies that, for any u ? L2(X ),?(1 + t2/?L)?lu?L2 (X ) ? ?(l)?1? ?0e?s sl?1?e?st2/?Lu?L2 (X ) ds. (2.70)Since ? has full support, L2(X )?M (X ) is dense in M (X ) (where Lp (X ) is alwayswith respect to ?), and the claimed continuity of (1 + t2/?L)?l : M (X ) ? L2(X )follows from (2.68). In particular, the pointwise bound for pt implies that forl > ?/4,?(1 + t2/?L)?l?x ?L2 (X ) ? ?(l)?1? ?0e?s sl?1?e?st2/?L?x ?L2 (X ) ds (2.71)? ?(l)?1??(x)t??/2?? ?0e?s sl?1??/4 ds= C??(x)t??/2? .Let ?t (?) = Wt (?)1/2. Then (2.61) and the spectral theorem also imply that??t (L)(1 + t2/?L)l ?L2 (X )?L2 (X ) = sup?>0?t (?)(1 + t2/??)l ? Cl , (2.72)uniformly in t > 0. It follows from (2.71) that ?t (L) : M (X ) ? L2(X ) with??t (L)?x ?L2 ? C??(x)t??/2? . (2.73)Finally, by the Cauchy-Schwarz inequality,|?t (x , y) | = t2/? (?t (L)?y , ?t (L)?x ) ? t2/? ??t (L)?y ?L2 (X ) ??t (L)?x ?L2 (X )(2.74)which, with (2.73), proves (2.28). The continuity of ?t is implied by the continuityof ?t (L) : M (X ) ? L2(X ) and of ?x in x ? X (in the weak-* topology). Remark 2.2.3. The decay for ?s claimed in (2.57) can be obtained by a straightfor-ward generalization of the above argument, replacing (2.69) by(1 + t2/? z2? + t2/?m2)?l = ?(l)?1? ?0e?s sl?1e?st2/?m2 e?sz2t2/?? ds. (2.75)542.2. Proof of main resultRemark 2.2.4. Furthermore, by (2.61), the operators Wt (L) are smoothing for t >0, in the general sense that, for any t > 0,Wt (L) : L2(X ) ? C? (L), where C? (L) :=??n=0D(Ln ) ? L2(X ) (2.76)is the set of C?-vectors for L; see [95]. Standard elliptic regularity estimates implye.g. that C? (L) = C? (X ) when E is the quadratic form associated to an ellipticoperator with smooth coefficients.2.2.2 Proof of Lemma 2.2.1To complete the proof of Theorem 2.1.2, it remains to demonstrate Lemma 2.2.1.We first prove it under condition (P?,?) in Lemma 2.2.5 below; this proof is quitestraightforward using the assumption and (2.2). Then we prove Lemma 2.2.1 in thesituation of condition (P??,B) in Lemma 2.2.7; here additional ideas are required.To fix conventions, let us define the Fourier transform of an integrable function? : R? R by??(k) = (2pi)?1?R?(x)e?ik x dx for all k ? R. (2.77)Lemma 2.2.5 (Lemma 2.2.1 under (P?,?)). For any ? : R? [0,?) such that ?? issmooth and symmetric with supp(??) ? [?1, 1], and for any ? > 0, there is C > 0such thatWt (?) := C?(? 12?t) (2.78)satisfies (2.59), (2.60), (2.61), and also (2.63), for all ? > 0, t > 0; and if (P?,?)holds, then (Wt ) also satisfies (2.62).Remark 2.2.6. It is not difficult to see that such ? exist. For example, if ?? is asmooth real-valued function with support in [? 12 , 12 ], then ? = |? |2 satisfies theassumptions. For simplicity, let us assume sometimes in the following that ? ischosen such that C = 1 when Lemma 2.2.1 is applied.Proof. Note that for any ? : [0,?) ? R with t?(t) integrable, there is C > 0 suchthat??1 = C? ?0t2? ?(? 12?t) dttfor all ? > 0. (2.79)This simply follows (as in (2.2)) because the right-hand side is homogeneousin ? of degree ?1, which is immediate by rescaling of the integration variable.This shows (2.59); (2.60) is obvious by assumption; and (2.61) follows since ??552.2. Proof of main resultis smooth. The improved estimate (2.63) follows from the chain rule (or Fa? diBruno?s formula) and?m?12???????m??m ?12??????? C?,m (2.80)for non-negative integers m, using that supp(??) ? [?1, 1] implies that ? is smooth.Moreover, since supp(??) ? [?1, 1], and since ?? is smooth,Wt (L)u = C? 1?1??(s) cos(L 12?ts)u ds for all u ? L2(X ), (2.81)where the integral is the Riemann integral, i.e., the strong limit of its Riemann sums(with values in L2). Therefore (2.62) follows from (P?,?). The previous proof makes essential use of the finite propagation speed of thewave equation (P?,?) to prove (2.62). This property fails for discrete Dirichletforms such as (2.30) where we instead know the property (P??,B) that polynomialsof degree n of the generator have finite range ?(n).This leads to the following problem. Find polynomials W ?t , t > 0, of degreeat most t satisfying the properties (2.60), (2.61), (2.63) such that the decomposi-tion formula (2.59) for 1/? holds. In the proof of Lemma 2.2.5, the verificationof (2.61) (and (2.63)) and of the decomposition formula (2.59) are directly linkedto the ?ballistic? scaling of the wave equation: Wt (?) = W1(?t2). To constructpolynomials satisfying such ?ballistic? estimates, we are led by the following re-markable discovery of Carne [39]: The Chebyshev polynomials Tk , k ? Z, definedbyTk (?) = cos(k arccos(?)) for all ? ? [?1, 1], k ? Z, (2.82)are solutions to the discrete (in space and time) wave equation in the followingsense: Let ?+ f (n) = f (n + 1) ? f (n) and ?? f (n) = f (n ? 1) ? f (n) be thediscrete (forward and backward) time differences. Then, as polynomials in X ,???+Tn (X ) = ?+??Tn (X ) = 2(X ? 1)Tn (X ). (2.83)In particular, when 2(X ? 1) = ?L or equivalently X = 1 ? 12 L, then v(n, x) =[Tn (1 ? 12 L)u](x) solves the following ?Cauchy problem? for the discrete waveequation:??+??v + Lv = 0, v(0) = u, (?+v ? ??v)(0) = 0. (2.84)The analogy between the discrete- and the continuous-time wave equations is likethat between the discrete- and the continuous-time random walk. It turns out thatthe structure of Chebyshev polynomials allows to prove the following lemma.562.2. Proof of main resultLemma 2.2.7 (Lemma 2.2.1 under (P??,B)). Let ? : R ? [0,?) satisfy the as-sumptions of Lemma 2.2.5. Then W ?t : [0, 4] ? [0,?), defined byW ?t (?) :=?n?Z?(arccos(1 ? 12?)t ? 2pint) for all ? ? [0, 4], t > 0, (2.85)is the restriction of a polynomial in ? of degree at most t to [0, 4], with coefficientssmooth in t, and, for any ? > 0, (2.59), (2.60), (2.61), (2.62), and (2.63) hold forall ? ? (0, 4 ? ?], t > 0.Proof. The proof verifies that W ?t as defined in (2.85) has the asserted properties.Let??t (x) :=?n?Z?(xt ? 2pint) =?k?Zt?1??(k/t) cos(k x) (2.86)where the second equality follows by symmetry of ??, the change of variables for-mula, and a version of the Poisson summation formula which is easily verified, forsufficiently nice ?. Then the claim (2.59) can be expressed as??1 =? ?0t2??t (arccos(1 ? 12?))dttfor all ? ? (0, 4]. (2.87)Let x = arccos(1? 12?) or equivalently ? = 2(1?cos x) = 4 sin2( 12 x). In termsof this change of variables, (2.87) and thus the claim (2.85) are then equivalent to14 sin?2( 12 x) =? ?0t2??t (x)dttfor all x ? (0, pi]. (2.88)The left-hand side defines a meromorphic function on C with poles at 2piZ. Itsdevelopment into partial fractions is (see e.g. [5, page 204])14 sin?2( 12 x) =?n?Z(x ? 2pin)?2 for all x ? C \ 2piZ. (2.89)It follows, by (2.79) with ? = 1 and ? = (x ? 2pin)2, assuming C = 1, that14 sin?2( 12 x) =?n?Z? ?0t2?((x ? 2pin)t) dttfor all x ? (0, pi]. (2.90)The order of the sum and the integral can be exchanged, by non-negativity of theintegrand, thus showing (2.88) and therefore (2.59).To verify that W ?t is the restriction of a polynomial, we note that by (2.85),(2.86), and supp(??) ? [?1, 1],W ?t (?) = ??t (arccos(1 ? 12?)) =?k?Zt?1??(k/t) cos(k arccos(1 ? 12?)) (2.91)=?k?Z?[?t ,t]t?1??(k/t)Tk (1 ? 12?)572.2. Proof of main resultwhere Tk , k ? Z, are the Chebyshev polynomials defined by (2.82). This showsthat W ?t (?) is indeed the restriction of a polynomial in ? of degree at most t to theinterval ? ? [0, 4]. In particular, (2.62) is a trivial consequence of (P??,B) whichstates that polynomials in L of degree n have range at most ?(n).Finally, we verify the estimate (2.63) and thus in particular (2.61). To this end,we note that, in analogy to (2.80), for ? ? [0, 4 ? ?] and non-negative integers m,?m?12??????m??m arccos(1 ?12?)?????? C?,m . (2.92)For example, for m = 1,??? arccos(1 ?12?) = 12 (? ? 14?2)?12 ? ??12 ??12 for ? ? [0, 4 ? ?]. (2.93)Therefore (2.63) follows, by the chain rule (or Fa? di Bruno?s formula), from(1 + t2(1 ? cos(x))l t?m??????m?xm ??t (x)?????? Cl ,m (2.94)which we will now show. The argument is essentially a discrete version of theclassic fact that the Fourier transform acts continuously on the Schwartz space ofsmooth and rapidly decaying functions on R. To show (2.94), first note that(1 ? cos(x))eik x = eik x ? 12 ei (k+1)x ? 12 ei (k?1)x =: ?keik x (2.95)and thus by induction, for any l ? N,(1 ? cos(x))leik x = (1 ? cos(x))l?1?keik x= ?k (1 ? cos(x))l?1eik x = ?lkeik x . (2.96)It follows by (2.86) and summation by parts that(1 + t2(1 ? cos(x))l t?m ?m?xm ??t (x) =?k?Zt?1??(k/t)(ik/t)m[(1 + t2?k )leik x](2.97)=?k?Z[(1 + t2?k )l t?1??(k/t)(ik/t)m]eik x .Let h(s) = 12 (|s | ? 1)1|s |?1 for s ? R. Then, for any smooth f : R? R,?nk f (k) = (h?n ? D2n f )(k), (2.98)582.3. Extensionswhere ? denotes convolution of two functions on R, h?n = h ? h ? ? ? ? ? h, and D fis the derivative of f . Indeed,?k f (k) = ? 12? 10[D f (k + t) ? D f (k ? t)] dt= ? 12? 10? t?tD2 f (k + s) ds dt=?RD2 f (s)h(s ? k) ds = (h ? D2 f )(k), (2.99)and (2.98) then follows by induction:?n+1k f = ?(h?n ? D2n f ) = h ? D2(h?n ? D2n f ) = h ? h?n ? D2D2n f . (2.100)It then follows using the facts that?k?Z |h?n (k ? s) | ? Cn , uniformly in s ? R,and that ?? is smooth and of rapid decay,t?1?k?Z????(1 + t2?2k )l [??(k/t)(ik/t)m]????(2.101)=l?n=0Cl ,nt?1?k?Z?R|h?n (k ? s) | |[D2n ((?)m ??)](s/t) | ds?l?n=0Cl ,nt?1?R|[D2n ((?)m ??)](s/t) | ds=l?n=0Cl ,n?R|[D2n ((?)m ??)](s) | ds ? Cm ,land thus (2.94), and therefore (2.63), follow from this inequality and (2.97). Proof of Lemma 2.2.1. Lemma 2.2.1 under (P?,?) is an immediate consequence ofLemma 2.2.5; under (P??,B), it follows from Lemma 2.2.7 with appropriate rescal-ing to achieve ? ? 3, i.e., by setting Wt (?) = c?1W ?t (c?) for some c > 0. 2.3 Extensions2.3.1 Discrete approximationIn view of the discussion about Chebyshev polynomials before Lemma 2.2.7, itis not surprising that the functions W ?t of Lemma 2.2.7 approximate the Wt ofLemma 2.2.5. In Proposition 2.3.1 below, we show that this is indeed the case withnatural error O(t?1) as t ? ?. This result is used in Section 2.3.2 to prove (2.36).592.3. ExtensionsProposition 2.3.1 (Discrete approximation). Let ? be as in Lemma 2.2.5 and 2.2.7,with associated functions Wt and W ?t for ? = 1. Then, for any integer l,|W ?t (?) ? Wt (?) | ? Cl (1 ? t)?1(1 + t2?)?l for all ? ? [0, 4]. (2.102)In particular, W ?t (?/t2) ? C?(?12 ) as t ? ?.Proof. Note that it suffices to restrict to t ? 1, since for t ? 1, the claim followsfrom (2.61). The left-hand side of (2.102) is then proportional to the absolute valueof?(arccos(1 ? 12?)t) ? ?(?12 t) +?n?Z\{0}?(arccos(1 ? 12?)t + 2pint). (2.103)We estimate the difference of the first two terms in (2.103) and the sum separately,and show that each of them satisfies (2.102). The first two terms can be written as?(arccos(1 ? 12?)t) ? ?(?12 t) = (arccos(1 ? 12?) ? ?12 )t?t (?) (2.104)with?t (?) =? 10??(s arccos(1 ? 12?)t + (1 ? s)?12 t) ds. (2.105)The bounds?2? = arccos(1 ? ?) + O(?) as ? ? 0+, (2.106)?2? ? arccos(1 ? ?) ? ?2?2? for all ? ? [0, 2], (2.107)and the rapid decay of ?? therefore imply that|?t (?) | ? Cl (1 + ?t2)?l (2.108)and?(arccos(1 ? 12?)t) ? ?(?12 t) ? Cl t?1(1 + t2?)?l . (2.109)To estimate the sum in (2.103), we can use the rapid decay of ? with the in-equality x + y ? 2(xy)1/2 to obtain that?n?Z\{0}?(xt + 2pint) ? Cl?n?Z\{0}(1 + xt + 2pint)?l (2.110)? Cl (1 + xt)?l/2t?l/2?n>0n?l/2 ? Cl (1 + xt)?l/2t?l/2for any l > 2, with the constant changing from line to line. In particular, uponsubstituting x = arccos(1 ? 12?), this bound and (2.107) imply?n?Z\{0}?(arccos(1 ? 12?)t + 2pint) ? Cl t?2l (1 + t2?)?l . (2.111)The claim then follows by adding (2.109) and (2.111). 602.3. Extensions2.3.2 Estimates for systems with constant coefficientsIn this section, we verify the assertions of Example 2.1.3. We work in the slightlymore general context of second-order elliptic systems (instead of operators) withconstant coefficients. These are defined as in Example 2.1.4, and we now show thatclaims of Example 2.1.3 hold mutadis mutandis. The analysis is straightforward,with aid of the Fourier transform. It reproduces several results of [4,31]. Note thatby writing L = 1c2[ 1c2L]?1 and considering 1c2L instead of L, we may assume thatthe coefficients, a, are bounded such that (P??,B) holds with B = 3 (for example).Spectral measuresThe spectral measures corresponding to the vector-valued case of (2.29) are givenin terms of the Fourier transform as follows. For F : [0,?) ? R,(v, F (L)u) =M?k ,l=1?Rd????????F????????d?i , j=1ai j ?i? j????????????????klv?k (?)u?l (?) d? (2.112)where u? = (u?1 , . . . , u?M ) is the Fourier transform of u = (u1 , . . . , uM ), separatelyfor each component,a(?) :=d?i , j=1ai j ?i? j =????????d?i , j=1akli j ?i? j????????k ,l=1,... ,M(2.113)are symmetric positive definite M ? M matrices, for all ? ? Rd , and the matricesF (a(?)) are defined in terms of the spectral decomposition of a(?). Similarly, forthe (vector-valued case of the) discrete Dirichlet form (2.30),(v, F (L)u) =M?k ,l=1?[??,?]d????????F????????d?i , j=1ai j (1 ? ei?i )(1 ? e?i? j )????????????????klv?k (?)u?l (?) d?(2.114)where here u? is the component-wise discrete Fourier transform. Let us also writea? (?) :=d?i , j=1ai j (1?ei?i )(1?e?i? j ) =????????d?i , j=1akli j (1 ? ei?i )(1 ? e?i? j )????????k ,l=1,...,M.(2.115)We will often use, without mentioning this further, that the spectra of a(?) anda? (?) are bounded from above and from below by |? |2.612.3. ExtensionsEstimatesLet us introduce the following notation for derivatives: For a function u : Rd ?R, we regard the lth derivative, Dlu(x), as an l-linear form, and |Dlu(x) | is anorm of the form Dlu(x). In terms of the Fourier transform, we denote by ?Dl (?)the corresponding ?multiplier? operator from functions to l-linear forms, and by| ?Dl (?) | its norm. Similarly, for a discrete function u : Zd ? R, the lth orderdiscrete difference in positive coordinate direction is denoted by ?lu(x) and hasFourier multiplier ??l (?). In particular, when l = 1,?D(?)  (i?1 , . . . , i?d ), ??(?)  (ei?1 ? 1, . . . , ei?d ? 1). (2.116)Furthermore, k and p will denote integers that may be chosen arbitrarily, and Cconstants that can change from instance to instance and may depend on k and p, aswell as l = (lx , ly , la , lm2 ), B+, B? , and M+, but not on x, ?, and m.Proof of (2.37),(2.33),(2.34). It follows by the change of variables ? 7? t?, fromthe fact that a(?) is homogeneous of degree 2, and from Wt (?) = W1(?t2) that?t (x , y; a,m2) = t2?RdWt (a(?) + m2)ei (x?y)?? d? (2.117)= t?(d?2) ??( x ? yt; a,m2t2)with??(x; a,m2) :=?RdW1(a(?) + m2)ei (x?y)?? d? (2.118)which is supported in |x | ? B+. This verifies (2.37). Furthermore, (2.33) is astraightforward consequence of (2.117) by differentiation and (2.63). Let us omitthe details and only verify them explicitly in the discrete case (2.34): The (deriva-tives of the) decomposition kernel ??t can here be expressed asDlaa Dlm2m2 ?lxx ?lyy ??t (x , y; a,m2) = t?(d?2)?lx?ly+2lm2 ???t ;l (x ? y; a,m2) (2.119)with???t ;l (x; a,m2) = td+lx+ly?2lm2?[??,?]dDlaa Dlm2m2 W ?t (a? (?) + m2) ??ly ??lx ei x ?? d?.(2.120)Thus (2.63), | ??(?) | ? C |? |, and ? ? a? (?)? ? C |? |2 |? |2 for ? ? RM imply| ???t ;l (x; a,m2) | ? C?[??,?]d(1 + C |? |2t2 + m2t2)?k?p (t |? |)lx+ly?2lm2 tdd?(2.121)? C(1 + m2t2)?k?Rd(1 + C |? |2)?p |? |lx+ly?2lm2 d?622.3. Extensionsand therefore that the integral converges if 12 (d + lx + ly ) > lm2 and p is chosensufficiently large. It follows that| ???t ;l (x; a,m2) | ? C(1 + m2t2)?k (2.122)verifying the claim. Proof of (2.36).?lxx ?lyy ??t (x , y) ? Dlxx Dlyy ?t (x , y) = t2?[??,?]dW ?t (a? (?)) ??lx ??ly ei? ?(x?y) d?(2.123)? t2?RdWt (a(?)) ?Dlx ?Dly ei? ?(x?y) d?.To simplify notation, we will write ?Dl = ?Dlx ?Dly = ?Dlx ? ?Dly if l = (lx , ly ),and similarly for ?. Then the difference (2.123) may be estimated as follows.Proposition 2.3.1 implies?[??,?]d|W ?t (a? (?) + m2) ? Wt (a? (?) + m2) | | ?Dl (?) | d?? Ct?1?Rd(1 + C |? |2t2 + m2t2)?p?k |? |l d? ? Ct?d?l?1(1 + m2t2)?k (2.124)where we have assumed in the second inequality above that p was chosen suffi-ciently large so that the integral is convergent. Similarly, we may proceed for theother differences, always choosing p large enough in the estimates. Using (2.63)with m = 1 and |a? (?) ? a(?) | = O(|? |3), which follows from Taylor?s theorem,we obtain?[??,?]d|Wt (a? (?) + m2) ? Wt (a(?) + m2) | | ?Dl (?) | d?? C?Rd|? |(1 + C |? |2t2 + m2t2)?p?k |? |l d? ? Ct?d?l?1(1 + m2t2)?k .(2.125)Taylor?s theorem similarly implies | ??l (?) ? ?Dl (?) | ? C |? |l+1 so that, by (2.61),?[??,?]d|W ?t (a? (?) + m2) | | ??l (?) ? ?Dl (?) | d?? C?Rd(1 + C |? |2t2 + m2t)?p?k |? |l+1 d? ? Ct?d?l?1(1 + m2t2)?k . (2.126)632.3. ExtensionsFinally, we obtain by (2.61) that?Rd\[??,?]d|Wt (a(?) + m2) | | ?Dl (?) | d?? C?Rd\[??,?]d(1 + C |? |2t2 + m2t2)?p?k |? |l d? ? Ct?2p (1 + m2t2)?k .(2.127)The combination of the previous four inequalities gives (2.36). 64Chapter 3Structural stability of a class ofdynamical systems3.1 Introduction and main result3.1.1 IntroductionLet V = R3 with elements V ? V written V = (g, z, ?) and considered as a columnvector for matrix multiplication. For each j ? N0 = {0, 1, 2, . . .}, we define thequadratic flow ?? j : V ? V by?? j (V ) =?????????1 0 00 1 0? j ? j ? j?????????V ????????????VT qgj VVT qzj VVT q?j V???????????, (3.1)with the quadratic terms of the formqgj =?????????? j 0 00 0 00 0 0?????????, qzj =??????????? j 12 ? j 012 ? j 0 00 0 0??????????, (3.2)andq?j =????????????ggj12?gzj12?g?j12?gzj ?zzj12?z?j12?g?j12?z?j 0???????????. (3.3)All entries in the above matrices are real numbers. We assume that there exists a? > 1 such that ? j ? ? for all j, together with assumptions that ensure that formost values of j we have ? j ? c > 0 and ? j ? 0. Our hypotheses on the parametersof ?? are stated precisely in Assumptions (A1?A2) below. The significance of theassumption c > 0 is explained in Section 3.1.3 below.The quadratic flow ?? defines a time-dependent discrete-time 3-dimensional dy-namical system. It is triangular, in the sense that the equation for g does not dependon z or ?, the equation for z depends only on g, and the equation for ? depends on653.1. Introduction and main resultg and z. Moreover, the equation for z is linear in z, and the equation for ? is linearin ?. This makes the analysis of the quadratic flow elementary.Our main result concerns structural stability of the dynamical system ?? under aclass of infinite-dimensional perturbations. Let (K j ) j?N0 be a sequence of Banachspaces and X j = K j ? V. We write x j ? X j as x j = (K j ,Vj ) = (K j , gj , z j , ? j ). Anorm on X j is given by?x j ?X j = max{?K j ?K j , ?Vj ?V} = max{?K j ?K j , |gj |, |z j |, |? j |}. (3.4)We identify K j and V with subspaces of X j , so that ?K j ?K j = ?K j ?X j and ?V ?V =?V ?X j with this norm on X j . However, we will only make use of the norm of theK- and V -components in X j separately, but never of ?x j ?X j . (The reason is thatthe two components will need to be re-weighted.) Suppose that we are given maps? j : X j ? K j+1 and ? j : X j ? V. Then we define ? j : X j ? X j+1 by? j (K j ,Vj ) = (? j (K j ,Vj ), ?? j (Vj ) + ? j (K j ,Vj )). (3.5)This is an infinite-dimensional perturbation of the 3-dimensional quadratic flow??, which breaks triangularity and which involves the spaces K j in a nontrivialway. We will impose estimates on ? j and ? j below, which make ? a third-orderperturbation of ??.We give hypotheses under which there exists a sequence (x j ) j?N0 with x j ? X jwhich is a global flow of ?, in the sense thatx j+1 = ? j (x j ) for all j ? N0 , (3.6)obeying the boundary conditions that (K0 , g0) is fixed, z j ? 0, and ? j ? 0.Moreover, within an appropriate space of sequences, this global flow is unique.As we have discussed in more detail in Chapter 1, this result provides an essen-tial ingredient in a renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk [9, 19, 38], where the boundary condition ? j ? 0is the appropriate boundary condition for the study of a critical trajectory. It is thisapplication that provides our immediate motivation to study the dynamical system?, but we expect that the methods developed here will have further applications todynamical systems arising in renormalisation group analyses in statistical mechan-ics.3.1.2 Dynamical systemWe think of ? = (? j ) j?N0 as the evolution map of a discrete time-dependent dy-namical system, although it is more usual in dynamical systems to have the spaces663.1. Introduction and main resultX j be identical. Our application in [9, 19, 38] requires the greater generality ofj-dependent spaces.In the case that ? is a time-independent dynamical system, i.e., when ? j = ?and X j = X for all j ? N0, its fixed points are of special interest: x? ? X is a fixedpoint of ? if x? = ?(x?). The dynamical system is called hyperbolic near a fixedpoint x? ? X if the spectrum of D?(x?) is disjoint from the unit circle [99]. It is aclassic result that for a hyperbolic system there exists a splitting X  Xs ? Xu intoa stable and an unstable manifold near x?. The stable manifold is a submanifoldXs ? X such that x j ? x? in X , exponentially fast, when (x j ) satisfies (3.6) andx0 ? Xs . This result can be generalised without much difficulty to the situationwhen the ? j and X j are not necessarily identical, viewing ?0? as a fixed point(although 0 is the origin in different spaces X j ). The hyperbolicity condition mustnow be imposed in a uniform way [25, Theorem 2.16].By definition, ?? j (0) = 0, and we will make assumptions below which can beinterpreted as a weak formulation of the fixed point equation ? j (0) = 0 for thedynamical system defined by (3.5). Despite this technical condition, will simplyrefer to 0 as a fixed point of ?. This fixed point 0 is not hyperbolic due to the twounit eigenvalues of the matrix in the first term of (3.1). Thus the g- and z-directionsare centre directions, which neither contract nor expand in a linear approximation.On the other hand, the hypothesis that ? j ? ? > 1 ensures that the ?-direction isexpanding, and we will assume below that ? j : X j ? K j+1 is such that the K-direction is contractive near the fixed point 0. The behaviour of dynamical systemsnear non-hyperbolic fixed points is much more subtle than for the hyperbolic case.A general classification does not exist, and a nonlinear analysis is required.3.1.3 Main resultIn Section 3.2, we give an elementary proof that there exists a unique global flow?V = (g?, z?, ??) of the quadratic flow ?? with boundary conditions g?0 = g0 (alwaysassumed sufficiently small) and ( z?? , ???) = (0, 0), where we are writing, e.g.,z?? = lim j?? z? j . Our main result is that, under the assumptions stated below,there exists a unique global flow of ? with small initial conditions (K0 , g0) andfinal conditions (z? , ??) = (0, 0), and that this flow is a small perturbation of ?V .The sequence g? = (g?j ) plays a prominent role in the analysis. Determined bythe sequence (? j ), it obeysg?j+1 = g?j ? ? j g?2j , g?0 = g0 > 0. (3.7)We regard g? as a known sequence (only dependent on the initial condition g0). Thefollowing examples are helpful to keep in mind.673.1. Introduction and main resultExample 3.1.1. (i) Constant ? j = b > 0. In this case, it is not difficult to showthat g?j ? g0(1 + g0bj)?1 ? (bj)?1 as j ? ? (e.g., by applying (3.41) below with?(t) = t?2).(ii) Abrupt cut-off, with ? j = b for j ? J and ? j = 0 for j > J, with J ? 1. Inthis case, g?j is approximately the constant (bJ)?1 for j > J. In particular, g?j doesnot go to zero as j ? ?.Example 3.1.1 prompts us to make the following general definition of a cut-offtime for bounded sequences ? j . Let ? ??? = sup j?0 | ? j | < ?, and let n+ = n ifn ? 0 and otherwise n+ = 0. Given a fixed ? > 1, we define the ?-cut-off time j?byj? = inf{k ? 0 : | ? j | ? ??( j?k )+ ? ??? for all j ? 0}. (3.8)The infimum of the empty set is defined to equal ?, e.g., if ? j = b for all j. Bydefinition, j? ? j?? if ? ? ??. To abbreviate the notation, we write? j = ??( j? j?)+ . (3.9)The evolution maps ? j are specified by the real parameters ? j , ? j , ? j , ? j ,? j , ? j , ???j , together with the maps ? j and ? j on X j . Throughout this paper, wefix ? > 1 and make Assumptions (A1?A2) on the real parameters and Assump-tion (A3) on the maps, all stated further below. The constants in all estimates arepermitted to depend on the constants in these assumptions, including ?, but not onj? and g0 > 0. Furthermore, we consider the situation when the parameters of ?? jare continuous maps from a metric space Mext of external parameters, m ? Mext,into R, that the maps ? j and ? j similarly have continuous dependence on m, andthat j? is allowed to depend on m, but that Assumptions (A1?A3) hold with theconstants independent of m. Corollary 3.1.7 below then shows that the solutions to(3.6) constructed in Theorem 3.1.4 below also depend continuously on m.In Section 3.2, as a preliminary result to the proof of the main result, weprove the following Proposition 3.1.2 concerning flows of the three-dimensionalquadratic dynamical system ??. Its proof is elementary.Assumption (A1). The sequence ?: The sequence (? j ) is bounded: ? ??? < ?.There exists c > 0 such that ? j ? c for all but c?1 values of j ? j?.Assumption (A2). The other parameters of ??: There exists ? > 1 such that ? j ? ?for all j ? 0. There exists c > 0 such that ? j ? 0 for all but c?1 values of j ? j?.Each of ? j , ? j , ? j , ? j , ? j , ???j is bounded in absolute value by O( ? j ), with aconstant that is independent of both j and j?.683.1. Introduction and main resultNote that when j? < ?, Assumption (A1) permits the possibility that eventu-ally ?k = 0 for large k. The simplest setting for the assumptions is in the situationwhen j? = ?, for which ? j = 1 for all j. Our applications include situations inwhich ? j approaches a positive limit as j ? ?, but also situations in which ? j isapproximately constant in j over a long initial interval j ? j? and then abruptlydecays to zero.Proposition 3.1.2. Assume (A1?A2). If g?0 > 0 is sufficiently small, then there existsa unique global flow ?V = ( ?V ) j?N0 = (g?j , z? j , ?? j ) j?N0 of ?? with initial condition g?0and ( z?? , ???) = (0, 0). This flow satisfies the estimates? j g?j = O(g?01 + g?0 j), z? j = O( ? j g?j ), ?? j = O( ? j g?j ), (3.10)with constants independent of j? and g?0. Furthermore, if the maps ?? j depend con-tinuously on an external parameter such that (A1?A2) hold with uniform constants,then ?Vj is continuous in this parameter, for every j ? N0.We now define domains D j ? X j on which we assume the perturbation (? j , ? j )to be defined, and an assumption which states estimates for (? j , ? j ). The domainand estimates depend on an initial condition g0 > 0 and a possible external param-eter m. Theorem 3.1.4 below shows existence and uniqueness of solutions to (3.6)with this initial condition, and existence and differentiability of solutions for initialconditions in a neighborhood of g0.For parameters r, u > 0 and sufficiently small g0 > 0, let (g?j , z? j , ?? j ) j?N0 bethe sequence determined by Proposition 3.1.2 with initial condition g?0 = g0, anddefine the domain D j = D j (g0 , r, u) ? X j byD j = {x j ? X j : ?K j ?K j ? r ? j g?3j ,|gj ? g?j | ? ug?2j | log g?j |,|z j ? z? j | ? u? j g?2j | log g?j |,|? j ? ?? j | ? u? j g?2j | log g?j |}. (3.11)Note that if ? j depends on an external parameter m, then D j also depends on thisparameter through g?j = g?j (m). For statements concerning continuity in m, we willassume that ? j is defined on the union of these domains over m ? Mext.Throughout this chapter, we denote by D?? the Fr?chet derivative of a map ?with respect to the component ?, and by Lm (X j , X j+1) the space of bounded m-linear maps from X j to X j+1. The following Assumption (A3) depends on positiveparameters (g0 , r, u, ?,?, R, M).693.1. Introduction and main resultAssumption (A3). The perturbation: The maps ? j : D j ? K j+1 ? X j+1 and ? j :D j ? V ? X j+1 are three times continuously Fr?chet differentiable, there exist? ? (0,??1), R ? (0, r (1? ??)), and M > 0 such that, for all x j = (K j ,Vj ) ? D j ,?? j (0,Vj )?K j+1 ? R? j+1g?3j+1, ?? j (x j )?V ? M ? j+1g?3j+1, (3.12)?DK? j (x j )?L(K j ,K j+1) ? ?, ?DK ? j (x j )?L(K j ,V) ? M, (3.13)and such that, for both ? = ? and ? = ? and 2 ? n + m ? 3,?DV ? j (x j )?L(V,X j+1) ? M ? j g?2j+1, (3.14)?DmV DnK? j (x j )?Ln+m (X j ,X j+1) ? M ( ? j g?3j+1)1?n (g?2j+1 | log g?j+1 |)?m . (3.15)The bounds (3.12) guarantee that ? is a third-order perturbation of ??. More-over, since ? < 1, the ?-part of (3.13) ensures that the K-direction is contractivefor ?. (3.15) imposes bounds on the second and third derivatives of ? and ? whichpermit these derivatives to be quite large.The following elementary Lemma 3.1.3 shows that a sequence ( ?K j ) j?N0 can bedefined inductively by ?K j+1 = ? j ( ?K j , ?Vj ), assuming that r is large enough. Denoteby piK D j the projection of D j onto K j , i.e.,piK D j = {K j ? K j : ?K j ?K j ? r ? j g?3j }. (3.16)Lemma 3.1.3. Assume Assumption (A3), let r? ? (R/(1? ??), r], and assume thatg0 > 0 is sufficiently small. Then ? j (D j (g0 , r? , u)) ? piK D j+1(g0 , r? , u).Proof. The triangle inequality and the first bounds of (3.12)?(3.13) imply?? j (K j ,Vj )?K j+1 ? ?? j (0,Vj )?K j+1 + ?? j (K j ,Vj ) ? ? j (0,Vj )?K j+1? R? j+1g?3j+1 + r???(1 + O(g0)) ? j+1g?3j+1? r? ? j+1g?3j+1, (3.17)where the last inequality uses that g?3j /g?3j+1 = 1+O(g0) whose elementary verifica-tion is given in Lemma 3.2.1(i) below, and that g0 > 0 is sufficiently small. The sequence x? = ( ?K j , ?Vj ) j?N0 is a flow of the dynamical system ?? = (?, ??)in the sense of (3.6), with initial condition ( ?K0 , g?0) = (K0 , g0) and final condition( z?? , ???) = (0, 0). We consider this sequence as a function x? : (K0 , g0) 7? x? =x?(K0 , g0) of the initial condition (K0 , g0). Our main result is the following Theo-rem 3.1.4 about flows x of the dynamical system ? = (?, ?? + ?) = ?? + (0, ?) ofinterest, as perturbations of the flows x? of ??.703.1. Introduction and main resultTheorem 3.1.4. Assume (A1?A3) with parameters (g0 , r, u, ?,?, R, M), let r? ?(R/(1???), r), b ? (0, 1). There exists u? > 0 such that for all u ? u?, there existsg? > 0 such that if g0 ? (0, g?] and ?K0?K0 ? r?g30 , the following conclusions hold.(i) There exists a global flow x = (K j ,Vj ) j?N0 of ? = (?, ?? + ?) with initialcondition (K0 , g0) and final condition (z? , ??) = (0, 0) such that, with x? =x?(K0 , g0), the following estimates hold:?K j ? ?K j ?K j ? b(r ? r?) ? j g?3j , (3.18)|gj ? g?j | ? bug?2j | log g?j |, (3.19)|z j ? z? j | ? bu? j g?2j | log g?j |, (3.20)|? j ? ?? j | ? bu? j g?2j | log g?j |. (3.21)The sequence x is the unique solution to (3.6) which obeys the boundaryconditions and the bounds (3.18)?(3.21).(ii) There is a neighbourhood I = I(K0 , g0) ? K0 ? R of (K0 , g0) such that,for initial conditions (K ?0 , g?0) ? I, there also exists a global flow x? of ?with (z?? , ???) = (0, 0), and (3.18)?(3.21) hold with x replaced by x? and x?replaced by x?? = x?(K ?0 , g?0). Moreover, for all j ? N0, the maps (K j ,Vj ) :I ? K j ? V are continuously Fr?chet differentiable, and?z0?g0= O(1), ??0?g0 = O(1). (3.22)Remark 3.1.5. (i) For j? = ? and with (3.10), the bounds (3.18) and (3.19)?(3.21)imply ?K j ?K j = O( j?3) and Vj = O( j?2 log j), respectively. However, the latterbounds do not reflect that K j ,Vj ? 0 as g0 ? 0, while the former do. Furthermore,(3.10) implies ? j g?j ? 0 as j ? ? (also when j? < ?), and thus (3.18) and(3.20)?(3.21) imply K j ? 0, z j ? 0, ? j ? 0 as j ? ?. More precisely, theseestimates imply z j , ? j = O( ? j g?j ) so that z j and ? j decay exponentially afterthe ?-cut-off time j?; we interpret this as indicating that the boundary condition(z? , ??) = (0, 0) is essentially achieved already at j?.(ii) We do not give a proof, but we expect that the error bounds in (3.18)?(3.21)have optimal decay as j ? ?. Some indication of this can be found in Re-mark 3.3.2 below.Theorem 3.1.4 is an analogue of a stable manifold theorem for the non-hyper-bolic dynamical system defined by (3.5). It is inspired by [25, Theorem 2.16]which however holds only in the hyperbolic setting. Irwin [78] showed that thestable manifold theorem for hyperbolic dynamical systems is a consequence of the713.1. Introduction and main resultimplicit function theorem in Banach spaces (see also [99, 102]). Irwin?s approachwas inspired by Robbin [97], who showed that the local existence theorem forordinary differential equations is a consequence of the implicit function theorem.By contrast, in our proof of Theorem 3.1.4, we directly apply the local existencetheorem for ODEs, without explicit mention of the implicit function theorem. Thisturns out to be advantageous to deal with the lack of hyperbolicity.Our choice of ?? in (3.1) has a specific triangular form. One reason for this isthat (3.1) accommodates what is required in our application in [9,19,38]. A secondreason is that additional nonzero terms in ?? can lead to the failure of Theorem 3.1.4.The condition that ? j is mainly non-negative is important for the sequence g?j of(3.7) to remain bounded. The following example shows that for the ? j term in theflow of z?, our sign restriction on ? j is also important, since positive ? j can lead toviolation of a conclusion of Theorem 3.1.4.Example 3.1.6. Suppose that ? j = ? j = ? j = 1, that ? = 0, and that g?0 > 0 issmall. For this constant ? sequence, j? = ? (for any ? > 1) and hence ? j = 1 forall j. As in Example 3.1.1, g?j ? j?1. By (3.1) and (3.7),z? j+1 = z? j (1 ? g?j ) ? g?2j = z? jg?j+1g?j? g?2j . (3.23)Let y? j = z? j/g?j . Since g?j/g?j+1 = (1 ? g?j )?1 ? 1, we obtain y? j ? y? j+1 + g?j andhencey? j ? y?n+1 +n?l= jg?l . (3.24)Suppose that z? j = O(g?j ), as in (3.20). Then y? j = O(1) and hence by taking thelimit n ? ? we obtainy? j ? lim supn??????????y?n+1 +n?l= jg?l????????? ?C +??l= jg?l . (3.25)However, since g?j ? j?1, the last sum diverges. This contradiction implies that theconclusion z? j = O(g?j ) of (3.20) is impossible.Because of its triangularity, an exact analysis of the flows of ?? with the bound-ary conditions of interest is straightforward: the three equations for g, z, ? can besolved successively and we do this in Section 3.2 below. Triangularity does nothold for ?, and we prove that the flows of ? with the same boundary conditionsnevertheless remain close to the flow of ?? in Section 3.3.723.2. The quadratic flow3.1.4 Continuity in external parameterThe uniqueness statement of Theorem 3.1.4 implies the following Corollary 3.1.7regarding continuous dependence on an external parameter of the solution to (3.6)given by Theorem 3.1.4.Corollary 3.1.7. Assume that the ? j depend continuously on an external param-eter m ? Mext and that Assumptions (A1?A3) hold uniformly in m. Let x(m) =(K (m),V (m)) be the solution for external parameter m given by Theorem 3.1.4.Then x j (m) is continuous in m for each j ? N0.Proof. Theorem 3.1.4 implies that V0(m) is bounded uniformly in m ? Mext. Thisimplies that there exists some limit point V ?0 of V0(m?) as m? ? m. Let x?j =(V ?j , K?j ) be the flow of ?(m, ?) started with this V ?0 and K?0 = K0 independentof m. By Proposition 3.1.2, ?V0(m) is continuous in m ? Mext. The continuity of?K j (m) follows inductively from this and the assumed continuity of the ? j and ? j .This continuity and (3.18)?(3.21) imply that any limit point x? must satisfy?K?j ? ?K j (m)?K j ? b(r ? r?) ? j (m)g?j (m)3 , (3.26)|g?j ? g?j (m) | ? bug?j (m)2 | log g?j (m) |, (3.27)|??j ? ?? j (m) | ? bu? j (m)g?j (m)2 | log g?j (m) |, (3.28)|z?j ? z? j (m) | ? bu? j (m)g?j (m)2 | log g?j (m) |. (3.29)The uniqueness assertion of Theorem 3.1.4 implies that x?j = x j (m), and thereforethat V0 is continuous in m. The continuity of x j now follows immediately from thecontinuity of the ? j . 3.2 The quadratic flowIn this section, we prove that, for the quadratic approximation ??, there is a uniquesolution ?V = ( ?Vj ) j?N0 = (g?j , z? j , ?? j ) j?N0 to the flow equation?Vj+1 = ?? j ( ?Vj ) with fixed small g?0 > 0 and with ( z?? , ???) = (0, 0). (3.30)Due to the triangular nature of ??, we can obtain very detailed information aboutthe sequence ?V .3.2.1 Flow of g?We start with the analysis of the sequence g?, which obeys the recursiong?j+1 = g?j ? ? j g?2j , g?0 > 0. (3.31)733.2. The quadratic flowThe following lemma collects the information we will need about g?.Lemma 3.2.1. Assume (A1). The following statements hold if g?0 > 0 is sufficientlysmall, with all constants independent of j? and g?0.(i) For all j, g?j > 0,g?j = O( infk? jg?k ), and g?j g??1j+1 = 1 + O( ? j g?j ) = 1 + O(g?0). (3.32)Moreover, for any j and k, g?j is non-increasing in ?k .(ii) For n ? 1 and m ? 0, there exists Cn ,m > 0 such that for all k ? j ? 0,k?l= j?l g?nl | log g?l |m ? Cn ,m???????| log g?k |m+1 n = 1? j g?n?1j | log g?j |m n > 1,(3.33)and there exists C > 0 such that for all j ? 0,? j g?j ?Cg?01 + g?0 j . (3.34)(iii) (a) For ? ? 0 and j ? 0, there exist constants cj = 1 + O( ? j g?j ) (dependingon ?) such that, for all l ? j,l?k= j(1 ? ? ?k g?k )?1 =(g?jg?l+1)?(cj + O( ?l g?l )). (3.35)(b) For ? j ? 0 except for c?1 values of j ? j?, ? j = O( ? j ), and j ? l, (witha constant independent of j and l),l?k= j(1 ? ?k g?k )?1 ? O(1). (3.36)(iv) Suppose that g? and g? each satisfy (3.31). Let ? > 0. If |g?0 ? g?0 | ? ?g?0 then|g?j ? g?j | ? ?g?j (1 + O(g?0)) for all j.Proof. (i) By (3.31),g?j+1 = g?j (1 ? ? j g?j ). (3.37)Since ? j = O( ? j ), by (3.37) the second statement of (3.32) is a consequence thefirst, so it suffices to verify the first statement of (3.32). Assume inductively thatg?j > 0 and that g?j = O(infk? j g?k ). It is then immediate from (3.37) that g?j+1 > 0 if743.2. The quadratic flowg?0 is sufficiently small depending on ? ???, and that g?j+1 ? g?j if ? j ? 0. By (A1),there are at most c?1 values of j ? j? for which ? j < 0. Therefore, by choosingg?0 sufficiently small depending on ? ??? and c, it follows that g?j ? O(infk? j g?k )for all j ? j? with a constant that is independent of j?.To advance the inductive hypothesis for j > j?, we use 1 ? t ? e?t and??l= j? | ?l | ???n=1??n = O(1) to obtain, for j ? k ? j?,g?j ? g?k exp?????????j?1?l=k?l g?l????????? g?k exp????????Cg?kj?1?l=k| ?l |????????? O(g?k ). (3.38)This shows that g?j = O(inf j??k? j g?k ). However, by the inductive hypothesis,g?j? = O(infk? j? g?k ) for j ? j?, and hence for j > j? we have g?j = O(infk? j g?k )as claimed. This completes the verification of the first bound of (3.32) and thus, asalready noted, also of the second.The monotonicity of g?j in ?k can be proved as follows. Since it is obvious thatg?j does not depend on ?k if k ? j, we may assume that k < j. Moreover, byreplacing j by j + k, we can assume that k = 0. Let g??j = ?g?j/? ?0. Clearly, g??0 = 0and thereforeg??1 = ?g?20 < 0. (3.39)Assuming that g??j < 0 by induction, it follows that for j ? 1,g??j+1 = g??j (1 ? 2? j g?j ) < 0, (3.40)and the proof of monotonicity is complete.(ii) We first show that if ? : R+ ? R is absolutely continuous, thenk?l= j?l?(g?l )g?2l =? g? jg?k+1?(t) dt + O(? g? jg?k+1t2 |??(t) | dt). (3.41)To prove (3.41), we apply (3.31) to obtaink?l= j?l?(g?l )g?2l =k?l= j?(g?l )(g?l ? g?l+1) =k?l= j? g?lg?l+1?(g?l ) dt. (3.42)The integral can be written as? g?lg?l+1?(g?l ) dt =? g?lg?l+1?(t) dt +? g?lg?l+1? g?lt??(s) ds dt. (3.43)753.2. The quadratic flowThe first term on the right-hand side of (3.41) is then the sum over l of the firstterm on the right-hand side of (3.43), so it remains to estimate the double integral.By Fubini?s theorem,? g?lg?l+1? g?lt??(s) ds dt =? g?lg?l+1? sg?l+1??(s) dt ds=? g?lg?l+1(s ? g?l+1)??(s) ds. (3.44)By (3.31) and (3.32), for s ? [g?l+1 , g?l ] we have|s ? g?l+1 | ? |g?l ? g?l+1 | = | ?l |g?2l ? (1 + O(g?0)) | ?l |g?2l+1 ? O(s2). (3.45)This permits us to estimate (3.44) and conclude (3.41).Direct evaluation of the integrals in (3.41) with ?(t) = tn?2 | log t |m givesk?l= j?l g?nl | log g?l |m ? Cn ,m???????| log g?k+1 |m+1 n = 1g?n?1j | log g?j |m n > 1.(3.46)To deduce (3.33), we only consider the case n > 1, as the case n = 1 is similar.Suppose first that j ? j? (and j? < ?). Assumption (A1) implies1 ? ?lc+(1 + | ?l |c)1?l<c ? O(?l ) + O(1?l<c ) (3.47)and therefore thatk?l= j?l g?nl | log g?l |m ?j??l= jO(?l )g?nl | log g?l |m +j??l= jO(1?l<c )g?nl | log g?l |m+k?l= j?+1??(l? j?)+ g?nl | log g?l |m . (3.48)By (3.46), the first term is bounded by O(g?n?1j | log g?j |m ). The second term obeysthe same bound, by (A1) and (3.32), as does the last term due to the exponentialdecay. This proves (3.33) for the case j ? j?. On the other hand, if j > j?, thenagain using the exponential decay of ?l and (3.32), we obtaink?l= j?l g?nl | log g?l |m ? C ? j g?nj | log g?j |m ? Cg?0 ? j g?n?1j | log g?j |m . (3.49)This completes the proof of (3.33) for the case n > 1.763.2. The quadratic flowTo prove (3.34), let c > 0 be as in Assumption (A1) and set g?j+1 = g?j ? cg?jwith g?0 = g?0. Let j0 = ?1 and denote by 0 ? j1 < j2 < . . . the sequence of jsuch that ? j < c. By induction, we show that g?jk+1 ? (1 + O(g?0))k g?jk+1. This istrivial for k = 0. To advance the induction, note that, since g?j is monotone in ?,g?j ? (1 + O(g?0))k g?j for j ? jk+1, and thereforeg?jk+1+1 = g?jk+1 (1 ? ? jk+1 g?jk+1 ) ? (1 ? ? jk+1 g?jk+1 )(1 + O(g?0))k g?jk+1=1 ? ? jk+1 g?jk+11 ? cg?jk+1(1 + O(g?0))k g?jk+1+1. (3.50)The induction is advanced since1 ? ? jk+1 g?jk+11 ? cg?jk+1= 1 + O(g?0). (3.51)By Assumption (A1), m = max{k : jk ? j?} is bounded so that, for j ? j?,? j g?j = g?j ? (1 + O(g?0))m g?j ? (1 + O(g?0))g?j . (3.52)For j > j?, we use that, for g?0 sufficiently small,??1 ? 1 ? cg?0 ? 1 ? cg?j =g?j+1g?j(3.53)and that, by (3.32), g?j = O(g?j? ) which together imply? j g?j ? O(??( j? j?) g?j? ) ? O(??( j? j?) g?j? ) ? O????????j?1?l= j?g?l+1g?l????????g?j? = O(g?j ).(3.54)The proof of (3.34) is concluded by the observation that g?j satisfies the boundclaimed, as can be seen by applying (3.41) with ?(t) = t?2.(iii-a) By Taylor?s theorem and (3.31), there exists rk = O(?k g?k )2 such that(1 ? ? ?k g?k )?1 = (1 ? ?k g?k )?? (1 + rk ) =(g?kg?k+1)?(1 + rk ). (3.55)Letcj ,l =l?k= j(1 + rk ). (3.56)773.2. The quadratic flowWith the bounds 1 + t ? e |t | and ?k = O( ?k ), we obtain???cj ,l ? 1??? =????????l?k= jrkl?m=k+1(1 + rm )?????????l?k= jO( ?k g?2k ) exp???????l?m=k+1O( ?m g?2m )???????? O( ? j g?j ). (3.57)In particular, cj ,l = 1 + O(g?0) = O(1) uniformly in j and l. Similarly, we obtainthat, with r?k = (1 + rk )?1 ? 1 = O( ?k g?2k ), for n ? l,|cj ,l ? cj ,n | = cj ,n???????n?k=l(1 + rk )?1 ? 1???????= cj ,n???????n?k=lr?kn?m=k+1(1 + r?m )???????? O( ?l g?l ). (3.58)In particular, (cj ,l )l is a Cauchy sequence, cj = liml?? cj ,l exists, and with (3.57),cj = 1 + O( ? j g?j ). It also follows that |cj ,l ? cj | ? O( ?l g?l ) as claimed, and theproof is complete.(iii-b) Since ? j ? 0 for all but c?1 values of j ? j?, by (3.32) with g?0 sufficientlysmall,?lk= j (1 ? ?k g?k )?1 ? O(1) for l ? j?, with a constant independent of j?.For j ? j?, we use 1/(1 ? x) ? 2ex for x ? [? 12 , 12 ] to obtainl?k= j(1 ? ?k g?k )?1 ? 2 exp????????l?k= j?k g?k????????? 2 exp????????Cg?j??k= j?| ?k |????????? O(1). (3.59)The bounds for l ? j? and j ? j? together imply (3.36).(iv) If |g?j ? g?j | ? ? j g?j then by (3.31),|g?j+1 ? g?j+1 | = |g?j ? g?j |(1 ? ? j (g?j + g?j )) ? ? j+1g?j+1 (3.60)with? j+1 = ? j1 ? ? j (g?j + g?j )1 ? ? j g?j= ? j(1 ?? j g?j1 ? ? j g?j). (3.61)In particular, if ? j ? 0, then ? j+1 ? ? j . By (A1), there are at most c?1 values ofj ? j? for which ? j < 0, and hence ? j ? ?(1 + O(g?0)) for j ? j?. The desiredestimate therefore holds for j ? j?. For j ? l > j?, as in (3.38) we havej?k=l(1 + O(?k g?k )) ? exp????????O(g?l )j?k=l?k????????? 1 + O(g?0), (3.62)and thus the claim remains true also for j > j?. 783.2. The quadratic flow3.2.2 Flow of z? and ?? and proof of Proposition 3.1.2We now establish the existence of unique solutions to the z? and ?? recursions withboundary conditions z?? = ??? = 0, and obtain estimates on these solutions.Lemma 3.2.2. Assume (A1?A2). If g?0 is sufficiently small then there exists a uniquesolution to (3.30) obeying z?? = ??? = 0. This solution obeys z? j = O( ? j g?j ) and?? j = O( ? j g?j ). Furthermore, if the maps ?? j depend continuously on an externalparameter m ? Mext such that (A1?A2) hold with uniform constants, then z? j and?? j are continuous in Mext.Proof. By (3.1), z? j+1 = z? j ? ? j g?j z? j ? ? j g?2j , so thatz? j =n?k= j(1 ? ?k g?k )?1 z?n+1 +n?l= jl?k= j(1 ? ?k g?k )?1?l g?2l . (3.63)In view of (3.36), whose assumptions are satisfied by (A2), the unique solution tothe recursion for z? which obeys the boundary condition z?? = 0 isz? j =??l= jl?k= j(1 ? ?k g?k )?1?l g?2l , (3.64)and by (A2), (3.33), and (3.36),| z? j | ???l= jO( ?l )g?2l ? O( ? j g?j ). (3.65)To verify continuity of z? j in an external parameter, let z? j ,n =?nl= j?lk= j (1 ??k g?k )?1?l g?2l . Clearly, since g?j is continuous in Mext for any j ? 0, z? j ,n is alsocontinuous, for any j ? n. (3.33)?(3.34) of Lemma 3.2.1(ii) imply that | z? j? z? j ,n | ?O( ?n g?n ) ? 0 uniformly, as n ? ?, and thus, as a uniform limit of continuousfunctions, it follows that z? j must be continuous in Mext.For ??, we first define? j = +? j g?j + ? j z? j ? ?ggj g?2j ? ?gzj g?j z? j ? ?zzj z?2j , ?j = ?g?j g?j + ?z?j z? j , (3.66)so that the recursion for ?? can be written as?? j+1 = (? j ? ?j ) ?? j + ? j . (3.67)Alternatively,?? j = (? j ? ?j )?1( ?? j+1 ? ? j ). (3.68)793.2. The quadratic flowGiven ? ? (??1 , 1), we can choose g?0 sufficiently small that12??1 ? (? j ? ?j )?1 ? ?. (3.69)The limit of repeated iteration of (3.68) gives?? j = ???l= j????????l?k= j(?k ? ?k )?1?????????l (3.70)as the unique solution which obeys the boundary condition ?? = 0. Geometricconvergence of the sum is guaranteed by (3.69), together with the fact that ? j ?O( ? j g?j ) ? O(1). To estimate (3.70), we use| ?? j | ???l= j?l? j+1O( ?l g?l ). (3.71)Since ? < 1, the first bound of (3.32) and monotonicity of ? imply that| ?? j | ? O( ? j g?j ). (3.72)The proof of continuity of ?? j in Mext is analogous to that for z? j . The proof iscomplete. Proof of Proposition 3.1.2. (3.10) follows immediately from Lemma 3.2.1(ii) andLemma 3.2.2. Since g?j is defined by a finite recursion, its continuity in m ? Mextis trivial. The continuity of z? j and ?? j was proved in Lemma 3.2.2. 3.2.3 Differentiation of quadratic flowThe following lemma gives estimates on the derivatives of the components of ?Vjwith respect to the initial condition g?0. We write f ? for the derivative of f with re-spect to g?0. These estimates will be an ingredient in the proof of Theorem 3.1.4(ii).Lemma 3.2.3. For each j ? 0, ?Vj = (g?j , z? j , ?? j ) is twice differentiable with respectto the initial condition g?0 > 0, and the derivatives obeyg??j = O???????g?2jg?20???????, z??j = O???????? jg?2jg?20???????, ???j = O???????? jg?2jg?20???????, (3.73)g???j = O???????g?2jg?0???????, z???j = O???????? jg?2jg?30???????, ????j = O???????? jg?2jg?30???????. (3.74)803.2. The quadratic flowProof. Differentiation of (3.7) givesg??j+1 = g??j (1 ? 2? j g?j ), (3.75)from which we conclude by iteration and g??0 = 1 that for j ? 1,g??j =j?1?l=0(1 ? 2?l g?l ). (3.76)Therefore, by (3.35),g??j =(g?jg?0)2(1 + O(g?0)). (3.77)For the second derivative, we use g???0 = 0 and g???j+1 = g???j (1 ? 2? j g?j ) ? 2? j g??2j toobtaing???j = ?2j?1?l=0?l g??2lj?2?k=l(1 ? 2?k g?k ). (3.78)With the bounds of Lemma 3.2.1, this givesg???j = O(g?jg?0)2 j?1?l=0?l g?2l = O???????g?2jg?0???????. (3.79)For z?, we define ? j ,l =?lk= j (1 ? ?k g?k )?1. Then (3.64) becomes z? j =??l= j ? j ,l?l g?2l . By (3.36), ? j ,l = O(1). It then follows from (A2), (3.77), andLemma 3.2.1(ii,iii-b) that??j ,l = ? j ,ll?k= j(1 ? ?k g?k )?1?k g??k =l?k= jO(?k g??k ) = O??????? jg?jg?20??????. (3.80)We differentiate (3.64) and apply (3.77) and Lemma 3.2.1(ii) to obtainz??j =??l= j??j ,l?l g?2l + 2??l= j? j ,l?l g?l g??l = O???????? jg?2jg?20???????. (3.81)Similarly, ???j ,l = O(g?2j /g?40 ) andz???j =??l= j???j ,l?l g?2l + 4??l= j??j ,l?l g?l g??l + 2??l= j? j ,l?l (g?l g???l + g??2l ) = O???????? jg?2jg?30???????(3.82)813.3. Proof of main resultusing the fact that g?3j /g?40 = O(g?2j /g?30 ) by (3.32). It is straightforward to justify thedifferentiation under the sum in (3.81)?(3.82).For ?? j , we recall from (3.69)?(3.70) that?? j = ???l= j????????l?k= j(?k ? ?k )?1?????????l , (3.83)with ?j and ?l given by (3.66), and with 0 ? (? j ? ?j )?1 ? ? < 1. This gives???j = ???l= j????????l?k= j(?k ? ?k )?1??????????????????l +l?i= j(?i ? ?i )?1??i????????. (3.84)The first product is bounded by ?l? j+1, and this exponential decay, together with(3.66), (3.65), and the bounds just proved for g?? and z??, lead to the upper bound| ???j | ? O( ? j g?2j g??20 ) claimed in (3.73). Straightforward further calculation leads tothe bound on ????j claimed in (3.74) (the leading behaviour can be seen from the z???jcontribution to the ???l term). 3.3 Proof of main resultIn this section, we prove Theorem 3.1.4. We begin in Section 3.3.1 with a sketch ofthe main ideas, without entering into details. The remainder of Section 3.3 expandsthe sketch into a complete proof.3.3.1 Proof strategyTwo difficulties in proving Theorem 3.1.4 arflow-e: (i) from the point of view ofdynamical systems, the evolution map ? is not hyperbolic; and (ii) from the pointof view of nonlinear differential equations, a priori bounds that any solution to (3.6)must satisfy are not readily available due to the presence of both initial and finalboundary conditions.Our strategy is to consider the one-parameter family of evolution maps ? =(?t )t?[0,1] defined by?t (x) = ?(t , x) = (?(x), ??(x) + t?(x)) for t ? [0, 1], (3.85)with the t-independent boundary conditions that K0 and g0 are given and thatz? = 0 and ??0). This family interpolates between the problem ?1 = ? weare interested in, and the simpler problem ?0 = ?? = (?, ??). The unique solution823.3. Proof of main resultfor ?? is x? j = ( ?K j , ?Vj ), where ?V is the unique solution of ?? from Section 3.2, andwhere ?K j is defined inductively for j ? 0 by?K j+1 = ? j ( ?Vj , ?K j ), ?K0 = K0. (3.86)We refer to x? as the approximate flow.We seek a t-dependent global flow x which obeys the generalisation of (3.6)given byx j+1 = ?tj (x j ). (3.87)Assuming that x j = x j (t) is differentiable in t for each j ? N0, we setx? j =??t x j . (3.88)Then differentiation of (3.87) shows that a family of flows x = (x j (t)) j?N0 ,t?[0,1]must satisfy the infinite nonlinear system of ODEsx? j+1 ? Dx? j (t , x j ) x? j = ? j (x j ), x j (0) = x? j . (3.89)Conversely, any solution x(t) to (3.89), for which each x j is continuously differen-tiable in t, gives a global flow for each ?t .We claim that (3.89) can be reformulated as a well-posed nonlinear ODEx? = F (t , x), x(0) = x? , (3.90)in a Banach space of sequences x = (x0 , x1 , . . . ) with carefully chosen weights,and for a suitable nonlinear functional F. To see this, consider the linear equationy j+1 ? Dx? j (t , x j )y j = r j , (3.91)where the sequences x and r are held fixed. Its solution with the same boundaryconditions as stated below (3.85) is written as y = S(t , x)r . Then we define F,which we consider as a map on sequences, byF (t , x) = S(t , x)?(x). (3.92)Thus y = F (t , x) obeys the equation y j+1 ? Dx? j (t , x j )y j = ? j (x), and hence(3.90) is equivalent to (3.89) with the same boundary conditions.The main work in the proof is to obtain good estimates for S(t , x), in the Ba-nach space of weighted sequences, which allow us to treat (3.90) by the standardtheory of ODE. We establish bounds on the solution simultaneously with existence,via the weights in the norm. These weights are useful to obtain bounds on the so-lution, but they are also essential in the formulation of the problem as a well-posedODE.833.3. Proof of main resultAs we will see in more detail in Section 3.3.4, the occurrence of Dx? j (t , x j )in (3.89), rather than the naive linearisation Dx? j (0) at the ?fixed point? x = 0,replaces the eigenvalue 1 in the upper left corner of the square matrix in (3.1) bya smaller eigenvalue 1 ? 2? jgj < 1. This helps address difficulty (i) mentionedabove. Also, the weights guarantee that a solution in the Banach space obeys thefinal conditions (z? , ??) = (0, 0), thereby helping to solve difficulty (ii).3.3.2 Sequence spaces and weightsWe now introduce the Banach spaces of sequences used in the reformulation of(3.89) as an ODE. These are weighted l?-spaces.Definition 3.3.1. Let X ? be the space of sequences x = (x j ) j?N0 with x j ? X j .For each ? = K, g, z, ? and j ? N0, we fix a positive weight w?, j > 0. We writex j ? X j = K j ? V as x j = (x?, j )?=K ,g ,z ,? . Let?x j ?Xwj = max?=K ,g ,z ,? (w?, j )?1?x?, j ?X j , ?x?Xw = supj?N0?x j ?Xwj , (3.93)andXw = {x ? X ? : ?x?Xw < ?}. (3.94)It is not difficult to check that Xw is a Banach space for any positive weightsequence w. Different choices of weights w will be needed. These are all definedin terms of the sequence g? = (g?j ) j?N0 which is the same as the sequence g? for afixed g?0; i.e., given g?0 > 0, it satisfies g?j+1 = g?j ? ? j g?2j . We define the two weightsw = w(g?0 , r, u) and r = r(g?0 , r, u) byw?, j =?????????????(r ? r?)g?3j ? j ? = Kug?2j | log g?j | ? = gug?2j | log g?j | ? j ? = z, ?,r?, j =?????????????(r ? r?)g?3j ? j ? = Kug?3j ? j ? = gug?3j ? j ? = z, ?,(3.95)where ( ? j ) is the ?-dependent sequence defined by (3.9). Furthermore, we recallthat x? = ( ?K , ?V ) = x?(K0 , g0) denotes the sequence in X ? uniquely determined fromthe boundary conditions ( ?K0 , g?0) = (K0 , g0) and ( z?? , ???) = (0, 0) via ?Vj+1 =?? j ( ?Vj ) and ?K j+1 = ? j ( ?K j , ?Vj ), whenever the latter is well-defined. Given an initialcondition ( ?K0 , g?0), let x? = x?( ?K0 , g?0).Denoting the closed ball of radius s in Xw by sB, observe that, if g?0 = g0 and?K0 = K0, the bounds (3.18)?(3.21) are equivalent to x ? x? + bB, and that, bydefinition, the projection of x? + B onto the the jth sequence element is containedin the domain D j . We will always assume that g0 = g?0 and g?0 are close, but notnecessarily that they are equal. The use of g? rather than g? permits us to vary the843.3. Proof of main resultinitial condition g0 = g?0 without changing the Banach spaces Xw , X r. The useof g0-dependent weights rather than, e.g., the weight j?2 log j for j? = ? (seeRemark 3.1.5(i)) allows us to obtain estimates with good behaviour as g0 ? 0.Note that the weight wg , j does not include a factor ? j , and thus does not go to 0when j? < ? (see Example 3.1.1(ii)).Remark 3.3.2. The weights w apply to the sequence x? (see (3.88)). As motivationfor their definition, consider the explicit example of ? j (x j ) = ? jg3j . In this case,the g equation becomes simplygj+1 = gj ? ? jg2j + t ? jg3j . (3.96)With the notation g?j = ??t gtj , differentiation givesg?j+1 = g?j (1 ? 2? jgj + 3t ? jg2j ) + ? jg3j . (3.97)Thus, by iteration, using g?0 = 0, we obtaing?j =j?1?l=0?lg3lj?1?k=l+1(1 ? 2?kgk + 3t ?kg3k ). (3.98)For simplicity, consider the case t = 0, for which g = g?. In this case, it followsfrom (3.35), (3.32), and (3.46) thatg?j ? O(1)j?1?l=0(g?jg?l+1)2?l g?3l = O(1)g2jj?1?l=0?l g?l ? O(g?2j | log g?j |), (3.99)which produces the weight wg , j of (3.95). (It can also be verified using (3.41) thatif we replace ? j by ? j in the above then no smaller weight will work.)3.3.3 Reduction to a linear equation with nonlinear perturbationFor given sequences x , r ? X ?, we now consider the equationy j+1 ? Dx? j (t , x j )y j = r j . (3.100)For x and r fixed, (3.100) is an inhomogeneous linear equation in y. Lemma 3.3.3below, which lies at the heart of the proof of Theorem 3.1.4, obtains bounds onsolutions to (3.100), including bounds on its x-dependence. The latter will allow usto use the standard theory of ODE in Banach spaces to treat the original nonlinearequation, where x and r are both functionals of the solution y, as a perturbation ofthe linear equation.853.3. Proof of main resultIn addition to the decomposition X j = K j ? V j , with x j ? X j written x j =(K j ,Vj ), it will be convenient to also use the decomposition X j = E j ? Fj withE j = K j ? R and Fj = R ? R, for which we write x j = (u j , v j ) with u j = (K j , gj )and v j = (z j , ? j ). We denote by pi? the projection operator onto the ?-componentof the space in which it is applied, with ? in any of {K,V }, {u, v} = {(K, g), (z, ?)},or {K, g, z, ?}.Recall that the spaces of sequences Xw are defined in Definition 3.3.1 and thespecific weights w and r in (3.95).Lemma 3.3.3. Assume (A1?A3). There exists a constant CS , independent of r andu, and a constant C?S = C?S (r, u), such that if g?0 > 0 is sufficiently small, thefollowing hold for all t ? [0, 1], x ? x? + B.(i) For r ? X r, there exists a unique solution y = S(t , x)r ? Xw of (3.100) withboundary conditions piu y0 = 0, piv y? = 0.(ii) The linear solution operator S(t , x) satisfies?S(t , x)?L(X r ,Xw) ? CS . (3.101)(iii) As a map S : [0, 1] ? ( x? + B) ? L(Xw , X r), the solution operator is contin-uously Fr?chet differentiable and satisfies?DxS(t , x)?L(Xw ,L(X r ,Xw)) ? C?S . (3.102)Lemma 3.3.3 needs to be supplemented with information about the initial con-dition x? and the perturbation ? for the analysis of (3.90) with (3.92). (Note thatthe sequence x? serves as initial condition, at t = 0, for the ODE (3.89), not asinitial condition for the flow equation (3.5).) Some information about x? is alreadycontained in Lemma 3.2.2. For ?, we define ? : x? + B ? X ? by(?(x))0 = 0, (?(x)) j+1 = ? j (x j ), (3.103)where ? j is the map of (3.5). The map ? : x?+B ? X ? is defined analogously. Thenext lemma expresses immediate consequences of Assumption (A3) for ? and ? interms of the weighted spaces. Although the proof of Theorem 3.1.4 only directlyrequires the estimates for ?, we will also need bounds on ? to prove Lemma 3.3.3,so for convenience we combine both in a single lemma.Lemma 3.3.4. Assume (A3), let ? > ??, and assume that g?0 > 0 is sufficientlysmall. Then (?, ?) : x? + B ? X r is twice continuously Fr?chet differentiable,??(x)?X r ? M/u, (3.104)863.3. Proof of main resultand there exists a constant C = C(r, u) such that?DK ?(x)?L(Xw ,X r) ? C, ?DV ?(x)?L(Xw ,X r) ? O(g?0 | log g?0 |),?DK?(x)?L(Xw ,X r) ? ?, ?DV?(x)?L(Xw ,X r) ? O(g?0 | log g?0 |), (3.105)and?D2x ?(x)?L2 (Xw ,X r) ? C, ?D2x?(x)?L2 (Xw ,X r) ? C. (3.106)We defer the proofs of Lemmas 3.3.3?3.3.4 to Sections 3.3.4 and 3.3.5, respec-tively. Given these, we now prove Theorem 3.1.4(i).Proof of Theorem 3.1.4(i). Let CS be the constant of Lemma 3.3.3, define u? =CSM/( 12 b ? (1 ? b)), and assume u > u?. For t ? [0, 1] and x ? x? + B, letF (t , x) = S(t , x)?(x). (3.107)Let ( ?K0 , g?0) = (K0 , g0). Lemmas 3.3.3?3.3.4 imply that if g?0 > 0 is sufficientlysmall, F : [0, 1] ? ( x? + B) ? Xw is continuously Fr?chet differentiable and?F (t , x)?Xw ? ?S(t , x)?L(X r ,Xw) ??(x)?X r ? CSM/u ? 12 b ? (1 ? b). (3.108)Similarly, by the product rule, it follows that there is C such that?DxF (t , x)?L(Xw ,Xw) ? ?[DxS(t , x)]?(x)?L(Xw ,Xw)+ ?S(t , x)[Dx ?(x)]?L(Xw ,Xw) ? C, (3.109)and thus, in particular, that F is Lipschitz continuous on x ? x? + B.The theorem now follows from the well-known local existence theory for ODEin Banach spaces. Indeed, for y ? B, let?F (t , y) = F (t , x? + y). (3.110)Let Xw0 = {y ? Xw : piu y0 = 0}, B0 = B ? Xw0 . Then the statement about boundaryconditions of Lemma 3.3.3(i) and (3.108) imply that ?F (t , 12 bB0) ? ?F (t ,B0) ?12 bB0. With (3.108)?(3.109), the local existence theory for ODEs on Banach spaces[2, Chapter 2, Lemma 1] implies that the initial value problemy? = ?F (t , y), y(0) = 0 (3.111)has a unique C1-solution y : [0, 1] ? Xw0 with y(t) ? 12 bB0 for all t ? [0, 1].(The length of the existence interval of the initial value problem (3.111) in 12 bB is873.3. Proof of main resultbounded from below by 12 b/( 12 b ? (1 ? b)) ? 1 because ? ?F (t , y)? ? 12 b ? (1 ? b)when ?y? ? 12 b. It does not depend on the Lipschitz constant of ?F.)In particular, as discussed around (3.90), it follows that x = x? + y(1) is asolution to (3.6). By construction, piu x0 = piu x?0 = ( ?K0 , g?0) = (K0 , g0). Also,piv y? (1) = 0 because y(1) ? Xw, and since piv x?? = 0, it is also true that piv x? =0. Thus x satisfies the required boundary conditions. The estimates (3.18)?(3.21)are an immediate consequence of ?y?Xw ? 12 b, with (3.95).To prove uniqueness, suppose that x? is a solution to (3.6) with boundary con-ditions (K ?0 , g?0) = (K0 , g0) and (z?? , ???) = (0, 0), and such that (3.18)?(3.21)hold (with x replaced by x?, and with x? as before). Let x? = x? as before. By as-sumption, x? ? x? ? bB0. It follows that F : [0, 1] ? (x? + (1 ? b)B0) ? Xwis Fr?chet differentiable and ?F (t , x)?Xw ? 1 ? b for all t ? [0, 1] and for allx ? x? + (1 ? b)B0 ? x? + B0 as discussed around (3.107)?(3.109). In particularthere is a unique solution x?(t) for t ? [0, 1] to x?? = F (t , x?) with x?(1) = x? andx?(t) ? x? +B0, by considering the ODE backwards in time, which is equally well-posed. It follows that x?(0) is a flow of ?0 = ?? with the same boundary conditionsas x?. The uniqueness of such flows, by Lemma 3.2.2, implies that x?(0) = x?, andthe uniqueness of solutions to the initial value problem (3.111) in x? + B0 then alsothat x = x? as claimed. This completes the proof of Theorem 3.1.4(i). To prove Theorem 3.1.4(ii), we need to know that the initial condition x? isdifferentiable in a small ball x? + ?B. The smoothness of x? is addressed in thefollowing lemma, whose proof is deferred to Section 3.3.5.Lemma 3.3.5. Assume (A1?A3), and let ? > 0 and g?0 > 0 both be sufficientlysmall. Then there exists a neighbourhood ?I = ?I? ? K0 ? R+ of ( ?K0 , g?0) such thatx? : ?I ? x? + ?B is continuously Fr?chet differentiable with?Dg0 x??Xw ? O(g??20 | log g?0 |?1). (3.112)Proof of Theorem 3.1.4(ii). For fixed initial condition ( ?K0 , g?0) = (K0 , g0) = u0obeying the hypothesis of Theorem 3.1.4(i), let ?I be the neighbourhood of u0 de-fined by Lemma 3.3.5 with ? < 12 b. By Lemma 3.3.5, x? : ?I ? x? + ?B ? Xw iscontinuously Fr?chet differentiable. It follows from [2, Chapter 2, Lemma 4] thaty? = ?F (t , y), y(0) = x?(u0) ? x? (3.113)has a unique C1-solution y : [0, 1] ? ?I ? Xw0 with ?y(t)?Xw ? 12 b. [2, Chapter 2,Lemma 4] and Lemma 3.3.5 also imply???Dg0 y(t , K0 , g0)???Xw? C???Dg0 x?(K0 , g0)???Xw? O(g??20 | log g?0 |?1). (3.114)883.3. Proof of main resultLet x(u0) = x? + y(1, u0). It follows as previously that x(u0) = (u(u0), v(u0)) is asolution to (3.6) with boundary conditions u(u0) = u0 and v? (u0) = 0. Moreover,the differentiability in the sequence space Xw implies in particular that, as elementsof the spaces X j , each x j = (K j ,Vj ) is a C1 function of u0. Also (3.114) with (3.95)immediately implies that?z0?g0= O(1), ??0?g0 = O(1). (3.115)To prove (3.18)?(3.21) for x(u0) with u0 ? I ? ?I, we use that ?x(u0) ? x?? ? 12 band ? x?(u0) ? x??Xw ? ? imply?K j ? ?K j ?K j ? ?K j ? ?K j ?K j + ? ?K j ? ?K j ?K j ? ( 12 b + ?)(r ? r?)g?3j (3.116)and analogously that|gj ? g?j | ? ( 12 b + ?)ug?2j | log g?2j | (3.117)|z j ? z? j | ? ( 12 b + ?)u? j g?2j | log g?2j | (3.118)|? j ? ?? j | ? ( 12 b + ?)u? j g?2j | log g?2j |. (3.119)Since ( 12 b + ?) < b, by assuming that |g?0 ? g?0 | is sufficiently small, i.e., shrinking?I to a smaller neighborhood I if necessary, we obtain with (3.73) that( 12 b + ?)g?2j | log g?2j | ? bg?2j | log g?2j |. (3.120)This completes the proof of Theorem 3.1.4(ii). It now remains only to prove Lemmas 3.3.3?3.3.5. We begin with Lemma 3.3.3,which lies at the heart of the proof.3.3.4 Proof of Lemma 3.3.3The proof proceeds in three steps. The first two steps concern an approximateversion of the equation and the solution of the approximate equation, and the thirdstep treats (3.100) as a small perturbation of this approximation.Step 1. Approximation of the linear equationDefine ??0j : X j ? X j+1 by extending ?? j trivially to the K-component, i.e., ??0j =(0, ?? j ) with respect to the decomposition X j+1 = K j+1 ? V. Thus ?(t , x) =893.3. Proof of main result??0(x) + (?(x), t?(x)). Explicit computation of the derivative of ?? j of (3.5), using(3.1), shows thatD ??0j (x j ) =??????????????0 0 0 00 1 ? 2? jgj 0 00 ? ?? j 1 ? 2? jgj 00 ?? j ??? j ?? j??????????????, (3.121)with?? j = ? j ? 2?ggj gj ? ?gzj z j ? ?g?j ? j ,?? j = ? j ? ?gzj gj ? 2?zzj z j ? ?z?j ? j ,?? j = ? j ? ?g?j gj ? ?z?j z j ,?? j = 2? jgj + 2? j z j . (3.122)The block matrix structure in (3.121) is with respect to the decomposition X j =E j ? Fj introduced in Section 3.3.3.The matrix D ??0j (x j ) depends on x j ? X j , but it is convenient to approximateit by the constant matrixL j = D ??0j ( x? j ) =(Aj 0Bj Cj), (3.123)where the blocks Aj , Bj , and Cj of L j are defined by evaluating the blocks of thematrix (3.121) at x? j rather than at x j (given explicitly in (3.129) below). We willstudy the equationy j+1 = L j y j + r j , (3.124)which approximates (3.100). Lemma 3.3.6 below provides a useful reformulationof (3.124). For its statement, we define linear operators H : D(H) ? X ? andU : D(U) ? X ? (where D(H) and D(U) are the subspaces of X ? on which theinfinite sums converge) bypiuH = 0, (pivH x) j = ???l= jC?1j ? ? ?C?1l Blpiu xl , (3.125)and(piuU x) j =j?1?l=0Aj?1 ? ? ? Al+1piu xl ,(pivU x) j = ???l= jC?1j ? ? ?C?1l piv xl . (3.126)903.3. Proof of main resultIt follows from the definitions (recalling piK Aj = 0 = AjpiK ) thatpiK H = 0 = HpiK , piV H = H = HpiV , piKU = UpiK , piVU = UpiV .(3.127)The empty product in the formula for piuU x is interpreted as the identity, so theterm in the sum corresponding to l = j ? 1 is simply piu x j .Lemma 3.3.6. Assume (A1?A2) and that g?0 > 0 is sufficiently small. If r ? D(U)and y ? D(H) satisfies piu y0 = 0 and piv y? = 0, then (3.124) holds if and only ify = Hy + Ur, (3.128)holds.The proof is straightforward, but requires an estimate on the product of thematrices Cj which we will prove first. Products of the Cj and Aj will also play animportant role in the analysis of the operators H and U in the following section, sothat it is convenient to prove a more precise statement about them now than whatis needed for the proof of Lemma 3.3.6. Let us first record explicitly the blocks ofL j :Aj =(0 00 1 ? 2? j g?j), Bj =(0 ? ?? j0 ?? j), Cj =(1 ? 2? j g?j 0??? j ?? j)(3.129)with ?? j , ?? j , ?? j , and ?? j as in (3.122) with x replaced by x?.Lemma 3.3.7. Assume (A1?A2). Let ? ? (??1 , 1). Then for g?0 > 0 sufficientlysmall (depending on ?), the following hold.(i) Uniformly in all l ? j,Aj ? ? ? Al =(0 00 O(g?2j+1/g?2l )). (3.130)(ii) Uniformly in all j,Bj =(0 O(g?j ? j )0 O( ? j )). (3.131)(iii) Uniformly in all l ? j,C?1j ? ? ?C?1l =(O(1) 0O( ? j ) O(?l? j+1)). (3.132)913.3. Proof of main resultProof. (i) It follows immediately from (3.129) thatAj ? ? ? Al =j?k=l(1 ? 2?k g?k )pig , (3.133)and thus (3.35) implies (i).(ii) It follows directly from (3.129) and Lemma 3.2.2 that (3.131) holds.(iii) Note that(c1 0b1 a1)? ? ?(cn 0bn an)=(c? 0b? a?)(3.134)witha? = a1 ? ? ? an , b? =n?i=1a1 ? ? ? ai?1bici+1 ? ? ? cn , c? = c1 ? ? ? cn . (3.135)We apply this formula with the inverse matricesC?1j =( (1 ? 2? j g?j )?1 0(1 ? 2? j g?j )?1?? j ?? j ?? j)(3.136)where ?? j = ???1j . ThusC?1j ? ? ?C?1l =(??j ,l 0?? j ,l ?? j ,l)(3.137)with?? j ,l = ?? j ? ? ? ??l , ??j ,l =l?k= j(1 ? 2?k g?k )?1 , (3.138)?? j ,l =l? j+1?i=1????????l?k= j+i(1 ? 2?k g?k )?1?????????? j+i?1????????j+i?2?k= j??k????????. (3.139)The product defining ??j ,l is O(1) by (3.36). Assume that g?0 is sufficiently smallthat, with Lemma 3.2.2 and (A2), ??m < ? for all m. Then ?? j ,l ? O(?l? j+1).Similarly, since ??m ? O( ?m ),|?? j ,l | ?l? j+1?i=1?iO( ? j+i?1) ? O( ? j ). (3.140)This completes the proof. 923.3. Proof of main resultProof of Lemma 3.3.6. The u-component of (3.124) is given byu j+1 = Aju j + piur j . (3.141)By induction, under the initial condition u0 = 0 this recursion is equivalent tou j = piu y j =j?1?l=0Aj?1 ? ? ? Al+1piurl , (3.142)which is the same as the u-component of (3.128).The v-component of (3.124) states thatv j+1 = Bju j + Cjv j + pivr j , (3.143)and this is equivalent tov j = C?1j v j+1 ? C?1j Bju j ? C?1j pivr j . (3.144)By induction, for any k ? j, the latter is equivalent tov j = C?1j ? ? ?C?1k vk+1 ?k?l= jC?1j ? ? ?C?1l (Blul + pivrl ). (3.145)By Lemma 3.3.7(iii), with some ? ? (??1 , 1) and with g?0 sufficiently small,?C?10 ? ? ?C?1k ? is uniformly bounded. Thus, if y j = (u j , v j ) satisfies (3.124) andv j ? 0, then C?10 ? ? ?C?1k vk+1 ? 0 and hencev j = ???l= jC?1j ? ? ?C?1l (Blul + pivrl ), (3.146)which is the same as the v-component of (3.128). Conversely, suppose that y jsatisfies (3.128) and v j ? 0. It is also straightforward to conclude that (3.146)implies (3.145) and thus that the v-component of y satisfies (3.124). Step 2. Solution of the approximate equationWe now prove existence, uniqueness, and bounds for the solution to the approxi-mate equation (3.124).Lemma 3.3.8. Assume (A1?A2) and that g?0 > 0 is sufficiently small. For eachr ? X r and x ? x? + B, there exists a unique solution y = S0r ? Xw to (3.124)933.3. Proof of main resultobeying the boundary conditions piu y0 = 0, piv y? = 0. The solution operator S0 isblock diagonal w.r.t. the decomposition x = (K,V ), withS0 =(1 00 S0VV), (3.147)and there is a constant CS0 > 0 such that, uniformly in small g?0,?S0VV ?L(X r ,Xw) ? CS0 . (3.148)The constant CS0 is independent of u and r.Proof. According to Lemma 3.3.6, it suffices to prove that there is a unique solu-tion in Xw to (3.128) (instead of (3.124)) which obeys the required boundary condi-tions. Observe that as a block matrix with respect to the decomposition x = (u, v),with Hvu = pivHpiu , the operator 1 ? H is triangular of the form1 ? H =(1 0?Hvu 1). (3.149)We will prove that Hvu is a bounded operator in L(Xw , Xw). It follows that 1 ? Hhas a bounded inverse on Xw given by the block matrix(1 ? H)?1 =(1 0Hvu 1). (3.150)We further show that U is a bounded operator in L(X r , Xw). This implies that theunique solution in Xw of (3.124) is given byy = S0r = (1 ? H)?1Ur (3.151)and, since piu (1 ? H)?1 = piu and piKU = piK , that (3.147)?(3.148) hold.The boundary condition piv y? = 0 is a consequence of y ? Xw, and the initialcondition piu y0 = 0 is implicit in the equation (3.128). The claim that piK S0 =S0piK and piV S0 = S0piV then follows from (3.127). Since piuS0r = piuUr , thecases ? = K, g of (3.148) follow from the bounds claimed for U.To complete the proof, we require estimates for pi?U for ? ? {K, g, z, ?}, andon pi?H for ? = z, ?. Thus there are six estimates in all. Their treatment is similar,and uses Lemma 3.2.1(ii), which gives that for all k ? j ? 0 and m ? 0,k?l= j?l g?nl | log g?l |m ? Cn ,m???????| log g?k |m+1 n = 1? j g?n?1j | log g?j |m n > 1.(3.152)943.3. Proof of main result(i) Bound for K-component. By definition, since piK Al = 0, we have piKU = piK .Therefore,?piKUr ?Xw ? supj?piK r j ?Xwj ? supj[w?1K , j rK , j] ?r ?X r = ?r ?X r . (3.153)(ii) Bound for g-component. By Lemma 3.3.7(i), (3.95), (3.32), and (3.152),?pigUr ?Xw ? supjj?1?l=0?pig Aj?1 ? ? ? Al+1rl ?Xwj ? supjj?1?l=0w?1V , j rV ,lO(g?j/g?l )2?r ?X r? c?r ?X r supj| log g?j |?1j?1?l=0?l g?l ? c?r ?X r . (3.154)(iii) Bound for z-component. By Lemma 3.3.7(iii), (3.95), and (3.152),?pizUr ?Xw ? supj??l= j?pizC?1j ? ? ?C?1l rl ?Xwl? c supjhVw?1V , j??l= j?l g?3l ?r ?X r ? c | log g?0 |?1?r ?X r . (3.155)Similarly, by Lemma 3.3.7(ii-iii), (3.95), and (3.152),?pizH ?L(Xw ,Xw) ? supj??l= j?pizC?1j ? ? ?C?1l Bl ?L(Xwl ,Xwj )? c supjw?1V , j??l= j?l g?lwV ,l ? c. (3.156)(iv) Bound for ?-component. Using Lemma 3.3.7(iii), we obtain?pi?Ur ?Xw ? supj[??l= j?pi?C?1j ? ? ?C?1l rl ?Xwj]? c supjuw?1V , j[??l= j?l g?3l +??l= j?l? j+1 ?l g?3l]?r ?X r? c | log g?0 |?1?r ?X r , (3.157)where we used (3.152) and also that ??l= j ?l+1? j ?l g?3l ? c ? j g?3j in the last step. Tobound ?pi?H ?L(Xw ,Xw) , we argue similarly as for pi?Ur , and use Lemma 3.3.7 to953.3. Proof of main resultobtain?pi?H ?L(Xw ,Xw) ? supj??l= j?pi?C?1j ? ? ?C?1l Bl ?L(Xwl ,Xwj )? c supjw?1V , j??????????l= jg?j ? jwV ,l +??l= j?l+1? j ? jwV ,l????????? c. (3.158)This proves the required bounds for ? = ? and thus completes the proof. Step 3. Solution of the linear equationWe now prove Lemma 3.3.3, which involves solving the equation (3.100).Proof of Lemma 3.3.3. Fix ? ? (??, 1).(i) We defineW j (t , x j ) = Dx? j (t , x j ) ? L j= [Dx ??0j (x j ) ? Dx ??0j ( x?)] + Dx (? j (x j ), t? j (x j )), (3.159)and rewrite (3.100) asy j+1 = Dx? j (t , x j )y j + r j = L j y j + W j (t , x j )y j + r j . (3.160)It will be convenient to combine the W j (t , x) to an operator on sequences via(W (t , x))0 = 0 and (W (t , x)) j+1 = W j (t , x). This operator can be written as ablock matrix with respect to the decomposition x = (K,V ) asW (t , x) =(WKK WKVWVK WVV), (3.161)with W?? = pi?W (t , x)pi? . We claim that W : [0, 1] ? ( x? + B) ? L(Xw , X r), thatW is continuously Fr?chet differentiable, and that if x ? x? + B then,?WKK ?L(Xw ,X r) ? ?, ?WVK ?L(Xw ,X r) ? C,?WKV ?L(Xw ,X r) ? o(1), ?WVV ?L(Xw ,X r) ? o(1), (3.162)as g?0 ? 0, and?DxW j (t , x j )?L(Xwj ,L(Xwj ,X rj+1)) ? C. (3.163)To see this, note that the first term on the right-hand side of (3.159) only dependson the V -components, and is continuously Fr?chet differentiable since, by (3.121),963.3. Proof of main resultD2 ??0j is a constant matrix for each j with coefficients bounded by O( ? j ). There-fore, for x ? x? + B,?[D ??0j ( x? j ) ? D ??0j (x j )]piV ?L(Xwj ,X rj+1) ? c ? j r?1V , j+1w2V , j ? x? j ? x j ?Xwj= O(ug?0 | log g?0 |2). (3.164)This contributes to the bounds (3.162), with g?0 taken small enough. The secondterm on the right-hand side of (3.159), as well as its derivative, have been boundedin Lemma 3.3.4, completing the proof of (3.163).By the assumption that y ? Xw, Lemma 3.3.8, and (3.162), the equation (3.160)with the boundary conditions of Lemma 3.3.3(i) is equivalent toy = S0(W (t , x)y + r). (3.165)(ii) To solve this equation, we use that if A and B are bounded operators on aBanach space such that A has a bounded inverse A?1 and ?A?1B? < 1, then A? Bhas a bounded inverse. (Indeed, A? B = A(1? A?1B) and the inverse of 1? A?1Bis given by the Neumann series.) As in (3.147), we write S0 as a block matrix withrespect to the decomposition x = (K,V ) asS0 =(1 00 S0VV). (3.166)LetA =(1 ? WKK 0?S0VVWVK 1 ? S0VVWVV), B =(0 WKV0 0)(3.167)such that 1 ? S0W (t , x) = A ? B. Then (3.162) with g?0 sufficiently small implies?WKK ?L(Xw ,Xw) < 1 and ?S0VVWVV ?L(Xw ,Xw) < 1. Thus A is a block matrix ofthe formA =(AKK 0AVK AVV)(3.168)where AKK and AVV have inverses in L(Xw , Xw), and it follows that A has thebounded inverse on Xw given by the block matrixA?1 =(A?1KK 0A?1VV AVK A?1KK A?1VV). (3.169)Moreover, (3.162) with g?0 sufficiently small implies that ?A?1B?L(Xw ,Xw) < 1 andthus that 1 ? S0W (t , x) has a bounded inverse in L(Xw , Xw). It follows that thesolution operator is given byS(t , x) = (1 ? S0W (t , x))?1S0. (3.170)973.3. Proof of main result(iii) By (3.170), continuous Fr?chet differentiability in x of S(t , x) follows from thecontinuous Fr?chet differentiability of S0W (t , x), which itself follows from part (i)and from DxS0W (t , x) = S0DxW (t , x) by linearity of S0. Explicitly,DxS(t , x) = (1 ? S0W (t , x))?1DxS0W (t , x)(1 ? S0W (t , x))?1S0. (3.171)By (3.163),?DxS0W (t , x)?L(Xw ,L(Xw ,Xw)) ? C?DxW (t , x)]?L(Xw ,L(Xw ,X r)) ? C. (3.172)Together with the boundedness of the operators (1 ? S0W (t , x))?1 and S0, thisproves (3.102) and completes the proof. 3.3.5 Proofs of Lemmas 3.3.4?3.3.5Proof of Lemma 3.3.4. We begin with the verification of the bounds on the firstderivatives in (3.104). By assumptions (3.13)?(3.14), together with (3.32), thedefinition of the weights (3.95), and for (3.174) also the fact that ? j/? j+1 ? ? by(3.9), we obtain for x ? x? + B,?DV? j (x j )?L(Xwj ,X rj+1) ? M ? j g?2j r?1K , j+1wV , j ? O(g?0 | log g?0 |), (3.173)?DK? j (x j )?L(Xwj ,X rj+1) ? ?r?1K , j+1wK , j ? ??(1 + O(g?0)), (3.174)?DV ? j (x j )?L(Xwj ,X rj+1) ? M ? j g?2j r?1V , j+1wV , j ? O(g?0 | log g?0 |), (3.175)?DK ? j (x j )?L(Xwj ,X rj+1) ? Mr?1V , j+1wK , j ? O(1), (3.176)which establishes the bounds on the first derivatives in (3.104), choosing g?0 smallenough. The bounds on the second derivatives are also immediate consequences ofAssumption (A3). Let ? denote either ? or ?. Then (3.15) and the definition of theweights (3.95) imply that, for 2 ? n + m ? 3,?DnK DmV ??Ln+m (Xw ,X r) ? C. (3.177)In addition, these bounds on the second and third derivatives imply that??(x + y) ? ?(x) ? D?(x)y?X r ? C?y?2Xw , (3.178)?D?(x + y) ? D?(x) ? D2?(x)y?L(Xw ,X r) ? C?y?2Xw , (3.179)and hence that ? : x? + B ? X r is indeed twice Fr?chet differentiable. The abovebound on the third derivatives also implies continuity of this differentiability. The?-bound is equivalent to Assumption (A3) since?? j (x j )?X rj+1 = r?1V , j+1M ? j+1g?3j+1 = M/u. (3.180)This completes the proof. 983.3. Proof of main resultProof of Lemma 3.3.5. Let?I = ([ 12 g?0 , 2g?0] ? K0) ? x??1( x? + ?B). (3.181)We will show that ?I is a neighbourhood of ( ?K0 , g?0) and that x? : ?I ? x? + ?B is con-tinuously Fr?chet differentiable. Since x??1( x?+?B) = ?V?1( x?+?B)? ?K?1( x?+?B),it suffices to show that each of ?V?1( x? + ?B) and ?K?1( x? + ?B) is a neighbourhoodof ( ?K0 , g?0), and that each of ?V and ?K is continuously Fr?chet differentiable on ?I asmaps with values in subspaces of Xw.We begin with ?V . Let ?V ?j denote the derivative of ?Vj with respect to g0, andlet ?V ? = ( ?V ?j ) denote the sequence of derivatives. It is straightforward to concludefrom Lemmas 3.2.3 and 3.2.1(iv) and (3.95) that? ?V ??Xw ? O(g??20 | log g?0 |?1), (3.182)and hence that ?V ? ? Xw if g0 ? ?Ig ? [ 12 g?0 , 2g?0], and similarly that ?V?1( x? + ?B)contains a neighbourhood of g?. That ?V ? is actually the derivative of ?V in the spaceXw can be deduced from the fact that the sequence ?V ??(g0) is uniformly boundedin Xw for g0 ? ?Ig (though not uniform in g?0). In fact, by Lemma 3.2.3,? ?Vj (g0 + ?) ? ?Vj (g0) ? ? ?V ?j (g0)?Xwj ? O(?2) sup0<??<?? ?V ??j (g + ??)?Xwj . (3.183)The continuity of ?V ? in Xw follows similarly.For ?K , we first note that ?DK0 ?K0?L(K0 ,K0) = 1, ?Dg0 ?K0?K0 = 0. By (A3) andinduction,?DK0 ?K j+1?L(K0 ,K j+1) ? ??DK0 ?K j ?L(K0 ,K j ) ? ? j+1. (3.184)Since ? < ??1 < 1, and since g?j+1/g?j ? 1 by (3.32), we obtain?DK0 ?K j+1?L(K0 ,K j+1) ? O(g??30 wK , j+1). (3.185)Similarly, by (3.14) and Lemma 3.2.3,?Dg0 ?K j+1?K j+1 ? ??Dg0 ?K j ?K j + O( ? j g?2j )?Dg0 ?Vj ?V? ??Dg0 ?K j ?K j + O( ? j g?4j /g?20 ). (3.186)By induction as in the proof of Lemma 3.1.3, again using ? < ??1, we conclude?Dg0 ?K j+1?K j+1 ? O( ? j g?4j /g?20 ) ? O(g??10 wK , j+1). (3.187)993.3. Proof of main resultThese bounds imply that ?K?1( x? + ?B) contains a neighbourhood of ( ?K0 , g?0) andalso that the component-wise derivatives of ?K with respect to g0 and K0 are respec-tively in Xw  L(R, Xw) and L(K0 , Xw).To verify that the component-wise derivative of ?K is the Fr?chet derivative inXw, it again suffices to obtain bounds on the second derivatives in Xw, as in (3.183).For example, since D2K0 ?K0 = 0, DK0 ?Vj = 0, andD2K0 ?K j+1 = DK?( ?K j , ?Vj )D2K0 ?K j + D2K?( ?K j , ?Vj )DK0 ?K j DK0 ?K j , (3.188)it follows from (3.184) and induction that, for (K0 , g0) ? ?I with ?I ? K0 ? R chosensufficiently small,?D2K0 ?K j+1? ? ??D2K0?K j ? + C?2 j ? C(1 + j?)? j ? O(g??30 wK , j+1). (3.189)Thus the component-wise derivative D2K0 ?K is uniformly bounded L2(K0 , Xw) for(K0 , g0) ? ?I. Similarly, slightly more complicated recursion relations than (3.188)for D2g0 ?K j and Dg0 DK0 ?K j show that the component-wise second derivative of ?Kis uniformly bounded in L2(K0 ? R, Xw) for ?I sufficiently small. This shows as in(3.183) that ?K is continuously Fr?chet differentiable from ?I to Xw.We have thus shown that x? is continuously Fr?chet differentiable from a neigh-bourhood ?I of ( ?K0 , g?0) to Xw, and (3.112) follows from (3.182), (3.187). 100Chapter 4Outlook4.1 The weakly self-avoiding walk with contact attractionIn Section 1.2, the weakly self-avoiding walk with additional contact self-attractionwas introduced, see (1.9), but the subsequent discussion focused on the special casewithout self-attraction, ? = 0. For the model with self-attractive interaction, thereis the conjectured phase diagram of Figure 1.3 which, in particular, predicts thesame behavior as for ? = 0 also for sufficiently small ? > 0. However, even smallself-attraction makes the analysis more difficult than the weakly self-avoiding walkalready is because the energy functional then loses the superadditivity property. For? = 0,H (L + L?) = ??x(Lx + L?x )2 ? ??x(L2x + L?2x ) = H (L) + H (L?). (4.1)This superadditivity implies, for example, that ct =?x ct (x) is submultiplicative,i.e., ct+s ? ctcs , and therefore that there is ?c such that 1t log ct ? ?c ; see e.g. [88]or [12]. The subadditivity (4.1) does not hold if ? > 0.As a result of the failure of (4.1), little is known if ? > 0. For example, the re-sults about the (weakly or strictly) self-avoiding walk in dimension five and higherobtained with the lace expansion do not easily extend to small ? > 0. The uniqueexception is a result by Ueltschi [109] who studies a model of the strictly self-avoiding walk with additional small self-attraction, in dimension five and higher,but relies on very particular exponentially decaying step weights (instead of nearestneighbor steps). The special step distribution helps in the analysis, for example bymaking ct submultiplicative, but is an undesirable feature otherwise.Although superadditivity of H fails for ? > 0, it has been observed [110] thatthe attractive force can be written as?x?y:y?xLtxLty = 2d?x(Ltx )2 +?xLtx (?Lt )x= 2d?x(Ltx )2 ?12?x(?Lt )2x (4.2)1014.2. Logarithmic corrections to scaling behaviorso thatH?,? (L) = (? ? 2d?)?xL2x +?2?x(?L)2x ? H??2d?,0(L). (4.3)In terms of the renormalization group approach sketched in Section 1.4, the term(?L)2 is irrelevant. This is the basis for our work in preparation, with Brydges andSlade, in which we extend the result [38] to small ? > 0, thus showing that thetwo-point function is asymptotic to a multiple for |x |?(d?2) in dimension d ? 4.4.2 Logarithmic corrections to scaling behaviorA long-term goal of the renormalization group program for four dimensional weak-ly self-avoiding walks is to prove the conjecture (1.13) for the weakly self-avoidingwalk, or more generally that, for any p ? 0,(EHt |wt |p)1p ? cpt12 (log t) 18 (t ? ?). (4.4)A step towards this goal, interesting in itself, is to establish that the so-called sus-ceptibility ?(?) = ?x G? (x) has a related logarithmic correction,?(?c + T?1) ? cT (log T ) 14 (T ? ?) (4.5)where ?c is the smallest real number such that ?(?) < ? for ? > ?c . In work inpreparation with Brydges and Slade, we utilize results from Chapters 2?3, togetherwith [10, 34?37], to establish (4.5).4.2.1 End-to-end distance and Laplace transformsA heuristic argument (a version of Fisher?s scaling relation for the critical expo-nents that applies in the critical dimension, see e.g. [15, 88]) predicts that ifEHt |wt |2 ? ct(log t)2? (t ? ?), (4.6)?(? + ?) ? ??1(? log ?)? (? ? 0), (4.7)G?c (x) ? c |x |?(d?2) (log |x |)?? (|x | ? ?), (4.8)then the exponents of the logarithms should be related by? = 2? ? ?. (4.9)It has been proved that ? = 0 [38] and we can prove that ? = 14 . Then (4.9) leadsto the prediction ? = 18 as in (4.4).1024.2. Logarithmic corrections to scaling behaviorLet us give some indication in which way (4.5) is a natural step in the directionof proving (4.4). The left-hand side of (4.4) is(EHt |wt |p)1p =(?x ct (x) |x |p?x ct (x))1p. (4.10)It would suffice to establish the more general claim thate??c t?xct (x) |x |p ? cptp2 (log t) 14+ p8 (t ? ?). (4.11)This would in particular include?xct (x) ? ce?c t (log t) 14 (t ? ?). (4.12)An approach to proving (4.11) is given by proving related asymptotic behavior ofits Laplace transform, which is given in terms of the two-point function (1.16) by? ?0????????xct (x) |x |p???????e??t dt =?xG? (x) |x |p . (4.13)The asymptotics (4.11) are related to the asymptotics of the Laplace transform nearits the critical point ?c . For example, equation (4.11) implies that?xG?c+ 1T (x) |x |p ? c?p(T (log T ) 14)1+ p2 (T ? ?). (4.14)For p = 0, this is the same as (4.5). Equation (4.14) follows from (4.11) by a directcalculation: indeed, with t = sT ,? ?0e?(?c+1T )t????????xct (x) |x |p???????dt = T? ?0e?se??c sT????????xcsT (x) |x |p???????ds,(4.15)and, using (4.11), it is possible to conclude thate??c sT?xcsT (x) |x |p ? cpTp2 sp2 (log T ) 14+ p8 (T ? ?). (4.16)This implies (4.14) with c?p given byc?p = cp? ?0e?s sp2 ds = cp?(1 + p2). (4.17)1034.2. Logarithmic corrections to scaling behaviorThe converse, that (4.14) implies (4.11), is not true in general. However, Tauberiantheory [59, Chapter XIII] shows that (4.14) implies that (4.11) holds asymptoticallyin Cesaro mean, i.e.,1T? T0e??c t????????xct (x) |x |p???????dt ? cpTp2 (log T ) 14+ p8 (T ? ?). (4.18)To conclude (4.11) rather than the averaged version (4.18), further informa-tion is needed such as, e.g., eventual monotonicity of the integrand in (4.18), orrelated asymptotics as z = 1T ? 0 for z in a region of the complex plane. The lat-ter approach presumably requires major extensions to the argument which shows(4.5), but in the simpler case of weakly self-avoiding walks on a four dimensionalhierarchical lattice, this was successfully carried by Brydges and Imbrie [28].4.2.2 The renormalization group approachThe renormalization group method can be used to establish that the long-distancebehavior of the weakly self-avoiding walk is, in a suitable sense, related to thatof a free field. The critical model, ? = ?c , is described by a massless free field,m2 = 0, and subcritical models, ? > ?c , are related to massive free fields, m2 > 0.For example, we can show that there is a function ? = ?(m2) such that?(?(m2)) ? cm2(m2 ? 0), (4.19)i.e., the susceptibility of the weakly self-avoiding walk with parameter ? = ?(m2)is similar to that of the free field with mass m2. It turns out important to establishthe relation between ? and m2 in the non-critical case. 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Kogut, The renormalization group and the ? expan-sion, Physics Reports 12 (1974), no. 2, 75?200.113Appendix APerturbation theory andcoordinates of therenormalization groupIn this appendix, the second-order part of the renormalization group map for theweakly self-avoiding walk model is considered, i.e., the map ?? of Section 1.4.5.The map ?? is defined in terms of a map ?pt that arises from formal perturbationtheory, but does not satisfy the condition (3.1) imposed on the map ?? of Chapter 3itself. The remedy to this issue is an (explicit) coordinate change, exhibited in thisappendix, that transforms ?pt into a map ?? to which Chapter 3 can be applied. Themaps are defined in terms of the decomposition of the Green function of Chapter 2.This provides an explicit connection between Chapters 2 and 3.A.1 Flow of coupling constantsLet C = C1 + C2 + ? ? ? be a positive definite decomposition of the Green function,and use the convenient short-hand notation, with j fixed,C = Cj , w = w j =j?l=1Cl . (A.1)By translation-invariance, we can identify C and w with functions of one variable,for example, Cx = C0x . LetVx = g?2x + ??x + z??,x (A.2)be the (local) interaction polynomial for the weakly self-avoiding walk model. (Forthe definitions of ? and ??, see (1.43) and (1.50).) In [10], a new local interactionpolynomial Vpt,x is defined, in terms of V , C, and w, describing the effect of (for-mal) second-order perturbation theory. The details of the specification of Vpt arenot important for the current discussion, so we only state the result: Vpt is essen-tially of the same form as (A.2) with coefficients gpt , ?pt , zpt given by polynomials114A.2. Bounds on the coefficientsof degree two in g, ?, z. To express the coefficients of the polynomials, it is con-venient to introduce the following abbreviations: for a function f = f (?,w), set?[ f ] = f (? + 2C0g,w + C) ? f (?,w). (A.3)Moreover, for a function q : Zd ? R, set(?q)x = 12?e?Zd :|e |1=1(qx+e ? qx ), (A.4)(?q)2x = 12?e?Zd :|e |1=1(qx+e ? qx )2 , (A.5)andq(n) =?xqnx . (A.6)All functions q below arise in terms of the covariance decomposition, e.g., q = w,and satisfy:?xqx xi = 0,?xqx xi x j = q(??)?i j (i, j = 1, . . . , d). (A.7)Then the coefficients are given by:???????????????????????????gpt = g ? 8g2?[w(2)] ? 4g?[?w(1)],?pt = ? + 2C0g ? 4g2(?[w(3)] ? 3w(2)C0) ? 2g(? + 2C0g)?[w(2)]? ?[?2w(1)] + 2g(z + y)?[(w?w)(1)] + 8g?w(1)C0 ,zpt = z ? 2g2?[(w3)(??)] ? 12?[?2w(??)] ? 2z?[?w(1)].(A.8)A.2 Bounds on the coefficientsFrom now on, assume that the covariance decomposition C = ??j=1 Cj is given by[Cj ]x =?????????????????? 12 L0??t (x)dtt( j = 1)? 12 Lj12 Lj?1??t (x)dtt( j > 1)(A.9)115A.2. Bounds on the coefficientswhere ??t is as given in Example 2.1.3. In particular, (A.9) implies the finite rangeproperty[Cj ]x = 0 if d(x , 0) > 12 L j (A.10)and the bounds|[??Cj ]x | ? O(L?(d?2+|? |1)( j?1)). (A.11)Natural estimates on the coefficients in (A.8) are given in terms of the variable? = L2 j? instead of ? and ?pt = L2( j+1)?pt. Let ?pt(g, z, ?) = (gpt , zpt , ?pt).Proposition A.2.1. The coefficients of the polynomials ?pt are bounded by O((1 +m2L2 j )?k ) for any k ? R and continuous in m2 ? [0, ?) for some ? > 0.Proof. The proof uses (A.10)?(A.11) and is given in reference [10]. The previous result is similar to Assumption (A2) of Chapter 3. (We will showbelow that O((1 + m2L2 j )?k ) can be bounded by O( ? j ).) However, the map ?? ofChapter 3 is assumed to be triangular which ?pt is not. This is will be addressed inthe next subsection. In addition, for the applicability of the result of Theorem 3.1.4,a positive lower bound on the coefficient of the g2-term in the g-equation is crucialto satisfy assumption (A1). This is a consequence of Lemma A.2.2 below, in whichwe verify that the sequence of coefficients has a positive limit if m2 = 0.Lemma A.2.2. Let d = 4, m2 = 0. Then there is ?? > 0 such that? j := 8? j [w(2)] = ?? + O(L? j ). (A.12)Remark A.2.3. The constant ?? can be determined exactly:?? =log(L)pi2. (A.13)Proof of Lemma A.2.2. Denote the covariance decomposition by Cj (x), x ? Z4.By (2.36), there is c0 ? Cc (R4) such that with cj (x) = L?2 jc0(L? j x),Cj (x) = cj (x) + O(L?3 j ). (A.14)Let us first verify(Cj ,Cj+l ) ? ?c0 , cl ? = O(L? jL?2l ) (A.15)where we use the notation (F,G) = ?x?Z4 F (x)G(x) whenever F,G : Z4 ? Rand ? f , g? =?R4 f g dx for f , g : R4 ? R. Let Rj = Cj ? cj . Then:(Cj ,Cj+l ) = (cj , cj+l ) + (cj , Rj+l ) + (cj+l , Rj ) + (Rj , Rj+l ). (A.16)116A.2. Bounds on the coefficientsRiemann sum approximation shows(cj , cj+l ) ? ?c0 , cl ? = L?4 j?y?L? jZdc(y)cl (y) ??Rdc(y)cl (y) dy= O(L? j )??(ccl )?L? = O(L?2l? j ). (A.17)The remaining terms are easily bounded using |supp(Cj ) |, |supp(Rj ) | = O(L4 j ):(cj , Rj+l ) ? O(L4 j )?cj ?L? (Z4) ?Rj+l ?L? (Z4) ? O(L? jL?3l ), (A.18)(cj+l , Rj ) ? O(L4 j )?cj+l ?L? (Z4) ?Rj ?L? (Z4) ? O(L? jL?2l ), (A.19)(Rj , Rj+l ) ? O(L4 j )?Rj ?L? (Z4) ?Rj+l ?L? (Z4) ? O(L?2 jL?3l ), (A.20)and (A.15) follows. From this we can now deduce:j?k=1(Ck ,Cj+1) =j?k=1?c0 , cj+1?k ? +j?k=1O(L?k L?2( j?k ))=j?k=1?c0 , ck ? + O(L? j ), (A.21)(Cj+1,Cj+1) = ?c0 , c0? + O(L? j ), (A.22)and thus, using ?c0 , ck ? = ?c0 , c?k ?,w(2)j+1 ? w(2)j = 2(w j ,Cj+1) + (Cj+1 ,Cj+1) (A.23)=j?k=? j?c0 , ck ? + O(L? j ). (A.24)Note that with ?c?k ?L? ? L2k ?c0?L? and supp(c?k ) ? BCL?k ,??k= j+1|?c0 , ck ?| =??k= j+1|?c0 , c?k ?| ? ?c0?L???k= j+1L2k?BCL?k|c0(x) | dx? ?c0?2L???k= j+1O(L?2k ) ? O(L?2 j ). (A.25)Thus, with ?? = 8??k=???c0 , ck ?, we have obtained8(w(2)j+1 ? w(2)j ) = ?? + O(L? j ). (A.26)That ?? > 0 can be seen from the fact that c?k ? 0 and Plancherel?s theorem. 117A.2. Bounds on the coefficientsProof of Remark A.2.3. By (A.21), it follows that?? = 8?c0 , v? with v =?k?Zck . (A.27)The Fourier transforms of c and v arec?0(?) = 1|? |2? |? |L?1 |? |?(t) dt , v?(?) = 1|? |2(A.28)where ? is a non-negative function with? ?0 ? dt = 1. Observe that the claim forv? follows from the claim for c?; the latter claim is verified at the end of the proof.(A.28) implies, by Plancherel?s theorem, radial symmetry, and Fubini?s theorem,?c0 , v? =1(2pi)4?R4|? |?4(? |? |L?1 |? |?(t) dt)d?= ?3(2pi)4? ?0(? rL?1r?(t) dt)drr= ?3(2pi)4? ?0(? Lttdrr)?(t) dt (A.29)where ?3 = 2pi2 is the surface measure of the 3-sphere (? R4). The inner integralin the last equation is equal to log(L). Thus, with? ?0 ? dt = 1,?? =8?3(2pi)4 log(L) =log(L)pi2(A.30)as claimed. To verify (A.28), use that by (2.36)?(2.37), there is k > 0 such that??t (x) = (t/k)?(d?2) ??(k x/t) + O(t?(d?2+1)) (A.31)where, denoting the Fourier transform of ?? by ??, see (2.118), (2.78),??(?) =? ?0t2?(|? |t) dtt,? ?0t2?(t) dtt= 1. (A.32)In particular, the function c in (A.14) is more explicitly given byc?0(?) = 1k2? 1212 L?1t2?(|? |tk)dtt= 1|? |2? |? |L?1 |? |?(t) dt (A.33)as claimed where ? is given by?(t) =( t2k)2?( t2k) 1t. (A.34)This completes the proof. 118A.3. TransformationA.3 TransformationAs discussed, the map ?pt(g, z, ?) = (gpt , zpt , ?pt) does not have the right form toapply the result of Chapter 3. In Proposition A.3.1, we show that the coordinatescan be brought to the form expected in Chapter 3 by a simple transformation.Proposition A.3.1. Define ?? : R3 ? R3 by (g?, z?, ??) = ??(g, z, ?) with ?? =L2( j+1) ??, ? = L2 j?, andg? = g ? 8g2?[w(2)], (A.35)z? = z ? 2g2?[(w3)(??)], (A.36)?? = ? + 2C0,0g ? 4g2(?[w(3)] ? 3w(2)C0,0 + C0,0?[w(2)])? 2g??[w(2)] + 2gz?[(w?w)(1)]. (A.37)Then the coefficients of the polynomials ?? are bounded by O((1 + m2L2 j )?k ) foran arbitrary k and m2 ? [0, ?). Define T : R3 ? R3 by T (g, z, ?) = (gT , zT , ?T ),with ? = L2 j?, ?T = L2 j?, wheregT = g + 4g?w(1) , (A.38)zT = z + 2z?w(1) + 12 ?2w(??) , (A.39)?T = ? + ?2w(1) . (A.40)Then T (V ) = V + O(|V |2). Let T+ = Tj+1. There exists a ball B ? R3 independentof j and m2 ? [0, ?) such that, on B,T+ ? ?pt ? T?1 = ?? + ?pt (A.41)where ?pt is an analytic function on B with ?pt(g, z, ?) = O((1 + m2L2 j )?k (|g | +|z | + |?|)3) uniformly in j and m2 ? [0, ?), for any k.Remark A.3.2. The transformation T is simple and explicit, but we believe thatits existence may have a deeper origin that we have not unravelled. Formally, i.e.,without consideration of the formal third-order error, different covariance decom-positions induce dynamical systems like (1.92) whose three-dimensional parts canbe of slightly different form. Some of the monomials that appear in the polynomi-als ?? j are essentially independent of the decomposition. On the other hand, somedecompositions of the Green function have the special property that?x[Cj ]x = 0 (A.42)119A.3. Transformationwhich is not true for the finite range decomposition discussed in Chapter 2. It canbe seen that the terms in (1.43) involving w(1) would thus vanish with such a de-composition. Is it possible that the existence of such a transformation expresses aninvariance property of the dynamical system under coordinates induced by differentcovariance decompositions?Note that the map ?? has the form assumed for ?? in Chapter 3. The next corol-lary illustrates how the result of Chapter 3 is used in the study of the weakly self-avoiding walk, except that in the real application, the error coordinate is non-trivial.Corollary A.3.3. Fix any ? > 1. The maps ?? then satisfy Assumptions (A1?A2) ofChapter 3. Moreover, Assumption (A3) can be satisfied with ? = ?pt and ? = 0.Sketch of proof. (i) Set jm = [logL m]. We first show that for any c < log L/pi2,there is n < ? such that the number of j ? jm with ? j < c is bounded by n,uniformly in m2 ? [0, ?). To prove this, we first note that (A.13) implies that, ifm2 = 0, for every c + ? < log L/pi2, there is n0 such that the number of j such that? j < c + ? is bounded by n0. We now prove the claim for m2 > 0. It can be shownusing Example 2.1.3 that there are constants c? and q independent of L such that???????m2? j (m2)?????? c?LqL2 j , (A.43)but we omit the proof. This implies that for j ? jm ? q ? p, with p large enough,| ? j (0) ? ? j (m2) | ? c?LqL2 jm2 ? c?L?p ? ?. (A.44)It follows that the number of j ? jm such that ? j < c can be bounded by n0+p+q.(ii) We now verify Assumptions (A1)?(A2) of Chapter 3 for ??. Let ? > 1,j? = inf{k ? 0 : | ? j | ? ??( j?k ) ? ??? for all j}, and ? j = ??( j? j?)+ . (A.45)Let k be such that L2k ? ?. Then(1 + m2L2 j )?k ? L?2k ( j? jm )+ ? ??( j? jm )+ . (A.46)(i) implies that ? ??? > c > 0 uniformly in m2 ? (0, ?). By Proposition A.3.1 and(A.46), there is a constant C such that| ? j | ? C??( j? jm )+ ?Cc??( j? jm )+ ? ??? ? ??( j? j?)+ ? ??? (A.47)with j? ? jm + log?C ? log? c. In particular, the number of j ? j? with ? j < cis bounded by n? = n + log?C ? log? c where n is as in (i), uniformly in m2 ?[0, ?). This proves Assumption (A1) and Assumption (A2) is then a consequenceof Proposition A.3.1 with (1 + m2L2 j )?k = O( ? j ). 120A.3. TransformationSketch of proof of Proposition A.3.1. The bounds on the coefficients of the maps??, given in (A.35)?(A.36), follow from Proposition A.2.1 and w(??) = O(L4 j ) andw(1) = O(L2 j ). The last two bounds are a straightforward with the properties ofthe covariance decomposition that |Cj | ? O(L?2 j ) and Cj (x) = 0 for x ? cL j .Indeed,w(1)j =j?l=1?x[Cl ]x =j?l=1O(L2l ) = O(L2 j ), (A.48)w(??)j =j?l=1?x|x |2[Cl ]x =j?l=1O(L4l ) = O(L4 j ). (A.49)These bounds similarly imply T = id + O((|g | + |z | + |?|)2) uniformly in j.Let w+ = w + C and ?+ = ? + 2C0g. Then (A.8) can be written asgpt + 4g?+w(1)+ = (g + 4g?w(1)) ? 82?[w(2)]g2, (A.50)?pt + ?2+w(1)+ = (? + ?2w(1)) + 2C0,0(g + 4g?w(1))? 4g2 (?[w(3)] ? 3w(2)C0,0)? 2g(? + 2C0g)?[w(2)]+ 2g(z + y)?[(w?w)(1)], (A.51)zpt + 2z?+w(1)+ + 12 ?2+w(??)+ = (z + 2z?w(1) + 12 ?2w(??)) ? 2g2?[(w3)(??)].(A.52)Expressing ? and ?pt as ? = L?2 j ? and ?pt = L?2( j+1)?pt, the right- and left-handsides of (A.50)?(A.52) equal ?? ? T (g, z, ?) + O((|g | + |z | + |?|)3) respectivelyT+ ? ??(g, z, ?) + O((|g | + |z | + |?|)3), with both bounds uniform in j. This andT+((g, z, ?) + r) = T+(g, z, ?) + O(r) imply the claim. 121


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