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Renormalisation group and critical correlation functions in dimension four Tomberg, Alexandre 2015

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Renormalisation group and critical correlationfunctions in dimension fourbyAlexandre TombergB.Sc. Honours Joint Mathematics and Computer Science, McGill, 2010M.Sc. Mathematics, McGill, 2011a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Mathematics)The University of British Columbia(Vancouver)August 2015c Alexandre Tomberg, 2015AbstractCritical phenomena and phase transitions are important subjects in statis-tical mechanics and probability theory. They are connected to the phe-nomenon of universality that makes the study of mathematically simplemodels physically relevant. Examples of such models include ferromagneticspin systems such as the Ising, O(n) and n-component j'j4models, but alsothe self-avoiding walk that has been observed to formally correspond to a\zero-component" spin model [39].Our subject in this thesis is the extension and application of a rigorousrenormalisation group method developed in [9, 10, 13] to study the criticalbehaviour of the continuous-time weakly self-avoiding walk and of the n-component j'j4model on the 4-dimensional lattice Z4. Although a \zero-component" vector is mathematically undened (at least naively), we areable to interpret the weakly self-avoiding walk in a mathematically rigorousmanner as the n = 0 case of the n-component j'j4model, and provide aunied treatment of both models.For the j'j4model, we determine the asymptotic decay of the criticalcorrelation functions including the logarithmic corrections to Gaussian scal-ing, for n  1. This extends previously known results for n = 1 to all n  1,and also observes new phenomena for n > 1, all with a new method of proof.For the continuous-time weakly self-avoiding walk, we determine the decayof the critical generating function for the \watermelon" network consistingof p weakly mutually- and self-avoiding walks, for all p  1, including thelogarithmic corrections. This extends a previously known result for p = 1,for which there is no logarithmic correction, to a much more general setting.iiPrefaceThis thesis is based on the joint work with Gordon Slade [64]. I closelycollaborated with Gordon Slade in the writing of [64], and my primarycontributions to that paper include reorganisation and streamlining of the proof of the integral represen-tation, perturbative calculations of logarithmic corrections, the tuning of theoperator Loc, and the existence of dierent renormalisation groupows depending on initial symmetries of the model, concepts of h-factorisation and reduction of symmetry, that link sym-metry to computations of exponents, dependence of norms on symmetry classes and logarithmic corrections,and inductive proof of the existence of observable renormalisation groupow and parts of the analysis of this ow, and the adaptation andgeneralisation of many arguments that are required to prove the mainresults from the simpler cases that appeared in [9, 10,13].These ideas and concepts appear throughout this thesis and are not re-stricted to one particular chapter. I will now discuss new material in thisthesis that is not in [64], and I will highlight in passing the sections wherethe above ideas are present.Chapter 1 is an introduction that motivates and denes the problems studied inthe remainder of the thesis. In Section 1.1, that was written for thisthesis, I explain many ideas from a number of references mentioned;iiino originality is claimed. Sections 1.2, 1.3 and 1.4, with the exceptionof Section 1.3.1, are adapted from [64].Chapter 2 discusses well-known ideas and is based on many dierent sectionsof [64] that I reorganised and adapted for this thesis. The stream-lined proof of integral representation from the Appendix of [64] is inSection 2.1.Chapter 3 is based on parts of [64], but discusses some material that appearedin [11, 21, 22]. Sections 3.1, 3.2 and 3.3 were expanded signicantlyfrom their counterparts in [64]. The tuning of the operator Loc is inSection 3.3 and the perturbative computations are in Section 3.4. Theconcept of h-factorisation and reduction of symmetry is introduced inSection 3.3.2.Chapter 4 is primarily taken from [64] with minimal modications. The proofsof h-factorisability are in Section 4.4.2 and the dependence of normparameters on the logarithmic correction exponents is discussed inSection 4.3.1. The inductive proof of observable renormalisation groupow is in 4.5.Chapter 5 contains many of my contributions to [64], but Section 5.2 and theproof of Theorem 1.8 in Section 5.4 were primarily the work of GordonSlade.Chapter 6 was written for this thesis. Parts of Section 6.1 are based on the dis-cussion of the main theorems from [64]. Section 6.2 is based on manydiscussions with Roland Bauerschmidt, David Brydges and GordonSlade.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction and main results . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 A model of a ferromagnet . . . . . . . . . . . . . . . . 11.1.2 Universality and critical exponents . . . . . . . . . . . 51.1.3 A brief overview of known results for the Ising model 61.2 The j'j4model . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Denition of the model . . . . . . . . . . . . . . . . . 81.2.2 Correlation functions . . . . . . . . . . . . . . . . . . . 111.3 The WSAW model . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Self-avoiding walk . . . . . . . . . . . . . . . . . . . . 121.3.2 Denition of the WSAW model . . . . . . . . . . . . . 141.3.3 Watermelon and star networks . . . . . . . . . . . . . 161.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.1 Statements of main theorems . . . . . . . . . . . . . . 171.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 20v1.4.3 A word about the proof . . . . . . . . . . . . . . . . . 222 Reformulation of the problem . . . . . . . . . . . . . . . . . 242.1 Integral representation of the WSAW . . . . . . . . . . . . . . 252.1.1 Innite volume limit for WSAW . . . . . . . . . . . . 252.1.2 Dierential forms . . . . . . . . . . . . . . . . . . . . . 272.1.3 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . 302.2 Change of variables and Gaussian approximation . . . . . . . 342.3 External elds: notation and generalities . . . . . . . . . . . . 362.3.1 Correlation functions as derivatives . . . . . . . . . . . 362.3.2 The eld shift operator . . . . . . . . . . . . . . . . . 372.3.3 Observable parameters and quotient spaces . . . . . . 382.4 Observable and external elds . . . . . . . . . . . . . . . . . . 402.4.1 Coupling the partition function to observable param-eters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.2 Derivatives of the pressure . . . . . . . . . . . . . . . . 413 Perturbative computations . . . . . . . . . . . . . . . . . . . 443.1 Progressive Gaussian integration . . . . . . . . . . . . . . . . 443.1.1 Covariance decomposition . . . . . . . . . . . . . . . . 453.1.2 Scales and corresponding eld sizes . . . . . . . . . . . 463.2 Renormalisation group strategy . . . . . . . . . . . . . . . . . 493.2.1 Classication of local eld monomials . . . . . . . . . 493.2.2 Cumulant expansion . . . . . . . . . . . . . . . . . . . 503.2.3 Space of local polynomials . . . . . . . . . . . . . . . . 523.3 Approximation by local polynomials . . . . . . . . . . . . . . 533.3.1 Localisation operator Loc . . . . . . . . . . . . . . . . 533.3.2 Symmetries and symmetry reduction . . . . . . . . . . 553.3.3 Range of Loc . . . . . . . . . . . . . . . . . . . . . . . 583.4 Perturbative calculations . . . . . . . . . . . . . . . . . . . . . 623.4.1 Iterating the cumulant expansion . . . . . . . . . . . . 623.4.2 Denition of Ij. . . . . . . . . . . . . . . . . . . . . . 633.4.3 Perturbative ow of coupling constants . . . . . . . . 65vi4 Renormalisation group flow . . . . . . . . . . . . . . . . . . . 734.1 Non-perturbative renormalisation group coordinate . . . . . . 734.1.1 Circle product . . . . . . . . . . . . . . . . . . . . . . 744.1.2 The denition of K . . . . . . . . . . . . . . . . . . . 754.1.3 The renormalisation group map . . . . . . . . . . . . . 764.2 Bulk ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.1 Existence of bulk ow . . . . . . . . . . . . . . . . . . 784.2.2 Properties of the bulk ow . . . . . . . . . . . . . . . 804.2.3 Change of variables . . . . . . . . . . . . . . . . . . . 844.3 Parameters and stability estimates . . . . . . . . . . . . . . . 854.3.1 Parameters, norms and domains . . . . . . . . . . . . 854.3.2 Stability estimates . . . . . . . . . . . . . . . . . . . . 884.4 A single renormalisation group step including observables . . 914.4.1 Modication to [24, Map 6] . . . . . . . . . . . . . . . 934.4.2 Reduced symmetry . . . . . . . . . . . . . . . . . . . . 944.5 Complete renormalisation group ow . . . . . . . . . . . . . . 975 Infinite volume limit and proofs of main results . . . . . . 1015.1 Inductive limit of observable ow . . . . . . . . . . . . . . . . 1015.2 Non-perturbative estimates . . . . . . . . . . . . . . . . . . . 1045.3 Proof of Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . 1055.4 Proof of Theorems 1.7{1.8 . . . . . . . . . . . . . . . . . . . . 1106 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117viiList of FiguresFigure 1.1 Two 6x6 congurations sampled from the Ising and O(2)models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 A simulation of d = 2 Ising model on a 200 400 box forT < Tc, T = Tcand T > Tc. . . . . . . . . . . . . . . . . . 4Figure 1.3 Plots of the magnetisationM(h; T ) and spontaneous mag-netisation M+(T ) for various choices of parameters. . . . 5Figure 1.4 A plot of the single spin distribution of the j'j4modelwhen n = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 1.5 Watermelon networks for p = 1; 2; 3; 5. . . . . . . . . . . . 16Figure 1.6 Star networks for p = 1; 2; 3; 5. . . . . . . . . . . . . . . . 17viiiAcknowledgementsI would like to thank my research adviser, Gordon Slade, for his supportthroughout my time at UBC. His guidance has been essential for the com-pletion of the work contained in this thesis.I would also like to thank David Brydges and Joel Feldman for serving onmy supervisory committee, and Roland Bauerschmidt for numerous helpfulcomments and discussions. Thanks are also due to the members of theProbability Group at UBC for providing an inspiring research environment.Part of the research contained in this thesis was carried out during my stay atLeiden University in the Netherlands and I am grateful for their hospitality.I would like to gratefully acknowledge the nancial support that I havereceived, in particular funding from the National Science and EngineeringResearch Council, the University of British Columbia, the Warwick EP-SRC Symposium on Statistical Mechanics, and the Heidelberg MathematicalPhysics, Analysis and Stochastics Summer School.The support of my family and friends has been very important to me,and I would like to thank everyone for all they have done. Lastly, and mostimportantly, I would like to thank Lucia for creating a very special atmo-sphere, for taking a real interest in my work and for asking very insightfulquestions.ixChapter 1Introduction and mainresults1.1 IntroductionThe quantitative description of phase transitions is an important topic instatistical mechanics. Critical phenomena are associated with continuousphase transitions [48]. For example, ferromagnetic, superconducting, andsuperuid transitions are in this class. Such transitions occur at criticalpoints { special combinations of parameters of a system where the distinc-tion between phases essentially vanishes. The characteristics of a system inthe vicinity of a critical point are captured by critical exponents that arerelated to power-law behaviour of certain observables.In this section, we provide an overview of these topics in more detail byusing the famous Ising model as an example. Later, in Sections 1.2 and 1.3,we give precise denitions of the models that we will work with in this thesis.Finally, in Section 1.4, we state and discuss our main theorems.1.1.1 A model of a ferromagnetA ferromagnet is a material that can be magnetised by an external magneticeld and keeps its magnetic properties long after the external eld has been11.1. IntroductionFigure 1.1: Two 6x6 congurations sampled from the Ising (left) andO(2) (right) models.removed. However, heating a magnet beyond a certain temperature alwaysremoves all magnetisation. To model this phenomenon, let us consider asample of some material whose atoms have a magnetic moment and arearranged in a regular crystalline structure.Let   Zd be a nite box. For each x 2 , we consider a randomvariable 'x, called the spin. The spin is a representation of the directionof the magnetic moment and the whole conguration is given by the eld' = ('x)x2. Two 2-dimensional representations of such congurations 'are given in Figure 1.1. On the left, the spins 'xwere only allowed to pointeither up or down; this corresponds to the Ising model. On the right, wesampled spins 'xin the 1-sphere S1; this is the classical XY model, a specialcase of the O(n) model with n = 2.For simplicity of exposition, let us now concentrate on the Ising modeland restrict 'x2 f1; 1g. Since neighbouring spins tend to point the sameway, we will impose an energy penalty on unaligned nearest neighbour spins.We dene the discrete gradient of a eld f on Zd by (ref)x= fx+e fx21.1. Introductionand setH0(') = H0;(') =12Xx2Xe:jej=1(re')2x: (1.1)This is the energy associated to the conguration ' and, up to an additiveconstant that is independent of the conguration itself, it simply countstwice the number of misaligned neighbouring spins.We assign to ' a probability proportional to its Boltzmann weight withthe parameter T representing the temperatureP(') = PT;(') / eH0;(')=T: (1.2)Note that this probability distribution remains the same if we set T = 1 inthe Boltzmann weight and instead regard the spins as having values T1=2rather than 1.There is a critical value Tcof the temperature parameter, such thattypical congurations drawn from the above distribution look quite dierentdepending on whether T is above or below Tc. Figure 1.2 shows a simulationwhere  is a 200  400 box in Z2, and the 1 values of the eld ' arerepresented by the colours white and blue. The spins on the boundary ofthe top half of  are constrained to be white, and blue for the bottom half.The Hamiltonian (1.1) favours local alignment of spins, and Figure 1.2suggests that this local property induces order on the global (macroscopic)scale for T < Tconly. We will now relate this qualitative description toa quantitative one: the persistent magnetisation of a ferromagnet. To thisend, we add an external eld h 2 R to the Hamiltonian in (1.1) byHh;(') = H0; hXx2'x=12Xx2Xe:jej=1(re')2x h'x: (1.3)Note that H0;(') is invariant under a global spin ip ' 7! ', but inHh;(') this symmetry is broken. Let be the set of possible congu-rations ' that satisfy some boundary conditions and dene the partition31.1. IntroductionFigure 1.2: A simulation of d = 2 Ising model on a 200  400 box.Boundary conditions are white for the top half and blue for thebottom half. From left to right: T < Tc, T = Tcand T > Tc.functionZh;T;=X'2eHh;(')=T: (1.4)The probability measure Ph;T;(') = Z1h;T;eHh;(')=Tis the nite volumeGibbs measure and we denote expectations with respect to that measureby hF ih;T;. Taking  " Zd, we obtain the innite volume Gibbs measurehF ih;T= lim"ZdX'2F (')Ph;T;('): (1.5)It can be shown that this limit exists, although the simulation from Fig-ure 1.2 suggests that for T < Tcand h = 0 it is not unique and dependson the boundary conditions. In fact, the non-uniqueness of the innite vol-ume Gibbs measures is closely related to the physical phenomenon of phasetransitions, but we will not discuss this here (see [40] for more information).We now dene the magnetisation1M(h; T ) = h'0ih;Tand spontaneousmagnetisation M+(T ) = limh#0M(h; T ). Figure 1.3 shows the behaviourof these functions as we vary h and T . In particular, note that below the1The total magnetisation of the sample isM(h; T )jj, but since we are sending  " Zd,we prefer to work with magnetisation density instead.41.1. IntroductionM+(T )T = TcT > TcT < TchM(h; T )T = 0 T = Tc1M+(T )Figure 1.3: Plots of the magnetisationM(h; T ) and spontaneous mag-netisation M+(T ) for various choices of parameters.critical temperature, the spontaneous magnetisation is positive, meaningthat placing a sample of the Ising model in a magnetic eld h and thenturning this external eld o by sending h # 0, leaves the material magne-tised. Therefore, the global order that we saw in the leftmost simulationin Figure 1.2 corresponds to the persistent magnetisation of a ferromagnetgiven by M+(T ).1.1.2 Universality and critical exponentsThe Ising model described in the previous section may seem very simplistic,but the graph of M(h; Tc) given in Figure 1.3 is surprisingly accurate [7].In fact, it has been found experimentally that it matches (up to scaling fac-tors) the critical behaviour of many magnetic materials even though theirinternal structures are very dierent (see for example the review [65]). Thisphenomenon is called universality and is not only limited to the magneti-sation.Other interesting quantities associated with the Ising model include themagnetic susceptibility (T ) =@@h0M(h; T ) and the two point functionGT(x; y) = h'x'yi0;T, related via (T ) =Px2Zd GT (0; x). According tothe rst graph on Figure 1.3, the susceptibility is innite at T = Tc, but51.1. Introductionmuch more can be said about the behaviour of these functions in the vicinityof the critical point:(T )  (T  Tc)(T # Tc); (1.6)GTc(x; y)  jx yj(d2+)(jx yj ! 1); (1.7)M+(T )  (Tc T )(T " Tc); (1.8)M(h; Tc)  h1=(h # 0): (1.9)The exponents ; ;  and  in the above asymptotic equations are calledcritical exponents and are predicted to be universal for large classes ofmodels. Physical systems are divided into universality classes accordingto their critical behaviour. For example, the universality class of the Isingmodel contains a variety of models of phase transitions, including ferro-magnetism and critical opalescence of liquids, sharing the same universalexponents.Universality explains how a relatively simple mathematical model canexhibit the same critical behaviour as a number of dierent and compli-cated \real world" systems. It also gives physical relevance to the study ofidealized statistical-mechanical models, since this theoretical investigationprovides insights into the critical behaviour of real systems as long as theyare in the same universality class. From the mathematical side, it also posesan important problem of actually computing the critical exponents for dif-ferent critical models and proving rigorously that the equations (1.6){(1.9)hold.1.1.3 A brief overview of known results for the Ising modelSince the Ising model is not the primary focus of this thesis, we will be verybrief and restrict our attention to the case of the lattice Zd. There has beensignicant progress in the case of d = 2,Theorem 1.1. For d = 2, T1c=12log(1+p2) and  =18,  = 15,  =14.The scaling limit of the interface between the two phases in the T = Tcsimulation of Figure 1.2 is SLE3.61.1. IntroductionThe computation of Tcand  is due to Onsager [60], the value of was computed by Camia, Garban and Newman [26], and  is attributed toWu [55]; the result about the interface is due to Chelkak, Duminil-Copin,Hongler, Kemppianen and Smirnov [28]. However, there are many othernames that are connected with the study of the Ising model in 2 dimensions(see [56] for a review).The physically most interesting case of d = 3 is the most dicult andthe least understood. Very recently, and about 70 years after Onsager'sexact solution of the 2-dimensional Ising model, it was proved that thespontaneous magnetisation of the 3-dimensional Ising model vanishes at thecritical temperature. It remains a major open problem to prove the existenceof critical exponents for d = 3.Theorem 1.2 (Aizenman, Duminil-Copin, Sidoravicius [4]). For d = 3,M+(Tc) = 0.Another case where much is known is in dimensions d > 4. There,the critical exponents take the same values as for the Ising model on thecomplete graph, which is called the the Curie{Weiss or mean-eld model.The values of the exponents for d > 4 are thus called the mean-eld values.Theorem 1.3 (Aizenman [2], Aizenman and Fernandez [5], Frohlich [36],Sakai [63]). For d > 4,  = 1,  = 12,  = 3,  = 0.The dimension d = 4 is called the upper critical dimension. There, thepredicted behaviour is mean-eld with logarithmic corrections.Theorem 1.4 (Aizenman and Fernandez [5], Aizenman and Graham [6]).For d = 4, for ; ; , deviations from mean-eld are at most logarith-mic.The precise behaviour for 4-dimensional Ising model remains open, al-though we would like to mention the rigorous analysis of the 4-dimensionalhierarchical Ising model by Hara, Hattori and Watanabe [43].This thesis addresses a closely related problem: the critical behaviourof the n-component j'j4model, and of the continuous-time weakly self-71.2. The j'j4modelavoiding walk (or WSAW) in 4-dimensions. We will now give precise def-initions of these models, their relationship to the Ising model and to eachother, followed by statements of our main results.1.2 The j'j4 modelThe n-component j'j4model is a relative of the Ising model in which thespins are allowed to be arbitrary vectors 'x2 Rn. In 2 dimensions, a con-guration of a 2-component j'j4model is similar to the O(2) congurationappearing in Figure 1.1, except that the arrows should also be of varyinglengths. It is conjectured that the j'j4model is in the same universalityclass as the Ising model for n = 1, and as O(n) model for all n [27].1.2.1 Definition of the modelWe work on a torus: let L > 1 be an integer, and let  = N= Zd=LNZdbe the d-dimensional discrete torus of side length LN. Ultimately we areinterested in the thermodynamic limit N !1. For convenience, we some-times consider  to be a box in Zd, approximately centred at the origin,without opposite sides identied to create the torus. We can then regardxed a; b 2 Zd as points in  provided that N is large enough, and we makethis identication throughout the thesis. In particular, we always assumethat N is suciently large that  contains the given a; b.The spin eld ' is a function ' :  ! Rn, or equivalently a vector' 2 (Rn). We use subscripts to index x 2  and superscripts for the com-ponents i = 1; : : : ; n. We write jvj for the Euclidean norm jvj2=Pni=1(vi)2and v  w =Pni=1viwifor the Euclidean inner product on Rn. For e 2 Zdwith jej = 1, we dene the discrete gradient by (re')x= 'x+e'x, and thediscrete Laplacian by  = 12Pe2Zd:jej1=1rere. We write 'x (')x=Pni=1'ix('i)x.Given g > 0;  2 R, we dene a function Ug;;Nof the eld that willreplace the Hamiltonian (1.1). The quartic term is j'xj4= ('x 'x)2.Ug;;N(') =Xx214gj'xj4+12j'xj2+12'x ('x): (1.10)81.2. The j'j4modeljjg12jjg1214gj'j4+12j'j2jjg12jjg12e(14gj'j4+12j'j2)Figure 1.4: A plot of potentials 14gj'j4+12j'j2and e(14gj'j4+12j'j2)as a function of j'j for  < 0. The rightmost graph correspondsto the single spin distribution in the j'j4model when n = 1.Note that the third term in Ug;;Nis the same as H0when n = 1. The rsttwo terms14gj'xj4+12j'xj2contribute to the single spin distribution (seeFigure 1.4). Analogously to (1.5), we dene the expectation of a randomvariable F : (Rn) ! R byhF ig;;N=1Zg;;NZF (')eUg;;N(')d'; (1.11)where d' is the Lebesgue measure on (Rn), and Zg;;Nis a normalisationconstant dened so that h1ig;;N= 1. Thus ' is a eld of classical continuousn-component spins on the torus , i.e., with periodic boundary conditions.In view of the comment below (1.2), we can think of the Ising spinsas taking values T12at temperature T . By the plots on Figure 1.4, for < 0, in the 1-component j'j4model, the spins are concentrated around(jjg)12. Since we like to think of g as xed, the parameter  plays the roleof temperature and this suggests that there is a phase transition associatedto a critical value c.91.2. The j'j4modelTo identify this critical point, we dene the susceptibility(g; ;n) = limN!1Xx2Nh'1a'1xig;;N= n1limN!1Xx2Nh'a 'xig;;N; (1.12)assuming the limit exists. By translation-invariance of the measure,  isindependent of a 2 Zd. For n = 1; 2, standard correlation inequalities [34]imply that for the case of free boundary conditions the limit dening thesusceptibility exists (possibly innite) and is monotone non-increasing in .Proofs are lacking for n > 2 due to a lack of correlation inequalities in thiscase (as is discussed in [34]), although one expects that these facts known forn  2 should remain true also for n > 2. In our theorems below, we provethe existence of the innite volume limit with periodic boundary conditionsdirectly in the situations covered by the theorems, without application ofany correlation inequalities.For d = 4, small g > 0, and for all n  1, it is proved in [13] that thereis a critical value c= c(g;n) such that, for  = c+ ", the susceptibilitydiverges according to the asymptotic formula(g; ;n)  Ag;n"1(log "1)(n+2)=(n+8)as " # 0; (1.13)for some amplitude Ag;n> 0. Here, and throughout the document, we writef  g to mean lim f=g = 1. In this thesis, we study correlation functionsboth exactly at the critical value c(g;n) and in the limit as  # c(g;n). Itis also shown in [13] that c(g; n) = ag +O(g2) with a = (n+ 2)G00> 0,where, for a; b 2 Z4, Gabdenotes the massless lattice Green function. Froman analytic perspective,Gab= (1Z4 )ab; (1.14)where the right-hand side is the matrix element of the inverse lattice Lapla-cian acting on square integrable scalar functions on Z4. From a probabilisticperspective, Gabequals12dtimes the expected number of visits to b of a sim-ple random walk on Z4 started from a (the extra factor 12d=18is due toour denition of the Laplacian). It is a standard fact (see, e.g., [50]) that,101.2. The j'j4modelas ja bj ! 1,Gab=1(2)2ja bj21 +O1ja bj2: (1.15)1.2.2 Correlation functionsWe study innite volume correlation functions. The existence of the in-nite volume limit is not known for general n, and it is part of our re-sults that the limit does exist for n  1, provided g is suciently small.We write hF ig;= limN!1hF ig;;Nwhen the limit exists. We also writehF ;Gi = hFGihF ihGi, both in nite and innite volume, for the correla-tion or truncated expectation of F;G. Our main results include the preciseasymptotic behaviour as ja bj ! 1, for all n  1 and for p = 1; 2, of the4-dimensional innite volume critical correlation functionsD('ia)p; ('jb)pEg;cfor 1  i; j  n: (1.16)By the O(n) symmetry of Ug;;Nfrom (1.10), it is sucient to consider thetwo special cases (i; j) = (1; 1) and (i; j) = (1; 2). The rst case turns outto be positive. Since the transformation '17! '1does not change themeasure, the second case is zero for all odd p. The second case only makessense for n  2, and it turns out to be negative for p = 2. In principle ourmethods could be used to study also p > 2, but new issues arise for p > 2and we have not pursued this case.We dene the critical correlation functions (1.16) as the limitD('ia)p; ('jb)pEg;c= lim"#0limN!1D('ia)p; ('jb)pEg;c+";N: (1.17)Similarly, for  > c, we writeXx1;x22Z4D'ix1'jx2; ('ka)2Eg;= limN!1Xx1;x22ND'ix1'jx2; ('ka)2Eg;;N: (1.18)It is part of the statement of our results that these limits exist for small111.3. The WSAW modelg > 0 and for n  1, p = 1; 2. However, we do require that the limit betaken through tori N= Z4=LNZ4 with L large, as this restriction is part ofthe hypotheses of results from [10,13,23,24] upon which our analysis relies.We therefore always tacitly assume that L is large, throughout the rest ofthe thesis, for both the j'j4and WSAW models. When we assume that gis small in theorems, g is chosen small depending on the value of L, anddepending also on n.1.3 The WSAW modelThe j'j4model is related to the well-known Ising model in much the sameway as the weakly self-avoiding walk model (WSAW from now on) relatesto the self-avoiding walk model.1.3.1 Self-avoiding walkThe self-avoiding walk is a probability measure on paths in a lattice that as-signs the same probability to every walk of n steps2without self-intersections.The primary physical application of this model that goes back to the poly-mer chemist Flory [35] is the excluded volume problem for a long polymerchain in a dilute solution. See [46] for an overview of its long history.Let x 2 Zd. An n-step self-avoiding walk from 0 to x is a sequence! : f0; 1; : : : ; ng ! Zd with !(0) = 0, !(n) = x, j!(i + 1)  !(i)j = 1, and!(i) 6= !(j) for all i 6= j. We let Sn(x) denote the set of all n-step self-avoiding walks from 0 to x and put Sn= [x2ZdSn(x). Let cn(x) = jSn(x)jand cn=Pxcn(x) = jSnj. We declare all walks in Snto be equally likely:each walk has probability c1n.It is easy to see that cm+n cmcn, and by Fekete's subadditivity lemma(see for example [54]), c1=nn! . Also, we dene the two-point functionby Gz(x) =P1n=0cn(x)zn; it can be shown that its radius of convergence iszc= 1for all x [54].2It is customary to use n to denote the number of steps of a walk. This n is in no wayrelated to the number of components of the spin eld ', that we also denote by n.121.3. The WSAW modelIt is predicted that there are universal critical exponents ; ;  such thatcn Ann1(n!1); (1.19)Enj!(n)j2 Dn2(n!1); (1.20)Gzc(x)  Cjxj(d2+)(jxj ! 1); (1.21)where f  g means lim f=g = 1 and Enis the expectation with respect tothe uniform measure on Sn.For dimensions d  5, (1.19){(1.21) have been proved using the laceexpansion.Theorem 1.5 (Brydges and Spencer [25], Hara and Slade [44], Hara [42]).For d  5,  = 1,  =12,  = 0 and the self-avoiding walk converges tothe Brownian motion in the scaling limit.For d < 4, very little has been proved. There is a complete set ofpredictions for d = 2:  =4332,  =34,  =524(Nienhuis [59]), and the scalinglimit is SLE8=3(Lawler, Schramm, Werner [52]). For d = 3 there are onlynumerical results.For d = 4, it is predicted that the scaling limit is again Brownian motion,the critical exponents take the same values as for d  5, but logarithmiccorrections appear:cn An(logn)1=4; Enj!(n)j2 Dn(logn)1=4; Gzc(x)  cjxj2: (1.22)Recall that for the Ising and j'j4models, the susceptibility can be obtainedby summing the two-point function over one of its endpoints. We do thesame here and provide the prediction for d = 4:(z) =Xx2ZdGz(x) =1Xn=0cnzn; (z) j log(zc z)j1=4zc z; as z " zc: (1.23)There is a connection, discovered by de Gennes [39], between the self-avoiding walk and the spin systems that we discussed earlier: one can con-sider the self-avoiding walk to be a \zero-component" ferromagnet. The131.3. The WSAW modelnumber n of components of the spin variable ' is 1 for the Ising model andcan take any value n  1 for the O(n) and the j'j4models, but it cannotbe set to 0 in the denitions of these models. However, by taking the n! 0limit of the two-point function h'ia'jbig;c, one can obtain the self-avoidingwalk two-point function Gzc(a; b) [54]. This connection suggests that thecritical exponents (1.19){(1.21) also correspond to (1.6){(1.9) in the n! 0limit.In this thesis, we study the d = 4 case for the continuous-time weaklyself-avoiding walk, which is predicted to be in same universality class as thestrictly self-avoiding walk, and is also be related to spin systems in a similarway.1.3.2 Definition of the WSAW modelNotation Since weakly and strictly self-avoiding walks can be consideredto be models of \zero-component" ferromagnets, as discussed at the endof Section 1.3.1, we will systematically use the n = 0 to denote WSAWquantities when they can be confused with those in the j'j4model. Forexample, the zeros in the notation for the susceptibility  and the criticalpoint c, in (1.26) and below, serve this purpose. This notation is especiallyuseful for the statement our main theorems in Section 1.4.Let X be the continuous-time simple random walk on the integer latticeZd, with d > 0. In more detail, X is the stochastic process with right-continuous sample paths that takes its steps at the times of the events of arate-2d Poisson process. Steps are taken uniformly at random to one of the2d nearest neighbours of the current position, and are independent both ofthe Poisson process and of all other steps. Let Eadenote the expectationfor the process with X(0) = a 2 Zd. The local time of X at x up to timeT is the random variable LT(x) =RT01X(t)=xdt, and the intersection local141.3. The WSAW modeltime up to time T is the random variableI(T ) =ZT0ZT01X(t1)=X(t2)dt1dt2=Xx2ZdLT(x)2: (1.24)Given g > 0,  2 R, and a; b 2 Zd, the continuous-time weakly self-avoiding walk two-point function is dened by the integral (possibly in-nite)W(1)ab(g; ) =Z10EaegI(T )1X(T )=beTdT: (1.25)In (1.25), self-intersections are suppressed by the factor egI(T ). The con-nection between (1.25) and the two-point function of the usual strictly self-avoiding walk is discussed in [16]. In dimension 4, (1.25) is also known as thetwo-point function of the lattice Edwards model (with continuous time).We dene the susceptibility by(g; ; 0) =Xb2ZdW(1)ab(g; ) =Z10EahegI(T )ieTdT: (1.26)By translation-invariance of the simple random walk and of (1.24),  isindependent of the point a 2 Zd. A standard subadditivity argument [10]shows that for all dimensions d > 0 there exists a critical value c=c(g; 0) 2 (1; 0] (depending also on d) such that(g; ; 0) <1 if and only if  > c: (1.27)It is shown in [10] that for d = 4, for small g > 0 and for  = c+ ", thesusceptibility diverges as(g; ; 0)  Ag;0"1(log "1)1=4as " # 0: (1.28)Moreover, c(g; 0) = ag + O(g2) with a = 2G00> 0. Those asymptoticformulas for the susceptibility and the critical point are both consistentwith setting n = 0 in the corresponding statements for the j'j4model inSection The WSAW modelbabababaFigure 1.5: Watermelon networks for p = 1; 2; 3; Watermelon and star networksFor p  1, consider the vector of p independent continuous-time simplerandom walks on Z4:X(T ) =X1(T1); : : : ; Xp(Tp)for T = (T1; : : : ; Tp) 2 Rp+: (1.29)We write Eafor the expectation of X with Xk(0) = a for all k. We denethe corresponding local times, for x 2 Z4, byLkTk(x) =ZTk01Xk(t)=xdt and LT(x) = L1T1(x) +   + LpTp(x): (1.30)Let Ip(T ) =Px2Z4 (LT (x))2. We write X(T ) = b to mean that Xk(Tk) = bfor all k = 1; : : : ; p, and write dT = dT1   dTp. The p-watermelon networkis then dened byW(p)ab(g; ) = p!ZRp+EaegIp(T )1X(T )=bekTk1dT: (1.31)The 1-watermelon network is simply the two-point function, which wasstudied in [9]. There it was proved that the critical two-point functionobeys W(1)ab(g; c)  Cja  bj2for small g. This is the same asymptoticbehaviour (1.15) for the Green function. By denition, Ip(T ) Ppi=1Ii1(Ti),where the superscript i indicates the self-intersection local time of Xi. Thisimplies that W(p)ab(g; c)  p!(W(1)ab(g; c))p O(ja  bj2p). In particular,the critical p-watermelon is nite for all p  1. Our main results provideprecise asymptotics for W(p)ab(g; c) for all p  1.161.4. Main resultsaaa aFigure 1.6: Star networks for p = 1; 2; 3; 5.For p  1 and a 2 Z4, we also deneS(p)(g; ) = p!ZRp+EaegIp(T )ekTk1dT= p!Xb1;:::;bp2ZdZRp+EaegIp(T )1(8k)(Xk(Tk)=bk)ekTk1dT:(1.32)The right-hand side is independent of a by translation invariance. By def-inition, S(1)is the susceptibility , while, for p  2, S(p)is the generatingfunction for a star network of weakly self- and mutually-avoiding walks asdepicted in Figure 1.6. By a similar argument to the one employed above forwatermelon networks, S(p)(g; ) < p!p(g; ). In particular, S(p)(g; ) < 1for  > c.1.4 Main results1.4.1 Statements of main theoremsLet n  0 and p  1 be integers. We x g > 0 small and drop it from thenotation. Exponents on logarithms turn out to be expressed in terms of+n;p= p2!n+ 2n+ 8; n;p= p2!2n+ 8; (1.33)with12= 0 so that in the degenerate case +n;1= n;1= 0. By denition,for n = 0 we have +0;p= 0;p=14p2. We also dene the constantb =n+ 8162: (1.34)171.4. Main resultsTheorem 1.6. Let d = 4. Let n  0 and p  2 be integers. Let g > 0 besuciently small, depending on n; p, and let " =  c(g;n) > 0. Thereare g-dependent constants An;p;> 0 such that the following hold as" # 0.(i) For p  2,1(; 0)pS(p)() A0;p;+(log "1)+0;p: (1.35)(ii) For p = 2, the left hand sides of the following equations are denedby limits (1.18), these limits exist and12Xx1;x22Z4D('x1 'x2) ; j'aj2EnAn;2;+(log "1)+n;2(n  1); (1.36)12Xx1;x22Z4D'1x1'1x2; ('1a)2En 1nAn;2;(log "1)n;2(n  2); (1.37)12Xx1;x22Z4D'1x1'1x2; ('2a)2E 1nAn;2;(log "1)n;2(n  2): (1.38)(iii) The amplitudes obey, as g # 0,An;p;= p!(bg)n;p(1 +O(g)): (1.39)For the case n  1, it is part of the statement of the following theoremthat the critical correlation functions on Z4 exist in the sense of (1.17). Wewrite error estimates as ja bj ! 1 in terms ofE(p)ab=8<:O(log ja bj)1(p = 1)O(log log ja bj)(log ja bj)1(p  2):(1.40)Theorem 1.7. Let d = 4. Let n  0 and p  1 be integers. Let g > 0 besuciently small, depending on n; p. There are g-dependent constantsA0n;p;> 0 such that the following hold as ja bj ! 1.181.4. Main results(i) For p  1,W(p)ab(c(0)) =A00;p;+(log ja bj)2+0;p1ja bj2p1 + E(p)ab: (1.41)(ii) For n  1,D'1a;'1bEc(n)=A0n;1;+ja bj21 + E(1)ab; (1.42)Dj'aj2; j'bj2Ec(n)=nA0n;2;+(log ja bj)2+n;21ja bj41 + E(2)ab: (1.43)(iii) For n  2,D('1a)2; ('1b)2Ec(n)=1n(n 1)A0n;2;(log ja bj)2n;2+A0n;2;+(log ja bj)2+n;21ja bj41 + E(2)ab;(1.44)D('1a)2; ('2b)2Ec(n)=1n0@A0n;2;(log ja bj)2n;2+A0n;2;+(log ja bj)2+n;21A1ja bj41 + E(2)ab:(1.45)(iv) The amplitudes obey, as g # 0,A0n;p;=p!(2)2p(bg)2n;p(1 +O(g)): (1.46)Concerning (1.46), the factor (2)2parises from the pthpower of theGreen function via (1.15). The power of g in (1.46) matches the power ofthe logarithm in the term where the amplitude appears. The combinationg log ja  bj is natural since there are no logarithmic corrections for theGaussian case g = 0.In Theorem 1.7, the interesting asymptotic behaviour as ja bj ! 1 isstressed. However, our proof applies more generally, and gives the followingresult for the case a = b, which provides a natural continuity statement asg # 0.191.4. Main resultsTheorem 1.8. Let d = 4. Let n  0 and p  1 be integers. Let g > 0 besuciently small, depending on n; p. Then, as g # 0,W(p)aa(c(0)) = Gpaa(p! +O(g)) (p  1); (1.47)D'1a;'1aEc(n)= Gaa(1 +O(g)) (n  1); (1.48)Dj'aj2; j'aj2Ec(n)= G2aa(2!n+O(g)) (n  1); (1.49)D('1a)2; ('2a)2Ec(n)= O(g) (n  2): (1.50)It is worth mentioning that even to prove that the left-hand sides of(1.41){(1.45) or (1.47){(1.50) are nite is a nontrivial result.1.4.2 DiscussionPrevious work Special cases of Theorem 1.7 have been proven previously.For (n; p) = (0; 1), (1.41) is the main result of [9]; a related result for a modelthat involves neither a lattice nor walks appears in [47]. For (n; p) = (1; 1),(1.42) is the main result of [37]. For (n; p) = (1; 2), (1.43) was proved in [38];in this case the leading behaviour is ja bj4(log ja bj)2=3. For a relatedmodel in which an ultraviolet cuto replaces the lattice setting, a version of(1.42) for the case (n; p) = (1; 1) appears in [33].The results: (i) (1.41) for p  2, (ii) (1.42){(1.43) for n  2 and p = 1; 2,and (iii) (1.44){(1.45) for n  2 and p = 2, are new as rigorous results.For the continuous-time weakly self-avoiding walk on a 4-dimensionalhierarchical lattice, much more has been proved [15,18,19,58]; in particular,the asymptotic behaviour of the end-to-end distance is identied in [18].The decay of the analogue of hj'aj2; j'bj2icbelow the critical dimensionhas been studied rigorously in a hierarchical setting of quantum elds overthe p-adics [1].Critical exponents For p = 1, the right-hand sides of (1.41){(1.42) givethe decay of the critical two-point function that is usually written in termsof the critical exponent  as jabj(d2+). This is a statement that  takes201.4. Main resultsits mean-eld value  = 0 for all n  0, with no logarithmic correction tothe leading behaviour.The exponents n;pin Theorem 1.6 and the exponents 2p; 2n;pin The-orem 1.7 are predicted to be universal. In particular, the n = 1 exponentsof (1.36) and (1.42){(1.43) are predicted to apply to the Ising model, andthe exponents of (1.36){(1.38) and (1.42){(1.45) for n  2 are predicted toapply to the O(n) model, including the classical XY (or rotor) model forn = 2, and the classical Heisenberg model for n = 3.Similarly, the n = 0, p  1 case of (1.35) and (1.41), namelyS(p)()()pA0;p;+(log "1)14(p2); W(p)ab(c) A00;p;+ja bj2p(log ja bj)24(p2); (1.51)are predicted to apply to the 4-dimensional strictly self-avoiding walk.For p  2 independent WSAWs, pS(p)is identically equal to 1, andW(p)ab(c) is asymptotic to a multiple of ja  bj2p. The logarithmic correc-tions in (1.51) for p weakly mutually-avoiding walks are consistent with theinterpretation that the intersection of each of thep2pairs of walks at a ver-tex gives rise to a penalty (log "1)14or (log jabj)14paid by each pair forjoining, despite their penchant to avoid. Related results were obtained viaa non-rigorous renormalisation analysis in [29], and a detailed non-rigorousgeneral treatment of polymer networks, including also dimensions below 4,can be found in [31].For the case of simple random walk, the formula for star networks in(1.51) is reminiscent of the fact, proved in [51], that p independent simplerandom walks started from the origin in Z4 do not have pairwise inter-sections before leaving the ball of radius n, with probability asymptotic to(logn)12(p2)(see [30] for a non-rigorous renormalisation analysis). A numberof authors have studied related matters for the case of two simple randomwalks [3, 32, 49, 62]. For spread-out models of strictly SAW in dimensionsd > 4, rigorous results for arbitrary graphical networks were obtained in [45].These results for d > 4 include a statement analogous to (1.51) for all p  1,but there is no logarithmic correction and the asymptotic behaviour is sim-211.4. Main resultsply constja  bjp(d2). See also [54, Theorem 1.5.5] for nearest-neighbourstrictly SAW for d  6.Reduction of symmetry For the case n  2 and p = 2, since n;2=2n+8<n+2n+8= +n;2, Theorem 1.7 gives (for i 6= j)D('ia)2; ('ib)2Ec(n)n 1nA0n;2;ja bj4(log ja bj)4=(n+8); (1.52)D('ia)2; ('jb)2Ec(n) 1nA0n;2;ja bj4(log ja bj)4=(n+8): (1.53)On the other hand, by (1.43),Dj'aj2; j'bj2Ec(n) nA0n;2;+ja bj4(log ja bj)2(n+2)=(n+8): (1.54)Thus, for an individual component, ('ia)2is more highly correlated with('ib)2, than is j'aj2with j'bj2, due to cancellations with the negative cor-relations of ('ia)2with ('jb)2for i 6= j. Negative correlations for dierentcomponents at the same point are to be expected since hj'aj2ic(n)<1 by(1.48), and therefore the eld has a typical size, so making one componentlarge must come at the cost of making one component small. This is similarto the fact that the squares of dierent components of a uniform randomvariable on the sphere are negatively correlated by the length constraint.Our results show how this eect persists over long distances at the criticalpoint.1.4.3 A word about the proofThe proof proceeds via second-order perturbative calculations [11] of thesort found in non-rigorous renormalisation group calculations in the physicsliterature, but here with all error terms rigorously controlled via a generalrenormalisation group method [23,24].First steps towards the application of the method to critical correlationfunctions were made in [9], where the case n = 0, p = 1 was studied. Here221.4. Main resultswe signicantly extend the methods applied in [9] to obtain a much moregeneral result, which identies logarithmic corrections that appear whenp  2 and reveals new phenomena for the case n  2, p = 2. We organisethe proof of our main theorems as follows.Chapter 2 deals with the unication of the n = 0 case with the case n  1despite the apparent dierences in the denitions of the WSAW andj'j4models.Chapter 3 is devoted to the calculation of the nite volume partition function tosecond order using a perturbative approach. The reduction of theO(n)symmetry, that is the origin of the dierent powers in the logarithmiccorrections in (1.52){(1.53), compared to the O(n) symmetric case of(1.54), is also discussed there.Chapter 4 adds a non-perturbative error coordinate to the perturbation theorycalculations and investigates the properties of the resulting renormal-isation group ow.Chapter 5 is the nal analysis of the renormalisation group ow, where we takethe thermodynamic limit and prove our main results.23Chapter 2Reformulation of theproblemInitially, the denitions of the j'j4and WSAW models appear quite dif-ferent. In this chapter, we develop a unied formulation of the problemsaddressed in our main theorems. We begin in Section 2.1 by recalling andextending the connection between the j'j4and WSAW models, which arisesfrom an integral representation of WSAW. Using the integral representa-tion, the WSAW star and watermelon networks are expressed in terms offunctional integrals which involve a complex boson eld ffi and a fermioneld  , with quartic self-interaction. The renormalisation group method weapply is well suited to the analysis of such problems with or without thefermion eld, and both models can be handled together, once we replace theGaussian expectation for the j'j4model by a Gaussian super-expectation,as discussed in Section 2.2. The specic correlation functions studied in ourmain theorems are obtained via the use of external elds, which we discussin Section 2.3. Finally, in Section 2.4 we reformulate the basic problem ina unied manner for both models in terms of these auxiliary elds.242.1. Integral representation of the WSAW2.1 Integral representation of the WSAWSuch integral representations are discussed at length in [20]. The particularapproach we present here arose in [15], but these ideas have a long historygoing back to [53,57,61,66].2.1.1 Infinite volume limit for WSAWThe integral representation for WSAW is for a nite volume version, and werst show how the watermelon and star networks on Zd can be approximatedby networks on a torus. Let ENadenote the expectation corresponding to pindependent continuous-time simple random walks on the torus N, startedat a 2 N. Let b 2 N. For p  1, we dene a nite volume p-watermelon(1.31), byW(p)ab;N(g; ) = p!ZRp+ENaegIp(T )1X(T )=bekTk1dT; (2.1)and a nite volume star network (1.32), byS(p)N(g; ) = p!ZRp+ENaegIp(T )ekTk1dT(2.2)(which is independent of a by translation invariance).By the argument under (1.31), W(p)ab;N p!(W(1)ab;N)p. By the Cauchy{Schwarz inequality, T1=Px2LT1(x)  (jjI1(T1))1=2, so I1(T1)  T21=jj,from which we conclude that W(1)ab;N, and hence W(p)ab;N, is nite for all g > 0and  2 R. Similarly, for all g > 0 and  2 R, S(p)N(g; ) <1.Proposition 2.1. For d > 0 and g > 0,W(p)ab(g; c) = lim#cW(p)ab(g; ) = lim#climN!1W(p)ab;N(g; ); (2.3)and, for  2 R,S(p)(g; ) = limN!1S(p)N(g; ): (2.4)Proof. A nearest-neighbour walk X on Zd, can be folded to obtain a walk252.1. Integral representation of the WSAWon N. Let LNT(x) be the local time of the folded walk, then we haveLNT(x) =Xy2Zd:kyk1LLN+1Tx+ yLN: (2.5)From this follows the monotonicity of the intersection local time INp(T ).IN+1p(T ) =Xx2Z4LN+1T(x)2=Xx2Z4XkykLLN+1Tx+ yLN2Xx2Z40@XkykLLN+1Tx+ yLN1A2=Xx2Z4LNT(x)2= INp(T ):(2.6)Since folding a walk can only increase the number of intersections, we haveINp(T )  IN+1p(T )  Ip(T ). We now letcT;N= ENaegIp(T ); cT= EaegIp(T ); (2.7)and observe that the monotonicity of INp(T ) implies cT;N cT;N+1 cT.Now, the only contribution to jcTcT;Nj comes from walks that wrap aroundthe torus Nand since INp(T ) > 0, for a rate-2d Poisson process Y (the clockof the continuous time walk),jcT cT;Nj  2pPnYT12diam(N)oand limN!1cT;N= cT: (2.8)The monotone convergence theorem and (2.8) yields (2.4), sinceS(p)N= p!ZRp+cT;NekTk1dT ! p!ZRp+cTekTk1dT = S(p): (2.9)For (2.3), we drop g and a; b from the notation, x  > 0> c, and lets= fT 2 Rp+: kTkk1 sg. We also set cT;N(a; b) = ENa[egIp(T )1X(T )=b]and let cT(a; b) denote the corresponding innite volume quantity. Then,262.1. Integral representation of the WSAWfor any 0  s <1,W(p)()W(p)N()W(p)()ZscT(a; b)ekTk1dT+W(p)N()ZscT;N(a; b)ekTk1dT+ZscT(a; b) cT;N(a; b)ekTk1dT;(2.10)Note that the argument that lead to (2.8) also applies to cT;N(a; b), so bydominated convergence, the last contribution to (2.10) vanishes. Let  =  0> 0, thenZRpnscTekTk1dT  esW(p)(0) andlim supN!1ZRpnscT;N(a; b)ekTk1dT  eslim supN!1W(p)N(0): (2.11)Therefore, since W(p)(0)  Sp(0) <1 andlim supN!1WpN(0)  lim supN!1SpN(0)  Sp(0) <1; (2.12)the rst two terms in (2.10) also vanish as s ! 1. This shows the secondequality in (2.3), the rst equality follows by monotone convergence.2.1.2 Differential formsLet M = jNj = LNd. Let u1; v1; : : : ; uM; vMbe standard coordinates onR2M . Then du1^dv1^  ^duM^dvMis the standard volume form on R2M ,where ^ denotes the anticommuting wedge product. The one-forms dux, dvygenerate the Grassmann algebra of dierential forms on R2M . We multiplydierential forms using the wedge product, but for notational simplicitywe do not display the wedge explicitly, and write, e.g., duxdvyin place ofdux^ dvy. The order of dierentials in a product therefore matters.A p-form is a function of u; v times a product of p dierentials, or asum of such. In general, a form K is a sum of p-forms for p  0, the largestsuch p is called the degree of K and the individual p-forms are called the272.1. Integral representation of the WSAWp-degree part of K. A form which is a sum of p-forms for even p only iscalled even. The integral of a dierential form K over R2M is dened to bezero unless K has degree 2M ; in that case, the 2M -degree part of K can bewritten as f(u; v)du1dv1   duMdvM, and we deneZK =ZR2Mf(u; v)du1dv1   duMdvM; (2.13)where the right-hand side is the Lebesgue integral of f over R2M .We set ffix= ux+ ivx,ffix= ux ivxand dffix= dux+ idvx, dffix=dux idvx, for x 2 . Since the wedge product is anticommutative, thefollowing pairs all anticommute for every x; y 2 : dffixand dffiy, dffixanddffiy, dffixand dffiy. Also,dffixdffix= 2iduxdvx: (2.14)Althoughffi is not an independent variable and is determined by ffi, we stillwrite f(ffi;ffi) because the Taylor series expansion of a smooth f around zerowill be in powers of both ffi andffi. The integralRf(ffi;ffi)Qx2dffixdffixisgiven by (2i)Mtimes the Lebesgue integral of f(u + iv; u  iv) over R2M .The product over x can be taken in any order, since each factor dffixdffixhaseven degree. We write x=1(2i)1=2dffix; x=1(2i)1=2dffix; (2.15)with a xed choice of the square root. Then x x=12idffixdffix=1duxdvx: (2.16)We refer to ffi;ffi as the boson eld and to  ; as the fermion eld.Let N∅denote the algebra of even dierential forms. An element K 2N∅can be written asK =2MXk=0Xp;q:p+q=2kXx1;:::;xp2Xy1;:::;yq2Kx;y x y; (2.17)282.1. Integral representation of the WSAWwhere x = (x1; : : : ; xp), y = (y1; : : : ; yq),  x=  x1   xp, y= y1   yp,and where each Kx;y(including the degenerate case p = q = 0) is a functionof (ffi;ffi). We x a positive integer pNand impose the smoothness conditionthat elements of N∅are such that the coecients Kx;yare in CpN(thechoice of pNis discussed in Section 4.3.2).Given a nite index set J , let K = (Kj)j2Jwith each Kj2 N∅. LetK0jdenote the degree-zero part of Kjthat we assume to be real. Given aC1function F : RJ ! C, we dene F (K) by its power series about thedegree-zero part of K, i.e.,F (K) =X1!F()(K0)(K K0): (2.18)Here  is a multi-index, with ! =Qj2Jj!, and (K K0)=Qj2J(KjK0j)j. The summation terminates as soon asPj2Jj= M since higher-order forms must vanish, and the order of the product on the right-handside does not matter since each Kjis assumed to be even.For x 2 , we dene the dierential forms with real degree-zero partfix= ffixffix+  x x; (2.19)fi;x=12ffix(ffi)x+ (ffi)xffix+  x( )x+ ( )x x; (2.20)where  = is the lattice Laplacian as dened above (1.10). The followingtheorem is a minor extension of [20, Theorem 5.1]. The integrand on theleft-hand side of (2.21) is dened as in (2.18), e.g., efix= ejffixj2(1+ x x),the integral is as in (2.13). On the right-hand side, Spdenotes the set ofpermutations of 1; : : : ; p.Proposition 2.2. For d > 0, g > 0,  2 R, p  1, and A = (a1; : : : ; ap),B = (b1; : : : ; bp) with each ai; bj2 N,ZePx2fi;x+gfi2x+fixffia1  ffiapffib1  ffibp=Xff2SpZRp+ENAegIp(T )1X(T )=ff(B)ekTk1dT; (2.21)292.1. Integral representation of the WSAWwhere on the right-hand side Xi(0) = aiand Xi(Ti) = ff(bi).Corollary 2.3. For d > 0, g > 0,  2 R, p  1, and a; b1; : : : ; bp2 N,S(p)N(g; ) =Xb1;:::;bp2NZePx2Nfi;x+gfi2x+fixffipaffib1  ffibp; (2.22)W(p)ab;N(g; ) =ZePx2Nfi;x+gfi2x+fixffipaffipb: (2.23)Proof. This is an immediate consequence of Proposition 2.2 and the deni-tions (2.2) of S(p)Nand (2.1) of W(p)ab;N.2.1.3 Proof of Proposition 2.2In this section, we prove Proposition 2.2 using ideas from [20], but organisethe proof in a more direct manner for our current goal. The proof of (2.21) isbased on three dierent formulas for the Green function (+V )1, whereV is a complex diagonal matrix whose diagonal entries vxobey Re(vx) > 0.The three formulas are presented in the following three lemmas.Lemma 2.4. Let Wnabdenote the set of nearest-neighbour n-step pathsfrom a to b. Then(+ V )1ab=1Xn=0XY 2WnabnYj=012d+ vYj: (2.24)Proof. We write  = 2d1 J and let U = 2d1+ V . Then (+ V )1isgiven by the Neumann series(+V )1= (UJ)1=U(1U1J)1=1Xn=0U1JnU1; (2.25)which converges since Re(V ) > 0. The ab matrix element of the right-handside is the right-hand side of (2.24), and the proof is complete.302.1. Integral representation of the WSAWLemma 2.5. Let X be a continuous time simple random walk on  withlocal time LT(x). Let V be a complex diagonal matrix with entries vxsuch that Re(vx) > 0, then(+ V )1ab=ZR+ENaePx2vxLT(x)1X(T )=bdT: (2.26)Proof. We think of X as a discrete time simple random walk Y with inde-pendent and identically distributed Exp(2d) holding times (ffi)i0. We setj=Pji=0ffi, and condition on Y to obtainZEahevLT1X(T )=bidT=1Xn=0XY 2Wnab12dnE"ePn1j=0vYjffjZnn1evYn(Tn1)dT#=1Xn=0XY 2Wnab12dnEePn1j=0vYjffj1vYnevYnffn 1;(2.27)where Wnabis the set of nearest-neighbour n-step paths from a to b. Sincethe ffiare i.i.d., the expectation factors into a product of n+1 expectationsthat can each be evaluated explicitly, with the result thatZEahevLT1X(T )=bidT=1Xn=0XY 2Wnab12dn0@n1Yj=02d2d+ vYj1A2d2d+ vYn 11vYn=1Xn=0XY 2WnabnYj=012d+ vYj:(2.28)By Lemma 2.4, this completes the proof.The next lemma uses the complex Gaussian probability measure on Cwith covariance C, dened bydC=detA(2i)MeffiAffidffidffi; (2.29)312.1. Integral representation of the WSAWwith A = C1and dffidffi is the Lebesgue measure dffi1dffi1   dffidffi(see,e.g., [20, Lemma 2.1] for a proof that this measure is properly normalised).The statement that dChas covariance C means thatRffiaffibdC= Cab.Integration by parts (see, e.g., [20, Lemma 2.2]) gives the formulaZCffiaFeffiAffidffidffi =XxCaxZC@F@ffixeffiAffidffidffi: (2.30)Lemma 2.6. Let V be a complex diagonal matrix with entries vxsuchthat Re(vx) > 0. Let A = + V and set C = A1= (+ V )1.ThenXff2SppYi=1(+ V )1aibff(i)=ZeffiAffi A ffia1  ffiapffib1  ffibp: (2.31)Proof. By denition,e A =MXn=0(1)nn! A n=(1)MM ! A M+ (forms of deg < 2M);(2.32)and only the rst (top degree) form on the right-hand side can contributeto the integral. Using  A =Px;yAxy x yand anti-symmetry, we obtain A M=Xx1;y1  XxM;yMAx1y1  AxMyM x1 y1   xM yM=X2SMXff2SMA(1)ff(1)  A(M)ff(M) (1) ff(1)   (M) ff(M)=M !Xff2SMA1ff(1)  AMff(M) 1 ff(1)   M ff(M)=M !Xff2SMsgn(ff)A1ff(1)  AMff(M) 1 1   M M= (1)MM ! (detA) 1 1   M M;(2.33)so the top degree part of e A is (detA) 1 1   M M. Since x x=322.1. Integral representation of the WSAW12idffixdffix, this givesZeffiAffi A ffia1  ffiapffib1  ffibp=ZCffia1  ffiapffib1  ffibpdC: (2.34)We apply the integration by parts formula (2.30) p times to see that theright-hand is equal to the left-hand side of (2.31), and the proof is complete.(The last step is an instance of Wick's Theorem [41].)Proof of Proposition 2.2. We prove (2.21). First, we dene F : RN ! RbyF (S) = ePx2NgS2x+(1)Sx(S 2 RN ): (2.35)Then, by the denition given in (2.1) and the fact thatPxLT(x) = kTk1,the summand on the right-hand side of (2.21) is equal toZRp+ENAeIp(T )1X(T )=ff(B)ekTk1dT=ZRp+ENAF (LT)1X(T )=ff(B)ekTk1dT: (2.36)Also,ZePx2fi;x+gfi2x+fixffipaffipb=ZF (fi )ePx2fi;x+fixffipaffipb: (2.37)We write F in terms of its Fourier transform^F asF (S) =ZeiPx2rxSx^F (r)dr: (2.38)With an appropriate argument to justify interchanges of integration (donecarefully in [20]), it therefore suces to show that for all sx2 C with332.2. Change of variables and Gaussian approximationRe(sx) > 0,ZePx2fi;x+sxfixffia1  ffiapffib1  ffibp=Xff2SpZRp+ENAePx2sxLT(x)1X(T )=ff(B)dT: (2.39)Let V be the diagonal matrix with entries sx. Since the components of X areindependent and identically distributed, the integral on the right-hand sideof (2.39) factors with each factor being (+V )1aiff(bi)by Lemma 2.5. ByLemma 2.6, the left-hand side of (2.39) is therefore equal to the right-handside, and the proof is complete.2.2 Change of variables and GaussianapproximationTo unify the treatment of the j'j4and WSAW models, for the j'j4modelinstead of (2.19){(2.20) we denefix=12j'xj2; fi2x=14j'xj4; fi;x=12'x (')x: (2.40)For either model, given g; ; z 2 R, we writeUg;;z;x= gfi2x+ fix+ zfi;x: (2.41)The polynomial Ug;;1;xappears in (1.11) with fi and fiinterpreted as in(2.40), and it appears in the right-hand sides of (2.22) and (2.23) with theinterpretation (2.19){(2.20). Given X   and g0; 0; z02 R, we deneU0(X) =Xx2XUg0;0;z0;x=Xx2Xg0fi2x+ 0fix+ z0fi;x: (2.42)To write our principal quantities as perturbations of a Gaussian, we makean appropriate change of variables. For j'j4, given z0> 1 and m2> 0, by342.2. Change of variables and Gaussian approximationdenition,Ug;;1;x(') = U0;m2;1;x((1 + z0)1=2') + Ug0;0;z0;x((1 + z0)1=2'); (2.43)withg0= g(1 + z0)2; 0= (1 + z0) m2: (2.44)The equations (2.44) can equivalently be written asg =g0(1 + z0)2;  =0+m21 + z0: (2.45)For the moment, we regard m2; z0as parameters that can be chosen arbi-trarily. In Section 4.1, we make careful choices of these, corresponding to\physical mass" and \wave function renormalisation" in the physics litera-ture. Let C = (+m2)1, with  the discrete Laplacian on N(actingon scalar functions). For j'j4, the Gaussian expectation with covariance Cis dened byECF = hF i0;m2;N: (2.46)Given a function F (') we write F0(') = F ((1 + z0)1=2'). Using (2.43) andthe change of variables 'x7! '0= (1 + z0)1=2'x, we obtainhF ig;;N=ECF0eU0()ECeU0(): (2.47)For WSAW, we use the Gaussian super-expectationECF =ZFePx2(fi;x+m2fix); (2.48)dened for F 2 N∅such that the integral exists. Such integrals are dis-cussed at length for our context in [20,21]. By Corollary 2.3 and an analogueof (2.43),W(p)ab;N(g; ) = (1 + z0)pECeU0()ffipaffipb: (2.49)Unlike in (2.47), there is no division by a partition function. In fact, as aresult of supersymmetry (see [20]), here ECeU0()= 1. In addition, since352.3. External elds: notation and generalitiesthe Gaussian super-expectation and the polynomial U0are invariant underthe transformation ffi 7! eiffi, EC(eU0()ffipa) = EC(eU0()ffipb) = 0, there isno subtracted term in (2.49), like there is in the truncated correlation (1.16)for the j'j4model.2.3 External fields: notation and generalitiesAs is often the case in statistical mechanics, we compute correlation func-tions as derivatives with respect to an external eld.2.3.1 Correlation functions as derivativesLet n  1 and J;H : ! Rn be functions that we refer to as external elds.We dene the inner product ( ; ) of two elds as ('; J) =Px2'xJx, where'x Jxis the standard dot product on Rn. For a non-negative integer k, wewrite DkJ(H) for the operation of k directional derivatives with respect to Jat J = 0, with each derivative taken in direction H. In other words,DkJ(H)F (J)=@@s10  @@sk0F (0 + s1H + : : :+ skH): (2.50)We say that the eld H is a constant eld, if there is no dependence onx, that is if Hx= H0for every x 2 . In particular, we let 1 denote theconstant eld 1x= 1 for all x 2 .For n = 0, we let J :  ! C and M = jj, then we can write F (J) asa function on R2M of the real and imaginary parts u1; v1; : : : ; uM; vMof J .We deneDJ=12MXk=1dduk iddvk; DJ=12MXk=1dduk+ iddvk: (2.51)Note that DJJ = 1, DJJ = 0 and the combinations obey the Leibniz rule,so that we can write F (J;J) and dierentiate polynomials in J andJ inthe intuitive way. We will specialise the notation for our purposes: we setDkJto be the operator of k directional derivatives with respect toJ in the362.3. External elds: notation and generalitiesdirection of 1 at (J;J) = (0; 0). That isDkJ= DkJ(1)F (J;J)=@@s10  @@sk0F (0; 0 + s11 + : : :+ sk1): (2.52)We illustrate the use of the above denitions with an example. Supposen = 1 and J is an external eld. Fix Hx= 1x=a, thenDJ(H)e('p;J)=@@s0esPx'pxHx=@@s0es'pa= 'pa: (2.53)Using (2.47) and interchanging the derivative with the expectation, we ob-tainh'paig;;N= (1 + z0)p=2ECh'paeU0()iECeU0()= (1 + z0)p=2DJ(H)ECheU0()+('p;J)iECeU0():(2.54)Similarly, if H = 1 (a constant eld 1x 1), DJ(1)e('p;J)=Px2'pxandXx2h'paig;;N= (1 + z0)1=2DJ(1)ECheU0()+('p;J)iECeU0(): (2.55)Since ECeU0()is the nite volume partition function, the above formulasallow us to compute correlation functions as derivatives of the partitionfunction coupled to an external eld J . However the computation of thepartition function coupled to an arbitrary external eld J as it appears inthe numerators of both (2.54) and (2.55) is not an easy task. Instead, werestrict to two very special cases of external eld dependence.2.3.2 The field shift operatorIf the dependence of the partition function on the external eld J is partic-ularly simple, we can absorb it into the Gaussian measure.Let n  1 and C = ( +m2)1Consider an integrable F 2 N that wecouple to an external eld J by a factor e(';J). Completing the square in372.3. External elds: notation and generalitiesthe Gaussian measure yields the following formulaECF (')e(';J)= e12(J;CJ)ECF ('+ CJ): (2.56)The expectation on the right hand side of (2.56) is eectively a conditionalexpectation that acts on the eld ', but leaves J xed. This promptsus to dene the shift operator  that adds to the eld ' a uctuationeld , so that the expectation ECacts on  and leaves ' xed. Then,(ECF )(') = ECF (' + ) and ECF is the convolution of F with theGaussian measure. With this notation, (2.56) becomesECF (')e(';J)= e12(J;CJ)ECF(CJ): (2.57)For n = 0, the denition of the map  is slightly more involved. Givenan additional boson eld ; and an additional fermion eld ; , with  =1p2id,  =1p2id, we consider the \doubled" algebraN (t0) containingthe original elds and also these additional elds. We dene a map  :N ()! N ( t 0) by making the replacement in an element of N of ffi byffi+ ,ffi byffi+,  by  + , and by +. Then for F 2 N (), ECFis obtained by regarding the expectation as an integral over the variables;; ;  which leaves the variables ffi;ffi;  ; xed. The equivalent of (2.57)isECF (ffi;ffi;  ; )e(ffi;J)+(ffi;J)= e(J;CJ)ECF(CJ;CJ; 0; 0): (2.58)2.3.3 Observable parameters and quotient spacesThe case of (2.54) is necessary for the proof of Theorem 1.7, but it is notcovered by (2.57) when p > 1. We rst extend (2.54) to an arbitrary n  1using the notation 'px, which is equal to 'xwhen p = 1, and to the vectorwhose components are ('ix)2for p = 2. Then for some h 2 Rn, let Hx=h1x=a, so thatDJ(H)e('p;J)=@@s0es('pah)= 'pa h: (2.59)382.3. External elds: notation and generalitiesAbove, we replaced the derivative DJ(H) with the partial derivative@@s0.This leads us to considerh'paig;;N= (1 + z0)p=2@@ffa0ECheU0()+ffa('pah)iECeU0()(2.60)instead of the right hand side of (2.54). We will refer to ffaas an observableparameter, since ffais a real number used to compute derivatives withrespect to the external (observable) eld Hx= h1x=a.To compute the derivative, we have no need to examine any dependenceon ffabeyond linear terms. We formalise this observation as follows: rst,we deneN∅= N∅() = CpN((Rn);R) (2.61)to be the space of real-valued functions of the elds ' 2 (Rn) having at leastpNcontinuous derivatives, where pNis xed as in Section 2.1.2. This is thespace of random variables of initial interest, but because of the introductionof the observable parameters to represent the correlation functions fromTheorems 1.6{1.7, these functions must also be able to depend linearly ontwo observable parameters ffaand ffb.We achieve this via the introduction of a quotient space, in which twofunctions of '; ffa; ffbbecome equivalent if their formal power series in theobservable elds agree to linear order in ffa; ffb. LetfN be the space of real-valued functions of '; ffa; ffbwhich are CpNin ' and C1in both ffa; ffb.Consider the elements offN whose formal power series expansion to linear-order in both ffaand ffbis zero. These elements form an ideal I infN , andthe quotient algebra N =fN=I has a direct sum decompositionN =fN=I = N∅NaNbNab: (2.62)The elements of Na;Nb;Nabare given by elements of N∅multiplied byffa, by ffb, and by ffaffbrespectively. As functions of the observable eld,elements of N are then identied with polynomials of degree at most 1 ineach ffaand ffb. For example, we identify e('ah)ffa+('bh)ffband 1 + ('a392.4. Observable and external eldsh)ffa+ ('b h)ffb+ ('a h)('b h)ffaffb. Any element F 2 N can be writtenasF = F∅ + ffaFa + ffbFb + ffaffbFab; (2.63)where F2 N∅for each  2 f∅; a; b; abg. We dene projections : N !Nby ∅F = F∅, aF = ffaFa, bF = ffbFb, and abF = ffaffbFab.For WSAW, the observable parameters are ffa; ffb2 C, and we extend(2.17) by now allowing the coecients Kx;yto be functions of ffa; ffbas wellas of the boson eld ffi;ffi. LetfN be the resulting algebra of dierentialforms, and let I again denote the ideal infN consisting of those elementsoffN whose formal power series expansion to rst-order in ffa; ffbis zero.The quotient algebra N =fN=I also has the direct sum decomposition(2.62). For example, ffixffiy x x2 N∅, and ffaffix2 Na. As functions of theobservable parameters, elements of N are again identied with polynomialswith terms of order 1; ffa; ffb; ffaffb. We use canonical projections also forWSAW.2.4 Observable and external fieldsWe are now ready to dene the observables for each of our two models: thej'j4and the WSAW.2.4.1 Coupling the partition function to observableparametersFor j'j4Given n  1, let Sij= h('ia)p; ('jb)pig;;N, which is what we wishto compute to prove Theorem 1.7. This denes a symmetric n  n matrixwhose diagonal elements are the same, and whose o-diagonal elements arealso the same.Following the ideas outlined in Section 2.3.3, we x h 2 Rn, and intro-duce two observable parameters ffa; ffb2 R. We dene a new polynomial V0(which depends on h; n; p) byV0;x= U0;x ffa('pa h)1x=a ffb('pb h)1x=b: (2.64)402.4. Observable and external eldsLet Dffadenote the operator@@ffaat ffa= ffb= 0, and similarly for higherderivatives. By (2.47) and calculation of the derivative similar to the onedone in (2.60),h  Sh ='pa h ;'pb hg;;N= (1 + z0)pD2ffaffblogECeV0(): (2.65)Given the values of h Sh for two choices of h, the matrix elements of S canbe computed easily.For WSAW LetV0;x= U0;x ffaffipa1x=a ffbffipb1x=b: (2.66)Then the expectation ECeV0()is well-dened for any p  1, including largep, since the supercially dangerous factor exp[ffaffipa+ ffbffipb] is equivalent toa polynomial in the elds, which is integrable. With this interpretation, forall p  1,W(p)ab;N(g; ) = (1 + z0)pD2ffaffbECeV0(): (2.67)In view of the observations below (2.49), we may equivalently writeW(p)ab;N(g; ) = (1 + z0)pD2ffaffblogECeV0(); (2.68)which has the same form as (2.65).2.4.2 Derivatives of the pressureWe nish this chapter with a lemma restating many of the nite volumeidentities obtained above.Definition 2.7. We dene the nite volume partition function ZNby theconvolutionZN= ECeV0(N)(2.69)For n  1, we write ZN(') to emphasise its dependence on the eld '.For n = 0, we write Z0N(ffi;ffi) for the degree-zero part of the form ZN. For412.4. Observable and external eldsn  1, we dene the (un-normalised) pressurePN(') = logZN('): (2.70)We use the notation used in Section 2.3 for derivatives with respectto external elds and observable parameters. For example, DffiZ0Nis thedirectional derivative of ZNwith respect toffi in the direction of the constanteld 1, evaluated at ffi =ffi = 0.Lemma 2.8. Fix m2 > 0 and z0> 1. For n  1, p = 1; 2 and aconstant external eld H,h'pa hig;;N= (1 + z0)p=2DffaPN(0); (2.71)'pa h ;'pb hg;;N= (1 + z0)pD2ffa;ffbPN(0); (2.72)h(';H)p;'pa hig;;N=(1 + z0)pm2pDp'(H)DffaPN: (2.73)For n = 0 and p  1,W(p)ab;N(g; ) = (1 + z0)pD2ffaffbZ0N(0); (2.74)S(p)N(g; ) =(1 + z0)pm2pDpffiDffaZ0N: (2.75)Note that the nite volume correlations as in (1.36){(1.38) can be writtenin the form (2.73) with appropriate choices of H;h 2 Rn.Proof. We rst prove (2.71){(2.73). The identity (2.72) is the same as(2.65), and (2.71) also follows similarly from explicit dierentiation. For(2.73), a direct computation of the derivative (using h(';H)ig;;N= 0) givesh(';H)p;'pa hig;;N= (1 + z0)pDpJ(H)DffalogECeV0()+(';J); (2.76)We let (J) = ECeV0()+(';J), and (2.76) becomesh(';H)p;'pa hig;;N= (1 + z0)pDpJ(H)Dffalog (J): (2.77)422.4. Observable and external eldsUsing (2.57), we get(J) = ECeV0()+(J;')= e12(J;CJ)ZN(CJ): (2.78)We thus obtain log (J) =12(J;CJ) + logZN(CJ). Since (J;CJ) is inde-pendent of the observable parameter ffa,DpJ(H)Dffalog (J) = DpJ(H)DffalogZN(CJ): (2.79)Since H is a constant eld, CH = m2H. The chain rule then givesDpJ(H)Dffalog (J) = m2pDp'(H)DffalogZN; (2.80)and the proof of (2.73) is complete.The case n = 0 is similar, except that the logarithm is superuous dueto the self-normalisation property of the Gaussian super-expectation (seediscussion below (2.48)). The identity (2.74) is a restatement of (2.67). For(2.75), direct computation givesS(p)N(g; ) = (1 + z0)pDpJDffaECeV0()+(J;ffi)+(J;ffi): (2.81)We dene (J;J) = ECeV0()+(J;ffi)+(J;ffi), and rewrite (2.81) asS(p)N(g; ) = (1 + z0)pDpJDffa(J;J): (2.82)Now (2.58) gives(J;J) = e(J;CJ)Z0N(CJ;CJ); (2.83)and (2.75) again follows by dierentiation and the chain rule.43Chapter 3Perturbative computationsIn this chapter, we calculate the nite volume partition function to leadingorder using a perturbative approach. In Section 3.1, we recall the covariancedecomposition and how a Gaussian expectation (or super-expectation) canbe evaluated progressively in an iterative fashion. This leads to the programof perturbative approximation using cumulant expansion in Section 3.2.To use this strategy, we dene a projection operator LocXin Section 3.3and identify its range in Lemma 3.9. We then implement the perturbativeprogram in Section 3.4, by computing in Proposition 3.12 the observablepart of the leading contribution to every step of the progressive integration.3.1 Progressive Gaussian integrationWe call C = (N+ m2)1the covariance. According to Lemma 2.8,our goal is the computation of the various derivatives of the nite volumepartition function given by the convolutionZN= ECeV0(N): (3.1)For n  1, V0is given by (2.64) and the expectation is the standard Gaussianexpectation (2.46). For n = 0, V0is given by (2.66) and the expectationis the Gaussian super-expectation (2.48). We compute these expectationsprogressively, using covariance decomposition.443.1. Progressive Gaussian integrationNotation Let fe1; : : : ; e4g be the standard basis for Z4 consisting of unitvectors ekwith 1 in the k-th position and 0's everywhere else. We dene thenite-dierence operators rkby rkf(x) = f(x+ek)f(x). If the functionf depends on more than one spatial variable, we specify the variable in theoperator notation, for example rx;kf(x; y) = f(x+ ek; y) f(x; y). Finally,for a multi-index  = (1; : : : ; 4), we write rx= r1x;1   r4x; Covariance decompositionWe use decompositions of the two covariances (Zd+m2)1and (N+m2)1. For Zd, the covariance exists for d > 2 for all m2  0, but for Nwemust restrict to m2> 0 since the nite-volume Laplacian is not invertible.In [11, Section 6.1], results from [8, 17] are applied to dene a sequence(Cj)1j<1(depending on m2 0) of positive denite covariances on Zdsuch that(Zd +m2)1=1Xj=1Cj(m2 0): (3.2)For j  0, we dene the partial sumswj=jXi=1Ci; w0= 0: (3.3)The covariances Cjare translation invariant, and have the nite-rangepropertyCj;xy= 0 if jx yj 12Lj: (3.4)For j < N , the covariances Cjcan therefore be identied with covarianceson  = N, and we use both interpretations. For m2> 0, there is also acovariance CN;Non  such that(N+m2)1=N1Xj=1Cj+ CN;N: (3.5)453.1. Progressive Gaussian integrationIt is shown in [11, Proposition 6.1] that for multi-indices ;  with `1normsjj1; jj1at most some xed value p, for j < N , and for any k 2 N,jrxryCj;x;yj  c(1 +m2L2(j1))kL(j1)(2+(jj1+jj1)); (3.6)where c = c(k) depends on k but is independent of j. The same boundholds for CN;Nif m2L2(N1)  for some  > 0, with c depending on  butnot on N .According to [21, Proposition 2.6], both the standard Gaussian expecta-tion and the Gaussian super-expectation can be expressed as a progressiveintegrationECF =ECN;N  ECN1      EC1F: (3.7)To compute the expectation ECeV0()of (3.1), we dene Z0= eV0()andZj+1= ECj+1Zj(0 < j < N); (3.8)with an abuse of notation in that we interpret CNas CN;N. By (3.7),ZN= ECZ0coincides with ZN= ECNZN1from the above, so the abovedenition is consistent with the formula for the partition function. Thus,we are lead to to study the recursion Zj7! Zj+1in order to compute ZN.3.1.2 Scales and corresponding field sizesFor simplicity, let us restrict the discussion in this section to the case ofn  1. See [21] for a more general exposition that extends to the case ofthe WSAW. We write Ej= ECj, and leave implicit the dependence of thecovariance Cjon the mass m.By denition of the operator  in Section 2.3.2, each EjZj1operationintegrates out a uctuation eld , a Gaussian random eld with covarianceCj. By the nite range property (3.4), (Cj)xy= 0 if jx  yj 12Lj. Sinceuncorrelated Gaussian random variables are independent, xis independentof yif x is suciently far from y depending on j, and we call j the scale.463.1. Progressive Gaussian integrationA very important scale is the coalescence scale jab, dened byjab=logL(2ja bj): (3.9)In other words, jabis the unique integer such that12Ljab ja bj <12Ljab+1: (3.10)By (3.4), the smallest j for which Cj;ab6= 0 is possible is j = jab+ 1, so jabis the last scale where band bare independent.The uctuation eld  has a magnitude (in an L2sense), given by(Ej(ix)2)12= (Cj)1=2xx, which by (3.6) is of the order of Lj. We write(Cj)1=2xx Ljto indicate that. Similarly, rixtypically has the magnitudej(rrCj)xxj1=2 Lj(1+jj). More generally, a functional of the eldM 2 N is called a monomial if for some integer m, xk2  and multi-indices k,M(') =mYk=1rk'ikxk; (3.11)where ikdenote components of the eld '. We say thatM is local if xk= xfor all k, and we write M = Mxin that case. We dene the dimension ofthe monomial M by [M ] =Pk(1+ jkj), so that by the above computation,the magnitude of M() is Lj[Mx].We make this computation precise for any eld functional F 2 N byintroducing the T0semi-norm as follows. We start with a convenient no-tation for partial derivatives with respect to 'ix. Let denote the set ofnite sequences of elements of   f1; : : : ; ng, that is z 2 means thatz =(x1; i1); : : : ; (xp; ip)is a nite sequence of tuples (xk; ik), where xk2 and 1  ik n. Given z 2 , we let jzj = p and deneFz(') =@p@'ipxp   @'i1x1F ('): (3.12)Consider test functions g : ! R. We dene a bilinear pairing of elements473.1. Progressive Gaussian integrationof N and the set of test functions.hF; gi0=Xz2Fz(0)jzj!g(z): (3.13)This allows us to think of the eld functionals F 2 N as linear maps g 7!hF; gi0and transform a norm on the space of test functions into a semi-normon N . The pairing of M and a test function g is easy to compute explicitly.For example, if n = 1 [21, Example 3.6] giveshmYk=1rk'xk; gi0= r1x1   rmxmXff2Sm1m!gxff(1); : : : ; xff(m): (3.14)To see that this is so, note that the bilinearity of the pairing allows thegradients to be moved out in front; then the only non zero derivatives ofproducts of m elds are the ones in (3.14). The symmetrisation also comesnaturally, since the order of the derivatives does not matter.We set `j= `0Ljfor an appropriate constant `0and x an integerp 0. The j(`j)-norm of a test function g is then dened bykgkj(`j)= supz2jzjpNsupPjkjp`jzjjLjPjkjjrg(z)j; (3.15)where rg(z) = r1x1   rpxpg(x1; i1); : : : ; (xp; ip). This induces a scaledependent semi-norm on N bykFkT0= supkgkj=1jhF; gi0j: (3.16)Note that kgkj= 1 implies that for all appropriate z 2 and all col-lections of multi-indices , jrg(z)j  `jzjjLjjj= `jzj0Lj(jzj+jj). Thisinequality is transformed into an equality by the supremum in (3.16) whenwe substitute (3.11) into (3.14) and obtainkMkT0= `m0Lj(m+Pjkj)= `m0Lj[M ]: (3.17)483.2. Renormalisation group strategyNotice that this is the same as the magnitude of jM()j that we estimatedearlier. However, when computing the expectation (EjM)('), we integrateM('+ ), meaning that the T0semi-norm with parameter `jis only usefulwhen the eld ' is small: for a local monomial kMxkT0 j(EjMx)(0)j. Weuse a dierent parameter~hjin the case of the eld ' being large, and manybounds require the control of both of these norms at the same time. We usethe notation hjto mean either `jor~hjin that case.3.2 Renormalisation group strategyIn this section, we develop a strategy for the study of the recursion Zj7!Zj+1via a perturbative expansion that we illustrate by approximately com-puting Z1from Z0. The rest of this chapter will be devoted to making thisstrategy into a precise approximation that can be iterated all the way fromZ0to ZN.3.2.1 Classification of local field monomialsBy the computation above (3.11), the typical magnitude of a uctuationeld decreases as j increases, but its range grows by the nite range prop-erty (3.4). Therefore, we dene the eective size of a local monomial, bysumming over blocks of appropriate size.Given j 2 f0; 1; : : : ; Ng, we partition  = Zd=LNZd into a disjoint unionof Ld(Nj)scale-j blocks of side length Lj, and denote the set of all suchblocks by Bj. One block contains the origin at its corner and is of the formfx 2  : jxj1< Ljg, and all other blocks are translates of this one byvectors in LjZd. By construction, the eld  in a block B is independentof the elds in all the other blocks that do not touch B. A scale-j polymeris a union of scale-j blocks, and we write Pj= Pj() for the set of scale-jpolymers.A local monomial Mxis said to be relevant if [Mx] < 4, marginalif [Mx] = 4, and irrelevant if [Mx] > 4. This stems from the following493.2. Renormalisation group strategycomputation: since a block B 2 Bjcontains L4jpoints,Xx2B(EjMx)(0)Xx2BkMxkT0 jBjL[Mx]j= L(4[Mx])j: (3.18)Therefore, relevant and marginal monomials grow or stay constant, underthe action of Ej, while irrelevant monomials shrink in size as j is incre-mented.Recall that Z0= eV0()with V0from (2.64) and (2.66). Note thatthe part without observables, ∅V0, contains only relevant and marginalmonomials, but this is not the case for the observable part of V0. Indeed,the observable elds only live at two points a and b, so they do not get thevolume factor jBj in (3.18).For observable subspaces in the decomposition (2.62), we introduce ob-servable norm parameters hff;j(that stand for either `ff;jor hff;j). For F 2 N ,F = F∅ + ffaFa + ffbFb + ffaffbFab, we denekFkT0(hj)= kF∅kT0(hj)+ hff;jkFakT0(hj)+ kFbkT0(hj)+ h2ff;jkFabkT0(hj):(3.19)The exact value of hff;jis discussed in Section 4.3.1. There, we pick hff;jsothat the monomials ffa('pa h), ffb('pb h), ffaffipa, and ffbffipbbecome marginalfor all j  jaband irrelevant for j > jab, where jabis the coalescence scalefrom (3.9).These denitions make sure that V0contains only marginal and relevantmonomials. In our analysis of the map Zj7! Zj+1, we will try and pre-serve this property by nding for each Zja local polynomial Vjhaving noirrelevant monomials such that Zj eVj().3.2.2 Cumulant expansionTo illustrate the ideas involved in the study of the recursion Zj7! Zj+1, weconsider the computation of Z1= E1eV0(), at the level of formal powerseries accurate to second order in the coupling constants of V0. This can be503.2. Renormalisation group strategydone by expansion of eV0()to second order, and the result can be writtenZ1= EC1eV0 expEC1V0+12EC1 (V0;V0); (3.20)where E(V0;V0) = EV 20 (EV0)2, and  denotes approximation accurateto second order in the sense of formal power series. This is an instance ofthe cumulant expansion. ThenZ1 eH1with H1= EC1V012EC1 (V0;V0) : (3.21)H1is a polynomial functional of the eld and can be computed explicitly.We dene the operatorsLC=8>>>><>>>>:12Xu;v2Cu;vnXi=1@@'iu@@'iv(n  1);Xu;v2Cu;v@@ffiu@@ffiv+@@ u@@ v(n = 0):(3.22)Then, for a polynomial P ,ECP = eLCP; (3.23)where the exponential on the right-hand side is dened by its power seriesexpansion (a nite series when applied to a polynomial); see [21, Lemma 4.2]for a proof. However, H1lacks many useful properties that V0had: forexample, H1contains irrelevant monomials and also relevant, but non-localmonomials.To be able to iterate the above construction using cumulant expansioneectively, we would like to be able to extract from H1another polynomialV1having the following properties(i) V1should contain only relevant and marginal monomials.(ii) For every X  , V1(X) =Px2XV1;xand each V1;xshould consist oflocal monomials only.(iii) V1should respect the symmetries present in V0. For example, since513.2. Renormalisation group strategy∅V0 only depends on j'j, it is invariant under any rotation of theeld '. We want ∅V1 to be rotation invariant as well.If a polynomial functional V 2 N satises the rst two properties above,we say that V is a local polynomial. We will now dene a space of localpolynomials that satisfy all of the required properties and that is sucientlylarge to parametrise Zjfor all j  N .3.2.3 Space of local polynomialsRecall that for n  1, V0from (2.64) depends on h 2 Rn. Given suchan h 6= 0, we dene a class of local polynomials Vhthat we will use toparametrise the result of progressive expectations. It is necessary to keeptrack of the dependence on the vector h for n  2, whereas for n = 0 andn = 1 we simply set h = 1. We deneax(h) =8<:ffipx(n = 0)'px h=jhj (n  1);bx(h) =8<:ffipx(n = 0)'px h=jhj (n  1):(3.24)Note that xdepends on the vector h only through its direction. We alsoneed the monomialfirr;x=8<:12Pe2Zd:jej1=1(reffi)x(reffi)x+ (re )x(re )x(n = 0)14Pe2Zd:jej1=1re'x re'x(n  1):(3.25)Then we dene the polynomials (for  = a; b)V∅;x = gfi2x+ fix+ zfi;x+ yfirr;x+ u; V;x=8<:x(h) (n = 0)x(h) + t(n  1);(3.26)and dene Vhto be the set of all functions x 7! Vx, where for all x 2 ,Vx= V∅;x  ffaVa;x1x=a  ffbVb;x1x=b  ffaffb12(qa1x=a+ qb1x=b): (3.27)523.3. Approximation by local polynomialsGiven X  , we also deneVh(X) = fV (X) =Px2XVx: V 2 Vhg: (3.28)The scalar coecients in the above polynomials are all real numbers forthe j'j4model. For the WSAW, all are real except a; b; qa; qbwhich arepermitted to be complex (this is discussed further in Section 4.3.1 below).Two useful subspaces of Vhare the subspace V(0)hconsisting of elementsof Vhwith u = y = ta= tb= qa= qb= 0, and the subspace V(1)hconsistingof elements with y = 0. The polynomial V0from (2.64) and (2.66) lies inthe subset of V(0)hwith a= b= 1.3.3 Approximation by local polynomialsIn this section, we discuss a projection operator Loc that will project thealgebra N onto the space of local polynomials. This operator will allowus to iterate the procedure outlined in Section 3.2.2 while also exploitingsymmetries of the original j'j4and WSAW models.3.3.1 Localisation operator LocDue to the presence of the nite-dierence operators, the condition of local-ity for monomials is not as strict as it may appear. Consider for n = 1 themonomial '0'e1, is it local? Yes, since'0'e1= '20+ '0'e1 '20= '20+ '0re1'0: (3.29)Similarly, '0'2e1= '20+2'0re1'0+'0(re1)2'0, and so on. Note that eachnext term in the expansion of '0'2e1is increasing in dimension: ['20] = 2,['0re1'0] = 3 and ['0(re1)2'0] = 4. This provides a natural way to ap-proximate any polynomial function of the eld ' by a linear combination ofrelevant and marginal local monomials. We illustrate this with an example:'0'4e1= '20+ 3'0re1'0| {z }relevant+3'0(re1)2'0| {z }marginal+'0(re1)3'0| {z }irrelevant(3.30)533.3. Approximation by local polynomialsThe localisation of '0'4e1at 0 2 Z4 consists of the rst three monomials onthe right hand side of the above equation, while the last one is discarded.The bilinearity of the pairing (3.13) allows us to extend these expansionsto any n and any F 2 N , but before we do that, let us introduce somenotation.Recall that is the set of all nite sequences z =(x1; i1); : : : ; (xp; ip),where xk2 , 1  ik n, and test functions are functions g : ! R.For X  , let N (X) consist of all F 2 N such that Fz(') = 0 wheneverany component of z 2 lies outside of X. We consider a set G of testfunctions and we say that F1; F22 N are G-equivalent if for all g 2 G,hF1; gi0= hF2; gi0, and G-separated otherwise.Given x = (x1; : : : ; x4) 2   Z4 and  = (1; : : : ; 4) 2 N4, we denex= (x1)1   (x4)4. To every monomial M of the form (3.11), we asso-ciate a test function gMby replacing each rk'ikxkwith xkk. More precisely,gM(z) =8<:Qmk=1xkkif z =(x1; i1); : : : ; (xm; im);0 otherwise:(3.31)Let Gd+be the vector space of test functions spanned by all gMassociatedto monomials of dimension at most d+, i.e. [M ] =Pk(1 + jkj)  d+. Thisdenition ensures that Gd+separates all monomials of the form (3.11) with[M ]  d+. Let Sd+(X)  N (X) be the set of local polynomials separatedby Gd+. In other words, if P 2 Sd+(X), then P =Px2XPx, where Pxcontains only monomials separated by Gd+.Proposition 3.1. For nonempty X   that does not wrap around thetorus and given d+, there exists a unique linear map locd+X: N∅() !Sd+(X) such thathlocd+XF; gi0= hF; gi0for F 2 N∅(), g 2 Gd+: (3.32)That is, locd+XF is the element of Sd+(X) that is Gd+-equivalent to F 2N∅.Proof. This is just a restatement of Proposition 1.5 of [22].543.3. Approximation by local polynomialsGiven X  , we will now construct the localisation operator LocXwhich projects N onto the vector space of local polynomials. Let F 2 N ,by the decomposition (2.63), F = F∅ + ffaFa + ffbFb + ffaffbFab. We deneLocXF using dierent specications for the operator locXfor each subspace.From Section 3.2.1, we know that marginal monomials have dimension 4in the bulk subspace N∅. By the remark following (3.19), in the observablesubspaces Naand Nb, we want ffa('pa h), ffb('pb h), ffaffipa, and ffbffipbtobe marginal for all j  jaband irrelevant for j > jab. Hence, we choosed+= p if j  jaband d+= 0 otherwise for those subspaces. By (3.19), theT0(hj) norm gets a factor of h2ff;jin the Nabsubspace. This suggests thatwe should make d+there to be twice its value in Naand Nb. However, by(3.4), Cj;ab= 0 for all j  jab, so the cumulant expansion cannot createterms in the Nabsubspace until after the coalescence scale. Therefore, weuse d+= 0 for all j in the Nabsubspace.With all these consideration in mind, we set dj+= p1jjaband make thefollowing scale dependent denition using the decomposition (2.63)Loc(j)XF = loc4XF∅ + ffa locdj+X\fagFa+ ffalocdj+X\fbgFb+ ffaffbloc0X\fa;bgFab:(3.33)We denote the range of LocXby V(X) = LocX(N ). Note that in general,this space contains more monomials than Vh(X). We will now investigatethe symmetries of the polynomial V0that Loc preserves and impose corre-sponding restrictions on the space N to produce a smaller space Nh. Wewill then show that LocX(Nh)  Vhfor certain choices of h.3.3.2 Symmetries and symmetry reductionThe operator locXfrom Proposition 3.1 and consequently LocXfrom (3.33)have many useful properties. In particular, they commute with certaintransformations of the eld functionals in N .Lattice symmetry. Let A denote the set of graph automorphisms of ,i.e., bijections that preserve nearest neighbours. An automorphismA 2 A acts on N via AF (') = F (A'), where (A')x= 'Ax. We say553.3. Approximation by local polynomialsthat a local monomial Mx2 N is Euclidean invariant if AMx=Mxfor all A 2 A that x x. Recall the denition of the set Pjof scale-j polymers in Setion 3.2.1. We say that a function F : Pj! N isEuclidean covariant if A(F (X)) = F (AX) for all automorphisms Aof  and all X 2 Pj.Field symmetry For n  1, an n  n real matrix m acts on F 2 N via(mF )(') = F (m'). There is no action of m on ffaor ffb. Givena group G of n  n matrices, we say that F 2 N is G-invariant ifmF = F for all m 2 G.For n = 0, let G = U(1) be the group fz 2 C : jzj = 1g withcomplex multiplication. We set ffa= ffpand ffb= ffp, with ff 2C. Then m 2 U(1) acts on F 2 N by (mF )(ff; ff; ffi; ffi;  ;  ) =F (mff; mff;mffi; mffi;m ; m ). We say that F is U(1)-invariant, orgauge invariant, if mF = F for all m 2 U(1).Another eld symmetry special to the n = 0 case is the supersymme-try. The supersymmetry operator Q is dened, e.g., in [11, Section 5]or [20, Section 6], and we say that F is supersymmetric if QF = 0.Roughly speaking, Q interchanges boson and fermion elds; it doesnot play a role for observables and the rest of this thesis can be readwithout delving into its precise meaning.By the next lemma, the operator LocXpreserves Euclidean invarianceand covariance, as well as any of the eld symmetries dened above. A proofis given in [13, Proposition 2.1] and [22, Sections 1.4, 1.6].Lemma 3.2. For every Euclidean automorphism A 2 A, for every ma-trix m 2M(n) if n  1, for any m 2 U(1) if n = 0, and for all F 2 N∅,AlocXF= locAX(AF ); mlocXF= locX(mF ): (3.34)Also for the case n = 0, locXcommutes with the supersymmetry oper-ator Q.563.3. Approximation by local polynomialsOur next step is to determine which symmetries are respected by theelements of Vh. By denition, for n = 0, elements of Vhof (3.27) are allU(1)-invariant, but for n  1, we use the following matrix groups: O(n), the group of n n orthogonal matrices. S(n), the permutation subgroup of O(n), consisting of the n! matricesobtained by permutations of the columns of the identity matrix. R(n), the reection subgroup of O(n) consisting of the 2ndiagonalmatrices with diagonal elements in f1;+1g.The bulk part V∅his O(n)-invariant for all n  1, but for n  2, the O(n)symmetry of the observable part of Vhcan be reduced by the choice of h.This can be seen already from the 'pa h term in V0;x, which is not R(n)invariant when p = 1, and which is not S(n)-invariant for p = 2 unless his a multiple of (1; : : : ; 1). We now dene a weaker property that replacesO(n)-invariance for the observable terms when n  2.Definition 3.3. Let n  1 and x h 2 Rn. We say that F 2 N is h-factorisable if for  = a; b:(i) there exists F2 (∅N )n(depending on h, not unique) such thatF = ff(F h), and(ii) (PF)(') = F(P') for all P 2 S(n), where by denition PFis theresult of permuting the components of Fwith the permutation P .We write Nh-fac= fF 2 N : F is h-factorisableg for the vector space ofh-factorisable elements of N . Of course, Vh Nh-facby denition.We are now ready to dene the space of symmetric eld functionals Nh.The vector h does not play a direct role when n = 0; 1, but we neverthelessuse it as a notational device. For n = 0, we say that F has no constant partif its degree-zero part (as a form) is equal to zero when evaluated at ffi =ffi = 0. Using a test function g that is only non-zero on the empty sequencez = ∅ in (3.32), we see that this property is preserved by localisation.Definition 3.4. For n  0, let Nhdenote the subspace of all F 2 N suchthat573.3. Approximation by local polynomials(i) If n = 0, ∅F is supersymmetric, F is U(1)-invariant, and F has noconstant part.(ii) If n  1, F 2 Nh-fac, ∅F is O(n)-invariant, and if in addition p = 2,then F is R(n)-invariant.Note that Vh Nh.3.3.3 Range of LocWe will now explore some elementary properties of h-factorisability andthen proceed to show that LocX(Nh)  Vhfor suitable choices of h.Lemma 3.5. Let n  1 and h 2 Rn. If F;K 2 Nh-fac, and if ∅F and∅K are S(n)-invariant, then FK 2 Nh-fac with (FK)= (∅F )K+F(∅K) for  = a; b.Proof. We write F∅ = ∅F and K∅ = ∅K. Since we work in a quotientspace with ff2= 0,FK = F∅(K) + (F )K∅ = ff ([F∅K+ FK∅]  h) ; (3.35)so the rst requirement of Denition 3.3 holds with (FK)as indicated.Secondly, by the hypotheses on F∅ and K, for P 2 S(n),(P (F∅K))(') = (F∅(PK))(') = F∅(')(PK)(')= F∅(P')K(P') = (F∅K)(P'):(3.36)The FK∅ term is similar, and the proof is complete.Lemma 3.6. Let n  1, X  , and F 2 Nh-fac. Then LocXF 2 Nh-facwith (LocXF )= LocXF. Also, EF 2 Nh-facwith (EF )= EF .Here LocXFand EF are dened component-wise.Proof. The statement has content only for n  2, so we write the proof forthis case. Since LocXcommutes with and is linear,LocXF = LocXF = LocXff(F h) = ff(LocXF h): (3.37)583.3. Approximation by local polynomialsThe invariance under permutations follows easily.Again by linearity, EF = EF = Eff(F h) = ff(EF  h).For the invariance under permutations P 2 S(n) of the elds, we use(P (EF ))(') = E(PF )(') = E(PF )('+ ) = EF (P ('+ ))= EF (P'+ ) = (EF )(P');(3.38)where  is the integration variable, and where the fourth equality follows bymaking the change of variables  7! P (with Jacobian equal to 1) in theintegral.Definition 3.7. For n  2, we write M2(n) for the set of n  n matricesof the form rI + sJ , with r; s 2 R, I the identity matrix, and J having allentries equal to 1.Let e+be the vector e+= (1; 1; : : : ; 1) 2 Rn. Every matrix in M2(n)has eigenspaces E, where E+= span(e+) with eigenvalue r + ns, and Eis the orthogonal complement E= (E+)?with eigenvalue r.Observe that the linear span of the permutation subgroup S(n) consistsof the setS(n) of nn matrices whose row and column sums are all equal.Given a set Z of matrices, we write Z0= fB : AB = BA for all A 2 Zg forits commutant. The following lemma states thatS(n) and M2(n) are eachother's commutant.Lemma 3.8. For n  1, M 02(n) =S(n) andS0(n) =M2(n).Proof. LetM 2M2(n) and suppose AM =MA. By denition,M = aI+bJfor some a; b 2 R, where J is the nn matrix whose entries are all 1. ThenAM = MA if and only if AJ = JA, which means that (AJ)ij= (JA)ijforall i; j, i.e.,PkAik=PkAkjfor all i; j. This shows that A 2S(n) andtherefore M02(n) =S(n).Let M 2 S0(n), and let Pijbe the permutation matrix that interchangesrows i; j. Then PijM = MPijand hence PijMPij= M . Let ekbe the593.3. Approximation by local polynomialsstandard basis column vectors. ThenMkl= hek;Meli = hek; PijMPijeli = hPijek;MPijeli: (3.39)With k = l = j 6= i, this gives Mkk= Mii. With i 6= k = j and l 6= i; j, itgives Mkl= Mil. With i 6= l = j and k 6= i; j, it gives Mkl= Mki. Thisshows that M  M2(n) and hence S0(n)  M2(n). Since, by inspectionevery matrix in M2(n) commutes with every matrix in S(n), we also haveM2(n)  S0(n), and the proof is complete.We are now ready to prove the main result of this subsection.Lemma 3.9. Let X   and  2 fa; b; abg. For n = 0, p  1, and forn  1 and p = 1; 2,LocX(Nh) [m2M2(n)Vmh(X): (3.40)In particular, if h 2 E, the right-hand side of (3.40) becomes simplyVh(X).Proof. We use properties of Loc from [22]. By (3.27), an element of abVhcan be written as ffaffb12(qa1x=a+qb1x=b) (independent of h). Thus (3.40)follows from our choice of d+= 0. Similarly, for  = a; b and j  jab,elements of LocX(Nh) are constant multiples of ff. Thus we assumehenceforth that j < jaband consider  = a; b, where d+= p.For n = 0 and p  1, h plays no role, and in this case ffa= ffpandffb= ffp. The only U(1)-invariant monomials containing ffa= ffpor ffb= ffp,and with dimension at most p, are fffpffip; ffpffipg. Since LocXpreserves U(1)invariance, this implies that, as required,LocX(aNh) = 1a2Xffpspanffipa	; LocX(bNh) = 1b2Xffpspanffipb	:(3.41)The appearance of the indicator functions on the right-hand sides of (3.41)follows directly from the denition of LocXin (3.33). For n  1 and p = 1; 2,603.3. Approximation by local polynomialssince d+= p,LocX(N )  12Xffspann1; 'ij 1  i  no(p = 1);(3.42)LocX(N )  12Xffspann1; 'i; 'i'k;r'ij 1  i; k  no(p = 2);(3.43)where the superscripts on 'indicate components.For the case p = 2, it follows from Lemma 3.2 that the R(n)-invarianceof Nhis preserved by LocX. The linear, mixed quadratic, and gradientmonomials from (3.43) are not invariant under replacement of one compo-nent of 'by its negative, and thus are not in LocX(Nh) when p = 2.Therefore, for both p = 1 and p = 2,LocX(Nh)  12Xffspann1; ('1)p; : : : ; ('n)po: (3.44)By Denition 3.4, if F 2 Nhthen F = ff(Fh), with F2 (∅N )nsuchthat (PF)(') = F(P') for all P 2 S(n). By (3.44), each component ofLocXFlies in span1; ('1)p; : : : ; ('n)p	. Therefore, there exist an n  nmatrix mand a vector v2 Rn such that LocXF= m'p+ v. WithLemma 3.6, this implies thatP (m'p+ v) = mP'p+ v(3.45)for every P 2 S(n), from which we conclude that Pv= vand Pm=mP for every P 2 S(n). The rst of these conclusions implies that v=se+for some s2 R (with the vector e+ from Denition 3.7), and byLemma 3.8 the second implies that m2 M2(n). Since mT= m for m 2M2(n),LocXF = 12Xffm'p h+ se+ h= 12Xff'p (mh) + se+ h:(3.46)The right-hand side lies in Vmh(X) (with t= se+ h), and this com-pletes the proof.613.4. Perturbative calculations3.4 Perturbative calculationsIn this section, armed with the projection operator LocX, we implementthe strategy outlined in Section 3.2.2. More precisely, for every step Zjof the progressive integration, we dene an approximation Ijaccurate tosecond order in perturbation theory, and then, we compute the relevant andmarginal contributions to Ij+1explicitly.3.4.1 Iterating the cumulant expansionIn Section 3.2.2, we have already discussed a procedure to approximate Z1using cumulant expansion. We recall that by (3.21), Z1 eH1, where H1is a non-local polynomial functional given by H1= EC1V012EC1 (V0;V0).The rst term of H1is easy to compute using (3.23) and the operator LCfrom (3.22). To simplify the computation of the second term, we dene forpolynomials A;B in the elds,FC(A;B) = eLCeLCAeLCB AB; (3.47)F;C(A;B) = FC(A; ∅B) + FC(A;B); (3.48)where we used the canonical projections as dened under (2.63), andthe abbreviation = 1 ∅ = a + b + ab. The F;C is an asymmetricversion of (3.47) that implements a very clever reorganisation of the observ-ables. Note that F;C(A;A) = FC(A;A), so the truncated expectation inthe second term in H1becomesEC(V0;V0) = EC(V20) (ECV0)2= F;C(ECV0;ECV0): (3.49)Now, we extract from H1a local polynomial using the projection Loc. Inthe following formula, we write F;C1instead of F;C1(EC1V0;EC1V0) for623.4. Perturbative calculationslack of space.H1= EC1V012F;C1EC1V0;EC1V0(3.50)= EC1V012LocF;C1| {z }local part, Vpt(V0)+121 LocF;C1| {z }non-local irrelevant part, W1(V0): (3.51)Since the non-local partW1(V0) is a quadratic function of V0, to second orderwe have eW1 1 +W1. This approximation is only good when the eld issmall, but the replacement is important since W1may contain monomialsthat are not integrable in the exponential. We now haveZ1 eH1 eVpt(V0)1 +W1(V0); (3.52)and this is the form that we will try to preserve for all j.3.4.2 Definition of IjIn this section, we give precise denitions of quantities discussed above andappearing in (3.52). Recall that the covariance wjis dened by (3.3). Fora local polynomial V in the elds, and for a polymer X 2 Pj, we setWj(V;X) =12Xx2X(1 Loc(j)x)F;wj(Vx; V ()): (3.53)The denition (3.53) is inapplicable for the nal scale j = N , since LocXrequires X not to wrap around the torus N; this special case is discussedin [23, Section 1.1.5]. Then, forX 2 Pj, we dene the interaction functionalIj(V;X) = eV (X)YB2Bj(X)1 +Wj(V;B): (3.54)633.4. Perturbative calculationsFor j = 0, where w0= 0, we interpret the above as I0(V;X) = eV (X). LetLj+1= LCj+1. Given V , we denePj;x=12Xy2Loc(j+1)xF;wj+1(eLj+1Vx; eLj+1Vy) eLj+1Loc(j+1)xF;wj(Vx; Vy); (3.55)and setVpt;j+1;x(V ) = eLj+1Vx Pj;x: (3.56)The denition (3.56) is equivalent to the denition in (3.51), except thatPj;xnow has two contributions: one from Loc(j+1)xF;Cj+1(: : :) and anotherfrom Loc(j+1)xWj(Vj).This denition transforms the idea behind (3.21) into a procedure wecan iterate. In fact, using the cumulant expansion argument, we can showthatZj+1= Ej+1Zj Ej+1Ij(V;)  Ij+1(Vpt(V );); (3.57)with \" as in Section 3.2.2. Equation (3.57) shows that, to second order,I enjoys a form of stability under expectation. Moreover, passing from Zjto Zj+1is like advancing from V to Vpt(V ), and this is something we cancompute explicitly. Of course, at this point there is no uniformity in scale jor volume  in the error estimate connected with the approximation , butwe need to go beyond perturbative arguments in Chapter 4 to get those.Before we proceed to the explicit computation of Vpt, let us prove that Ijand related quantities are Euclidean covariant and inherit h-factorisabilityfrom V . This will be useful for us later on.Lemma 3.10. Let V 2 Vh, X 2 Pj, and x 2 . Each of Wj(V;X),Ij(V;X), Pj;x(V ) and Vpt;x(V ) is in Nh-fac. Each of ∅Wj(V ), ∅Ij(V )(as functions of X 2 Pj), and ∅Pj(V ) and ∅Vpt(V ) (as functions ofx 2 ) is Euclidean covariant.Proof. Let A 2 Nh-facbe a polynomial in the elds, and let  = a; b.Then A = ff(A h), and we can assume that every component of643.4. Perturbative calculationsAis a polynomial. Recall the denition of LCin (3.22). Note thatLCA = ff(LCA h). Let P 2 S(n) be a permutation matrix. SinceLCacts component-wise, PLCA= LCPA, and hence, since A 2 Nh-fac,(PLCA)(') = (LCPA)(') = (LCA)(P'). This shows that LCA 2Nh-fac. Consequently,eLCA =deg(A)Xk=0(1)kk!LkCA 2 Nh-fac: (3.58)Let V 2 Vhand X 2 Pj. Then V 2 Nh-facby denition and everycomponent of Vis a polynomial. Using Lemmas 3.5{3.6 and the aboveobservations concerning LC, we see from (3.47){(3.48) that F;C(V; V ) ish-factorisable, as are Wj(V;X), Ij(V;X), Pj(V;X) and Vpt(V;X).The Euclidean covariance is a consequence of the denitions, the Eu-clidean invariance of wj, and the preservation of Euclidean covariance byLocX.3.4.3 Perturbative flow of coupling constantsWe nish this chapter with the explicit computation of the mapping V 7!Vpt. We assume that V 2 Vh, so as described in Section 3.2.3, it is a localpolynomial V (X) =Px2XVxand it is described by 11 coecients that wecall coupling constants. They are g; ; z; y; u for the bulk part ∅V anda; b; ta; tb; qa; qbfor the observable part V . To compute Vpt(V ), we onlyneed to compute how these 11 coupling constants change under the V 7! Vptmap. We distinguish the coecients from Vptby adding a subscript \pt".For  :  ! R, let (i) =Px2ix. Let C = Cj+1, w = wj, and, forg;  2 R, let+=  + g(n+ 2)C00; j[f(;w)] = f(+; w + C) f(;w); (3.59)j= (n+ 8)j[w(2)]: (3.60)The following proposition is proved in [11] for WSAW (n = 0) and in [13]653.4. Perturbative calculationsfor j'j4(n  1). The computation is mechanical, can be carried out on acomputer and yields a lot of terms. We have only listed the importantones below and hid the rest inside (: : :). Although this computation is animportant ingredient in the proof of Theorem 4.2, we will not use its resultselsewhere in this thesis.Proposition 3.11. Let d = 4 and V 2 Vh. Then ∅Vpt;j+1(V ) is givenby the new coupling constants (see the rst equation in (3.26))gpt= g  g2+ (: : :); (3.61)pt= ++ (: : :); (3.62)zpt= z + (: : :); (3.63)ypt= y + (: : :): (3.64)Moreover, uptfrom (3.26) can be calculated explicitly as well.On the other hand, the computation of Vptis of primary importanceto us, so we will provide more details. It is an extension of the proof givenin [11] for the specic case n = 0 and p = 1 to all p  1 for WSAW, and forp = 1; 2 for j'j4. We start with some preliminaries.Let I denote 1 2 R when n = 0 and the nn identity matrix for n  1.We dene a matrix T , which is in M2(n) (see Denition 3.7) for n  2, byT =8>>>><>>>>:p214I (n = 0)p2I13(n = 1)p22n+8I +p21n+8J (n  2):(3.65)The matrix T is the zero matrix for p = 1 (as12= 0), and otherwisehas eigenspace E+with eigenvalue +n;p=p2n+2n+8, and for n  2 also haseigenspace E= (E+)?with eigenvalue n;p=p22n+8. The correspondencebetween the matrix T for n  1 and the value we have assigned to n = 0should be understood via the eigenvalues, as0+20+8=20+8=14. For n = 0; 1there is only +and E+. For n  2 and p = 2, we have =2n+8<n+2n+8=+< 1, and this is the only setting where both eigenvalues play a role in663.4. Perturbative calculationsour analysis.We also dene the matrixAj=8<:(1 pj[w(1)])I  jgT (j + 1 < jab)I (j + 1  jab):(3.66)Thus Ajis nn for n  1 and 11 for n = 0. For n  2, Aj2M2(n). Theeigenvalues and eigenvectors of Ajplay an important role in identifying thelogarithmic corrections in Theorems 1.6 and 1.7. The eigenspaces are E,with eigenvaluesfj=8<:1 pj[w(1)] jgn;p(j + 1 < jab)1 (j + 1  jab):(3.67)The following proposition computes Vptas a function of V 2 Vh. We set&j= C0;0(1 1j+1<jab2w(1)) + 1j+1<jab+j[w(2)] + 1j+1jabj[w(2)]:(3.68)Proposition 3.12. Let d = 4. Let p  1 for WSAW, and p = 1; 2 forj'j4. Let V 2 Vhwith jhj = 1. Then Vpt;j+1(V ) 2 Vhptfor a new directionhptbelow, and for x = a; b, Vpt;j+1is given byhpt= (Ajh) =jAjhj; (3.69)pt;x= jAjhjx; (3.70)qpt;x= qx+ p!abj[wpab]; (3.71)tpt;x= tx+ 1n11p=2x(e+ h)&j: (3.72)In particular, if h 2 E, then hpt= h and pt;xhpt= fjxh.Note that by Proposition 3.9, we can only expect Vpt(V ) 2 Vhif wechoose h carefully, and it is clear from (3.69) that it is in general not thecase, when n  2. Instead, Vpt2 Vhptfor a new direction hpt. However, if his in one of the eigenspaces E, then hpt= h. To have hpt= h is a desirablesimplication, and this gives the eigenspaces Ea special signicance. The673.4. Perturbative calculationssymmetry restrictions of Nh, particularly h-factorisation, are used to carrythis perturbative fact over to the non-perturbative renormalisation groupcoordinate and show that Vj2 Vhfor all j. The two powers +n;pandn;pfor the logarithmic corrections in Theorems 1.6{1.7 will arise from thedistinction between h 2 E+and h 2 E.The proof of Proposition 3.12 involves similar but not identical calcula-tions for n = 0 and n  1. However, once Proposition 3.12 is proved, theremaining analysis for the proof of our main results is unied for all n  0.As noted below (3.9), j = jabis the smallest scale j for which Cj+1;ab6= 0is possible, and so i[wpab] can be nonzero for the rst time also when i = jab.Therefore the rst scale for which qpt q can be nonzero is qpt;jab+1.For the rest of this section, we write w = wj, C = Cj+1and L = LCj+1.The rst step in the proof of Proposition 3.12 is the computation of the rstterm in Vpt= eLV  P of (3.56), provided by the following lemma.Lemma 3.13. Let n = 0 and p  1, or let n  1 and p = 1; 2. ForV 2 Vh,eLVx= Vx+ g(n+ 2)C00fix+ 1n1upt 1p=2x(e+ h)C00ffx; (3.73)where uptis an explicit quadratic function of g; ; y + z.Proof. The computation of eL∅Vx is carried out in [10,13] and agrees withthe above formula. In particular, upt= 0 for n = 0, and uptis givenby [13, (3.27)] for n  1. For the observable part, for n = 0 we haveLV = 0 and hence eLVx= Vx, as in (3.73). For n  1 and p = 1; 2,we have L2V = 0, so eLCVx= Vx+ LVx. Direct calculation ofLVxgives the nal term of (3.73).To compute Pxwe use (3.55), i.e.,Px=12Xy2LocxF;w+C(eLVx; eLVy) eLLocxF;w(Vx; Vy);(3.74)683.4. Perturbative calculationsin conjunction with (3.48) which impliesF;w(Vx; Vy) = 2Fw(Vx; ∅Vy) + Fw(Vx; Vy): (3.75)For the following lemma, for each pair x; y 2 , we dene an n n matrix(1 1 if n = 0)Mxy= 1j+1<jabpwxyI + g(n+ 8)w2xyT: (3.76)Lemma 3.14. Let n = 0, p  1, or n  1, p = 1; 2. For V 2 Vh,LocxFw(Vx; ∅Vy) = ffxx(Mxy'px h) + 1n11p=2w2xy(e+ h);(3.77)LocxFw(Vx; Vy) = ffaffbp!abjhj2wpxy(1x=a1y=b+ 1x=b1y=a);(3.78)where for n = 0 we interpret 'xon the right-hand side of (3.77) asffiafor x = a and ffibfor x = b.Proof. We evaluate F using [11, Lemma 5.6], which implies that for n  1,FC(Ax; By)=DXk=11k!nXi1;:::;ik=1Xul;vl2(l=1;:::;k) kYl=1Cul;vl!@kAx@'i1u1   @'ikuk@kBy@'i1v1   @'ikvk; (3.79)with D = degA ^ degB. For n = 0, there is a related formula that alsoinvolves the fermions.For (3.77), we rst note that there is no contribution from the termsinvolving tor qin Vx, since the sum in (3.79) starts at k = 1 and hencealways involves dierentiation with respect to ', which is absent in theseterms. The cases aand bare symmetric, and we therefore only considera. It can be argued on the basis of dimensional considerations that thereis no contribution due to the terms yfirr+ zfiin ∅V . For the remainingcalculation, we use the notation appropriate for n  1 and comment on693.4. Perturbative calculationswhat is dierent for n = 0. To prove (3.77), we therefore computeLocaFw(aVa; ∅Vy) = anXi=1higLocxFw(ffa('ia)p; fi2y) + LocxFw(ffa('ia)p; fiy): (3.80)For n  1 and p = 1; 2,Fw('ix)p; j'yj2= Fw('ix)p; ('iy)2= 2pwxy('ix)p1('iy) + 2 p2!w2xy;(3.81)while for n = 0 and p  1,Fwffipx; jffiyj2= pwxyffip1xffiy: (3.82)Thus, for all (n; p) under consideration,LocaFwffa('ia)p; fiy= ffa1j+1<jabpwxy('ia)p+ 1n11p=2w2xy; (3.83)with the modication noted below (3.78) for n = 0. For n  1 and p = 1; 2,Fw('ix)p; j'yj4= Fw('ix)p; ('iy)4+ 2Fw('ix)p; ('iy)2Xj:j 6=i('jy)2;(3.84)Fw('ix)p; ('iy)4= 4pwxy('ix)p1('iy)3+ 12 p2!w2xy('iy)2; (3.85)while for n = 0 and p  1,Fwffipx; jffiyj4= 2pwxyffip1xffiyffi2y+ 2 p2!w2xyffip2xffi2y: (3.86)The terms of total degree above p are annihilated by Loc, andLoca24Fwffa('ix)p; ('iy)2Xj 6=i('jy)235= ffa1n12 p2!w2xyXj:j 6=i('ja)2: (3.87)703.4. Perturbative calculationsThus, for all (n; p) under consideration, we haveLocaFwffa('ix)p; fi2y= ffa1j+1<jab(n+ 8)w2xy(T'pa)i: (3.88)Assembly of the above completes the proof of (3.77). We omit the simplerproof of (3.78).Proof of Proposition 3.12. Equation (3.71) states thatqpt;x= qx+ p!ab[wpab]; (3.89)and this is an immediate consequence of (3.73), (3.74) and (3.78). To prove(3.69){(3.70), we usept;xhpt= xh 1j+1<jabp[w(1)]I  gTxh: (3.90)The rst term on the right-hand side arises from (3.73), and the rest of theright-hand side of (3.90) arises from Pa, via (3.74) and the rst term on theright-hand side of (3.77) (using also Lemma 3.13).Finally, we prove (3.72). By Lemma 3.13, the txterm in eLV is equalto tx 1n11p=2C00x(e+ h). The contribution due to P arises onlyfrom the rst term on the right-hand side of (3.75), and only for n  1 andp = 2, by Lemma 3.14. Thus we seek the contribution to tpt;xdue toXy2LocxFw+C(eLVx; eLVy) eLLocxFw(Vx; Vy): (3.91)We apply Lemma 3.13 and (3.77) to see that the rst term contributesa tx-term which is equal to 1n11p=2+w(2)j+1x(e+ h), and the secondcontributesXy2x(MxyeL'px h) 1n11p=2w(2): (3.92)713.4. Perturbative calculationsThe latter produces a tx-term 1n11p=21j+1<jab[pw(1)+ g(n+ 2)w(2)]C00+ w(2)x(e+ h);(3.93)where we have used Te+=n+2n+8e+for p = 2. This leads totpt;x= tx+ 1n11p=2x(e+ h)C0;0+ [w(2)]1j+1<jabC0;0[2w(1)+ g(n+ 2)w(2)];(3.94)which is equivalent to (3.72) by denition of +and & . This completes theproof.72Chapter 4Renormalisation group flowIn this chapter, we add a non-perturbative error coordinate to the pertur-bation theory calculations from Chapter 3. In section 4.1, we make thenecessary denitions and discuss the renormalisation group map that willallow us to control the errors in the cumulant expansion argument from Sec-tion 3.4.1. In Section 4.2, we restate some of the results of [10] for WSAWand [13] for j'j4that our construction will build upon, and in Section 4.3,we discuss the changes to norm parameters and stability estimates that areneeded to extend these results. In Section 4.4, we prove Theorem 4.9 whichprovides the non-perturbative counterpart to the perturbative statementof Proposition 3.12. Its proof requires many modications to argumentsin [23, 24] and is the most technical part of this thesis. In Section 4.5, weiterate the result of Theorem 4.9 and complete the program of (3.1), thatis computing the nite volume partition function.4.1 Non-perturbative renormalisation groupcoordinateProposition 3.12 gives the evolution of the observable coupling constants,as dened by the map V 7! Vpt. As discussed around (3.57), this mapdescribes the eect of taking the expectation at a single scale, but only at aperturbative level. In this section, we present aspects of the formalism of [10,734.1. Non-perturbative renormalisation group coordinate24], which introduces and employs a non-perturbative renormalisation groupcoordinate K. With this coordinate, Proposition 3.12 can be supplementedso as to obtain a rigorous non-perturbative analysis, including observables.4.1.1 Circle productRecall that the sets Bjand Pjof scale-j blocks and polymers in  aredened in Section 3.1. For maps F;G : Pj! N , we dene the circleproduct F G : Pj! N by(F G)(X) =XY 2Pj(X)F (X n Y )G(Y ) (X 2 Pj): (4.1)The empty set ∅ is a polymer, as is , so the sum over Y always includesY = ∅, and includes Y =  when X = . Every map F : Pj! Nthat we encounter obeys F (∅) = 1. The circle product is commutative andassociative, and has unit element 1∅ dened by 1∅(X) = 1 if X = ∅ andotherwise 1∅(X) = 0.We deneI0(X) = eV0(X); K0(X) = 1∅(X): (4.2)ThenZ0= eV0()= I0() = (I0K0)(): (4.3)We wish to maintain the form of (4.3) after each expectation in the progres-sive expectation (3.7). Namely, we seek to dene polynomials Uj2 V(0)h,constants uj, ta;j, tb;j, qa;j, qb;j, and a non-perturbative coordinate Kj:Pj! Nj, such that Zjof (3.8) is given byZj= ej(IjKj)(); j= ujjj+(ta;jffa+ tb;jffb)+12(qa;j+ qb;j)ffaffb;(4.4)with Ij= Ij(Uj) given by (3.54). We systematically use the symbol U forelements of V(0)hand V for other polynomials. Let j+1= j+1 j. Then(3.8) can equivalently be written asEj+1(IjKj)() = ej+1(Ij+1Kj+1)(): (4.5)744.1. Non-perturbative renormalisation group coordinateBy the denition in (2.69), ZN= ZN(') = (ECZ0)('). Since at thenal scale, PN= f∅;Ng, IN= eUN(1 +WN) by (3.54) andZN= eN(IN+KN) = eNeUN(1 +WN) +KN: (4.6)To prove Theorems 1.6 and 1.7, our goal is to achieve (4.6) with WNandKNas error terms, so that the pressure PN(') = logZN(') is to leadingorder equal to (N UN). Assuming this, we can evaluate the derivativesin Lemma 2.8 easily. For example, as UN(0) = 0, the derivatives in (2.72)and (2.74) are given by D2ffaffbN=12(qa;N+ qb;N). Similarly, since Nis aconstant, the derivative in (2.73) is Dp'(H)DffaUN= p!a;N(Hp0 h). Thus,for both models, the important information is ultimately encoded in theobservable coupling constants qx;Nand x;N.4.1.2 The definition of KA polymer X 2 Pjis connected if for any x; y 2 X there exists a path ofthe form x0= x; x1; : : : ; xn1; xn= y with kxi+1xik1= 1 for all i. Everypolymer can be partitioned into connected components, and we denote theset of connected components of X by Compj(X). Let Sj Pjdenote theset of connected polymers consisting of at most 2d= 16 blocks; elements ofSjare called small sets. (The specic number 16 plays a special role in [24],but not here.) The small set neighbourhood of X isX=[Y 2Sj:X\Y 6=∅Y: (4.7)For n  0, we dene N (X) to consist of those elements of N in (2.62)which depend on the boson, fermion (for n = 0), and external elds onlyat points in X, where we regard the external eld ffxas located at x forx = a; b. At scale j, K lies in the space Kjof maps from Pjto N , given inthe following denition.Definition 4.1. Let h = 1 for n = 0; 1 and h 2 Rn for n  1. LetKj= Kj(h;N) be the vector space of functions K : Pj! N with the754.1. Non-perturbative renormalisation group coordinateproperties: Field locality: K(X) 2 N (X) for each connected X 2 Pj. Also,aK(X) = 0 if a 62 X, bK(X) = 0 if b 62 X, and abK(X) = 0 ifeither (i) X 2 Sjand j < jabor (ii) a 62 X and b 62 X. Symmetry: K is Euclidean covariant and K(X) 2 Nhfor all X 2 Pj. Component factorisation: K(X) =QY 2Compj(X)K(Y ) for all X 2 Pj.4.1.3 The renormalisation group mapWe seek to dene a scale dependent renormalisation group map from adomain in V(0)hKjto V(1)hKj+1, which we write as(Uj;Kj) 7! (Vj+1;Kj+1); (4.8)that satises (4.5) with Ij= Ij(Uj) and Ij+1= Ij+1(Vj+1), and with theinitial condition (4.2). In [24, Section 1.8], this map is dened for the WSAWwith the observable having power p = 1. This construction applies in ourpresent more general setting with some modications. In this chapter, tosimplify the notation we sometimes drop labels j, and indicate scale j + 1simply by + as it is done in [24].The renormalisation group map (4.8) is composed of two maps: (U;K) 7!V+and (U;K) 7! K+. The former is explicit and relatively simple, and isdened as follows. Let LocY;Bdenote the operator dened by LocY;BF =PY(B), where PYis the polynomial determined by PY(Y ) = LocYF . Wedene a map V 7! V(1)from Vhto V(1)hby replacing zfi+ yfirrin V 2 Vhby (z + y)fiin V(1). We also dene a map V 7! V(0)from Vhto V(0)hbyreplacing zfi+ yfirrin V by (z + y)fiand replacing u; ta; tb; qa; qbin Vby zero. Then the map (U;K) 7! V+is given byV+(U;K) = V(1)pt(UQ) with Q(B) =XY 2S:YffBLocY;BK(Y )I(Y; V ); (4.9)where Vptis the explicit quadratic polynomial map V 7! Vptfrom Sec-tion 3.4.3. When K = 0, V+(U; 0) is simply V(1)pt(U). We split o the con-764.2. Bulk owstant part by writing V+= (+; U+), and in particular +(U; 0) = pt(U)and U+(U; 0) = V(0)pt(U). We express estimates on V+in terms of remaindersR+given byR+(U;K) = V+(U;K) V+(U; 0) = V+(U;K) V(1)pt(U) 2 V(1)h: (4.10)Therefore V+(U;K) = V(1)pt(U) +R+(U;K), so that R+contains correctionsto the perturbative computation in 3.4.3.A very important property of the renormalisation group map is thatthe evolution of the bulk coordinates (∅Vj ; ∅Kj) is independent of theobservables. Namely as in [24, (1.68)], the map (4.8) satises∅V+(U;K) = V+(∅U; ∅K); ∅K+(U;K) = K+(∅U; ∅K): (4.11)We denote this evolution by (V∅+;K∅+) and the bulk part of (4.8) becomes(∅V+; ∅K+) =V∅+(∅U; ∅K);K∅+(∅U; ∅K): (4.12)This property allows us to use the results from [10, 13] in the constructionof the map (4.8). We present these results in the next section.4.2 Bulk flowWe use the term renormalisation group ow or simply ow to refer to asequence (Vj;Kj) satisfying (4.5) for every j  N with the initial condition(4.2). In this section we present the results from [10, 13], where a criticalglobal renormalisation group ow of the bulk coordinates (∅Vj ; ∅Kj) isconstructed. By the property (4.11), this construction provides the bulkpart of our map (4.8), satisfying (4.5) for all j when ffa= ffb= 0.The bulk ow provides detailed information about the sequence ∅Vj ,and estimates on ∅Kj sucient for studying the innite volume limit atthe critical point.774.2. Bulk ow4.2.1 Existence of bulk flowFor the bulk ow, we change perspective on which variables are independent.Both j'j4and WSAW have parameters g; . In (2.42), additional parametersm2, g0, 0, z0are introduced. For the moment we consider m2; g0; 0; z0asfour independent variables and do not work with g;  directly. We relatem2; g0; 0; z0to the original parameters g;  in Section 4.2.3 below.To state the result about the bulk ow, let gjbe the (m2; g0)-dependentsequence determined by gj+1= gj jg2j, with g0= g0, and with j=j(m2) = (n+ 8)[w(2)j] as in (3.60). For m2> 0, we dene the mass scalejmto be the largest integer j such that mLj 1, and we set j0= 1. Bydenition, limm#0jm=1. Given  > 1 ( = 2 is a good choice), we denej= (jjm)+; (4.13)where x+= maxfx; 0g. By [11, Lemma 6.2], j= O(j) ( [11, Lemma 6.2]actually shows that j= O((jj)+) for another scale jused in [11,23, 24], but (jj)+and jare comparable by [11, Proposition 4.4].)By [10, Proposition 6.1] and [10, (8.22)] respectively, the boundsjgpj Og01 + g0jp(p  0);1Xk=jkgpk= O(jgp1j) (p > 1);(4.14)hold uniformly in (m2; g0) 2 [0; )2, for a small  > 0. The sequence gjconverges to 0 when m2= 0 but not when m2> 0.For WSAW, the following theorem is a consequence of [10, Proposi-tion 8.1]. For j'j4, it is [13, Theorem 3.6]. The latter also controls the owof the coupling constant uj, which is used for the analysis of the pressurein [13] but is not needed here. The domains D∅j, and the Wj-norms on thespace Kj, which appear in the theorem are discussed following its statement.Theorem 4.2. Let d = 4, n  0, and let  > 0 be suciently small.Let N  1. Let (m2; g0) 2 [0; )2and ffa= ffb= 0. There exist M > 0and an innite sequence of continuous functions Uj= (gcj; cj; zcj) of(m2; g0), independent of the volume parameter N , such that for initial784.2. Bulk owconditions U0= (g0; c0; zc0) and K0= 1∅, a ow (Uj ;Kj) 2 D∅jexistssuch that (4.12) holds for all j+1 < N , and, if m22 [L2(N1); ), alsofor j + 1 = N . Moreover, gcj= O(gj), zcj= O(jgj), j= O(jL2jgj),andkKjkWj= k∅KjkWjMjg3j(j  N): (4.15)In the remainder of the thesis, we often drop the superscripts and writesimplyUj= (gj; j; zj) (4.16)for the sequence provided by Theorem 4.2. The stated continuity of Ujisnot part of the statements of [10, Proposition 8.1] or [13, Theorem 3.6], butit is established in [10, Section 8.2].The denition of the Wjnorm on Kjin (4.15) is discussed at lengthin [24], and we do not repeat the details here. The inequality (4.15) providesvarious estimates on Kj(X) and on its derivatives with respect to elds, interms of the size of the polymer X. Some examples of its use are given inLemma 5.3 below. For example, as noted explicitly in [24, (1.64)], (4.15)with j = N implies that (with elds set equal to zero on the left-hand side)j∅KN ()j MN g3N; (4.17)uniformly in m22 [L2(N1); ).The Wj= Wj(~s) norm depends on a parameter ~s = ( ~m2; ~g) 2 [0; )2,whose signicance is discussed in [10, Section 6.3]. Useful choices of thisparameter depend on the scale j, as well as on approximate values of themass parameter m2of the covariance and the coupling constant gj. Weuse the convention that when the parameter ~s is omitted, it is given by~s = sj= (m2; ~gj(m2; g0)), where ~g = ~gjis dened in terms of the initialcondition g0by~gj= ~gj(m2; g0) = gj(0; g0)1jjm+ gjm(0; g0)1j>jm: (4.18)By [10, Lemma 7.4],~gj= gj+O(g2j); (4.19)794.2. Bulk owso the sequences (~gj) and (gj) are the same to leading order. Moreover,gj= gj(1 +O(gjj log gjj)); (4.20)this follows from [10, (6.1), (7.11)] for WSAW and the same result holds forn  1 according to [13]. Thus the sequences ~gj, gjand gjare essentiallyinterchangeable, and in particular error bounds expressed in terms of anyone of them are equivalent.The domain D∅j= D∅j(~s)  V∅h K∅jalso depends on ~s (with the con-vention mentioned above when ~s is omitted), is independent of h as we dealonly with the bulk here, and is dened as follows. For the universal constantCD 2 determined in [10], for j < N ,D∅j(~s) = f(g; ; z) 2 R3 : C1D~g < g < CD~g; L2jjj; jzj  CD~ggBW∅j(~j~g3): (4.21)The rst factor is the stability domain dened in [23, (1.55)] , restricted tothe bulk coordinates and real scalars. In the second factor, BX(a) denotesthe open ball of radius a centred at the origin of the Banach space X, and is as in [10, Theorem 6.3] and [13, Theorem 3.5]; for concreteness weuse  = 4M where M is the constant of Theorem 4.2. The space K∅isthe restriction of K to elements K with K(X) = 0 for all polymers X.Since, by (4.11), the renormalisation group acts triangularly, the distinctionbetween W and W∅is unimportant for the bulk ow, and W∅is denotedby W in [10].4.2.2 Properties of the bulk flowWe provide some details about the ow of bulk coupling constants, for lateruse.The bubble diagram is dened byBm2= (n+ 8)Z10Z10P (X(T ) = Y (S))em2Tem2SdT dS; (4.22)804.2. Bulk owwhere X;Y are independent continuous-time simple random walks (takingsteps at the events of a rate-(2d) Poisson process). For d = 4, it is an exercisein calculus (see [10, (1.8)]) to see thatBm2 b logm2 as m2 # 0, with b =n+ 8162. (4.23)We recall from [11, Lemma 6.3] thatj= b logL+O(Lj) for m2 = 0: (4.24)Lemma 4.3. For (m2; g0) 2 (0; )2, the limit g1= limj!1gjexists, iscontinuous in (m2; g0), and extends continuously to [0; )2. For g02(0; ),g11Bm2as m2# 0: (4.25)Proof. For n = 0, this is [10, Lemma 8.5], which is stated for a relatedsequence gj= gj+ O(g2j) (see at the end of [10, Section 8.3]). For n  1,(4.25) also holds, as indicated in [13, (4.28)].For the next lemma, recall that E(p)abis dened in (1.40).Lemma 4.4. As ja  bj ! 1, Ljab = 2ja  bj + O(1). If jm> jabtheng1jab= b(log ja bj)(1 + E(2)ab).Proof. It is an immediate consequence of (3.9) that Ljab= 2ja bj+O(1).Since jab< jm, we have ~gjab= gjabwith gjabdened by the sequence jgiven by m2= 0. By (4.19){(4.20) it suces to prove thatgjab(0)1= b(log ja bj)(1 + E(2)ab): (4.26)It is shown in the proof of [12, Lemma 2.1] that if  : R+! R is absolutelycontinuous thenkXl=jl (gl)g2l=Zgjgk+1 (t) dt+O Zgjgk+1t2j 0(t)j dt!: (4.27)814.2. Bulk owLet 1= b logL. We set  (t) = t2 in (4.27), and apply (4.24), to obtaing1k= g10+k1Xj=0j+O(j log gkj) = g10+ 1k+O(1)+O(j log gkj): (4.28)In particular, g1k= O(g10+ 1k) = O(k) (with g0-dependent constant).Therefore,g1k= 1k +O(log k): (4.29)This gives (4.26) and completes the proof.Lemma 4.5. Let j= j[w(1)], 0j= j+1w(1)j+1 jw(1)j. Then j=O(jgj) and jj 0jj = O(jg2j).Proof. By [11, Lemma 6.2], w(1)j= O(L2j) and by [11, Proposition 6.1],Cj+1;ab= O(jL2j). With (3.59) and (4.21), we therefore havej= (j+ (2 + n)gjCj+1;00)(w(1)j+ C(1)j+1) jw(1)j= jC(1)j+1+ (2 + n)gjCj+1;00w(1)j+1= O(~gjL2j)O(j) +O(gj)O(jL2j)O(L2j) = O(jgj):(4.30)For the second statement, by denition0j= (j+1 (j+ (2 + n)gjCj+1;00))w(1)j+1: (4.31)The subtracted terms in the dierence on the right-hand side cancel therst-order part of j+1(see [11, (3.31)]), leaving only the higher-order termswhich are bounded by O(jL2jg2j) according to [24, (1.80)]. This leads tothe desired bound on 0j.Recall from (3.67) that the eigenvalues of the matrix Ajdened in (3.66)are fj= 1  pj[w(1)]  jgj, where now gj; j(and also zj) are givenby the ow of the bulk coupling constants determined in Theorem 4.2. Theconstant  is given by  = n;pfrom (1.33), depending on the values of824.2. Bulk ow(n; p) and the choice of h 2 E. For j  J , we writej;J=JYi=jfi; j= 0;j: (4.32)The value of j;Jdepends on , and we write j;Jfor its values when = . The matrix product AJAJ1  Ajhas eigenvalues j;J, with theeigenvalue j;Jonly occurring for n  2 and p = 2. Error estimates in thefollowing lemma depend on , but this is unimportant since  is xed.Lemma 4.6. Let (m2; g0) 2 [0; ]. Let 0  j  J <1 and  2 R. Thereexists j= 1 +O(gj) such thatj;J= j gJ+1gj!(1 +O(JgJ)): (4.33)Proof. For i 2 N, let i= i[w(1)]. By Theorem 4.2, the sequence giobeysthe recursion relationgi+1= (1 ei)giwith ei= igi+ 4i+ ~ri; ~ri= O(ig2i): (4.34)As noted below (4.13), j= O(j). By Lemma 4.5, i= O(igi). Therefore,ei= O(igi). Let 0i= i+1w(1)i+1iw(1)i. By Lemma 4.5, ji0ij = O(ig2i).By (3.67),fi= (1 ei)(1 + di); (4.35)withdi= (1 ei)1((4  p)i+ ~ri) = (4  p)0i+O(jg2i): (4.36)By Taylor's theorem, for small t,1 t = (1 t)(1 +O(t2)): (4.37)834.2. Bulk owTherefore,fi= (1 ei)(1 +O(ig2i))(1 + di) =gi+1gi(1 + Ei); (4.38)withEi= di+O(ig2i) = (4  p)0i+O(ig2i) = O(ig2i): (4.39)Letj=1Yi=j(1 + Ei): (4.40)The innite product converges since Eiis summable by (4.39). Moreover,j= 1 +O(P1i=jEi) = 1 +O(jgj), by (4.14). With (4.32), we obtainj;J= j gJ+1gj!1J= j gJ+1gj!(1 +O(JgJ)); (4.41)and the proof is complete.For j  0, in view of Lemma 4.6 it is natural to dene, with  = n;pfrom (1.33),j= (gj=g0): (4.42)Lemma 4.7. As ja bj ! 1, if jm> jabthenjab=1bg0log ja bj(1 + E(p)ab): (4.43)Proof. Since Ljab= 2jabj+O(1) by Lemma 4.4, (4.43) follows from (4.42)and Lemma 4.4. The error estimate improves for p = 1 because in this case = 0; in fact j= 1 for all j when p = Change of variablesTheorem 4.2 is stated in terms of the parameters m2; g0, rather than theparameters g;  that dene the WSAW and j'j4models. The following844.3. Parameters and stability estimatesproposition, proved in [10, Proposition 4.2(ii)] for WSAW and [13, (4.23)] forj'j4, relates these sets of parameters via the functions zc0; c0of Theorem 4.2and (2.44). The critical value centers the analysis here, for the rst time.Proposition 4.8. Let d = 4, n  0, and 1> 0 be small enough.There exists a function [0; 1)2! [0; )2, that we denote by (g; ") 7!( ~m2(g; "); ~g0(g; ")), such that (2.44) holds with  = c(g) + ", if z0=zc0( ~m2; ~g0) and 0= c0( ~m2; ~g0). The functions ~m; ~g0are right-continuousas " # 0, and satisfy ~m2(g; 0) = 0, and ~m2(g; ") > 0 if " > 0.We also dene the right-continuous functions (as " # 0)~z0(g; ") = zc0( ~m2(g; "); ~g0(g; ")); ~0(g; ") = c0( ~m2(g; "); ~g0(g; ")):(4.44)Starting from (g; ), Proposition 4.8 provides ( ~m2; ~g0), and then Theo-rem 4.2 provides an initial condition U0= (~g0; ~z0; ~0) for which there ex-ists a global bulk ow of the renormalisation group map. This needs tobe supplemented by the observable ow, whose perturbative part is givenby Proposition 3.12. In the next section, we analyse the complete renor-malisation group ow, including the non-perturbative corrections for theobservable ow.4.3 Parameters and stability estimatesBefore we can extend the bulk ow provided by Theorem 4.2 to include theobservables, we need to discuss the extension of the stability domain D∅jandthe W∅jnorm to observable subspaces of N in Section 4.3.1. We also checkin Section 4.3.2 that the new denitions of norm parameters do not breakthe stability estimates in [23] on which [24] relies. This is an important,albeit technical step that is necessary for the proof of Theorem 4.9 later on.4.3.1 Parameters, norms and domainsWe use several norms, and domains dened via these norms. The normsextend those in [24, Section 1.7] where only the two-point function wasconsidered, to handle the new observables present here.854.3. Parameters and stability estimatesThe following sequences hjand hff;jeach have distinct values in twodistinct cases, which we identify as either the h = ` or h = ~h cases. This~h, which is called h in [10, 13, 23, 24], is not related to and should not beconfused with the vector h 2 Rn used to dene the space Vh. The twooptions for hj; hff;jare used to construct the Tffi;j(hj) norm in [24].For `0; k0> 0 as in [24, Section 1.7.1], and for j  0, lethj=8<:`j= `0Lj(h = `)~hj= k0~g1=4jLj(h = ~h):(4.45)With the notation x ^ y = minfx; yg and x+= maxfx; 0g, we also denehff;j= 1j^jab`pj^jab2p(jjab)+8<:~gj(h = `)~gp=4j(h = ~h):(4.46)The occurrence of  from (4.42) in (4.46) is a feature that is not visiblein [9], since if p = 1 then  = 0 and  = 1. The denition here is moresubtle, as it anticipates the ultimate appearance of logarithmic correctionsfor p  2. It plays an important role in Lemma 5.3 below.A j-dependent norm on Vhis dened, using the weights from the h = `case of (4.45){(4.46), bykV kVh= maxnjgj; L2jjjj; jzjj; jyjj; `pj`ff;j(jaj _ jbj);`ff;j(jtaj _ jtb); `2ff;j(jqaj _ jqbj); L4jjujo;(4.47)where x_y = maxfx; yg. We extend the domain in R3 appearing in (4.21) byincluding now the coupling constants a; b(for n = 0 these are permittedto be complex), and deneDj= fU 2 V(0)h: g > C1D~g; kUkVh< CD~gg: (4.48)The W norm is built from the Tffi= Tffi;j(hj) norm used in [24]. Concerning864.3. Parameters and stability estimatesobservables, the Tffinorm obeys (recall (3.19))kFkTffi= kF∅kTffi+ hffkFakTffi+ kFbkTffi+ h2ffkFabkTffi: (4.49)This is the same as what is used in [24], except we now dene hffby (4.46).We also need the following mass intervals. Given  > 0, letIj=8<:[0; ) (j < N)[L2(N1); ) (j = N);(4.50)and, for ~m22 Ij, let~Ij=~Ij( ~m2) =8<:[12~m2; 2 ~m2] \ Ij( ~m26= 0)[0; L2(j1)] \ Ij( ~m2= 0):(4.51)Let ~sj= ( ~m2; ~gj), and let ~jbe given by (4.13) with jmdetermined bymass ~m2rather than m2. We extend the bulk domain of (4.21) to a domainD∅j(~s)  V(0)h Kj, (with the same convention when the parameter ~s isomitted), dened byDj(~sj) = DjBWj(~j~g3): (4.52)The domain D also depends on the vector h, but we regard h as xed anddo not include it in the notation.Restriction to real coupling constants Complex coupling constants areused in [24] only to enable Cauchy estimates in the proof of Theorems 4.2and 4.9, but otherwise complex coupling constants are not used. In [24], real(V;K) does indeed yield real (R+;K+) for j'j4, as the vector space K is a realvector space, and when V is real there is no way to produce an imaginarypart in R+or K+. For WSAW, the complex eld can be reexpressed interms of a real eld, and the bulk coupling constants g; ; z; y can be seento remain real. For the observable coupling constants ; q, the complexityplays a more prominent role and we have not ruled out the possibility that874.3. Parameters and stability estimates; q become complex. We permit them to be complex here, and this createsno diculties.4.3.2 Stability estimatesIn this section, we check a number of stability estimates from [23] and showthat the new denitions of norm parameters can be accommodated. We alsoprovide any modications to arguments in [23], wherever needed. We writefjffi gjto mean that fj cgj: (4.53)Choice of hffOur choices of ~gjand hjin (4.18) and (4.45) are identical tothose used in [23,24], but the choice of hff;jin (4.46) diers by the appearanceof j(and thus  = n;p, see (4.42)) and by allowing all p  1. By [11,(6.101)],12~gj+1 ~gj 2~gj+1. Therefore, by (4.46),hff;j+1hff;j const8<:Lpj < jab1 j  jab;(4.54)where the improved bound occurs for j  jabsince the power of L in (4.46)stops changing at the coalescence scale. On the other hand, it is indicatedin [23, (1.79)] that what is required in [23,24] is that (4.54) should hold withLpreplaced by L1, which is a stronger requirement than (4.54).The Lpgrowth in (4.54) can be accommodated because now we taked+(a) = d+(b) = p1j<jab(see Section 3.3.1), rather than the choice 1j<jabused in [23, Section 4.2.2]. Because of this, in the proof of [23, Proposi-tions 2.8, 4.9], the computation of the small parameter ;(Y ) (not to beconfused with n;pdespite its similar name) gives exactly the same value;(Y ) = Ld1+ L11Y \fa;bg6=∅ present in [23, Proposition 2.8], and theanalysis of [23, 24] can continue to be based on the crucial contraction [23,Proposition 2.8] which remains unchanged.The Lpgrowth in (4.54) also violates the hypotheses of [23, Lemma 3.2],whose conclusion is used in several places in [23] (e.g., in the proofs of the884.3. Parameters and stability estimatesimportant results [23, Proposition 2.2, 2.6, 2.7]). However, the conclusionof [23, Lemma 3.2] continues to hold if its hypotheses are modied to use ourdenition of gauge invariance, and to use the bounds (4.54), h0jffi L1hj1,and h0ff;j+1(h0j+1)pffi hff;jhpjthat hold in our present context. Thus theconsequences of [23, Lemma 3.2] continue to hold in our present setting ofgeneral values of p.Choice of pNBy the denition of hff;jin (4.46),`ff;jhff;j= ~g1p=4j; (4.55)and this grows for p > 4. This plays a role in [24, Lemma 2.4], which is theplace that determines the choice pN= 10 used in [9,10,13]. We continue touse pN= 10 when p < 4. For p  4 (which we consider only for WSAW),we take a larger choice, as follows.First, [24, Lemma 2.4] is proved using [21, Proposition 3.17], which inturn relies on [24, Proposition 3.11]. We must choose pN A+ 1, where Aappears in the proof of [24, Proposition 3.11]. In the factor (A+1)in [24,Proposition 3.11], there can appear at most two bad ratios (4.55), since theworst case contains two observable elds, together with at least A 1 goodratios `j=hjwhich each yield a factor ~g1=4jby (4.45). Thus, at worst, (A+1)gives~g(A1)=4j~g2(1p=4)j; (4.56)and we require in [24, Lemma 2.4] that this is at most ~g10=4j. Therefore, theminimal pNwe can permit ispN= A+ 1 where14(A 1) + 2p2104; i.e., A = 2p+ 3: (4.57)Thus we can take any xed pN maxf10; 2p+ 4g.Stability estimate: value of VThe term (jaj+ jbj)hjhff;jappears in thedenition of V;jin [23, (1.80)], for the estimates of [23, Proposition 1.5].894.3. Parameters and stability estimatesThis term arises as the T0norm of affaffia+ bffbffib, and is suitable forp = 1. For general p  1, it needs replacement by (jaj + jbj)hpjhff;j. Thisreplacement has been incorporated into the denition (4.48) of Dj, so thatmembership in Djimplies that (jaj _ jbj)`pj`ff;j CD~gj. Also, by (4.45){(4.46), and since `0 1 and k0 1 (as chosen in [24, Section 1.7.1]),jxjhpjhff;j= jxj`pj`pff;j(hj=`j)p(hff;j=`ff;j)= jxj`pj`pff;j(k0=`0)p~gp=4j~gp=41j CD~gj(k0=`0)p~g1j CDkp0:(4.58)This fullls the required bound on V;jof [23, Proposition 1.5].Stability estimate: case of p  4 For p > 2, the proof of [23, Proposi-tion 5.1] must be modied. In particular, for p  4, we must justify placingsuch a large power in the exponent, as this appears to make the expectationof eVdivergent since the measure provides only exponentially quadraticdecay. Justication is possible because functions of ffaand ffbare equiv-alent to second-order polynomials, by denition of the quotient space in(2.62). Because of this, the placement of the observables in the exponent isan option that supercially appears worse than it actually is.In more detail, by denition of N , we have eaffaffipa= 1+affaffipa. There-fore,keaffaffiakTffi 1 + jajhffkffipakTffi 1 + jxjhffhp(1 + kffik)2p e2p(jxjhffhp)1=p(1+kffik2);(4.59)where in the second inequality we used [21, Proposition 3.10], and in thethird we used the elementary fact (see [23, Lemma 5.2]) that 1 + up(1 +x)2p e2pu(1+x2)for any x; u > 0 and p  maxf1; ug, with the choice u =(jxjhffhp)1=p. This modication permits the proof of [23, Proposition 5.1]to proceed as it is otherwise written.904.4. A single renormalisation group step including observables4.4 A single renormalisation group step includingobservablesThe following theorem is the centrepiece of the proof of Theorems 1.6{1.8. For observables, it provides the non-perturbative counterpart to theperturbative statement of Proposition 3.12. One of its consequences is thatif h 2 Rn is chosen to lie in one of the eigenspaces E, then Vj2 Vhfor allj. In other words, the complete renormalisation group ow keeps the vectorh 2 Rn xed for all j. Proposition 3.12 gives the perturbative version ofthis fact. The norms in Theorem 4.9 depend on the choice of the eigenspaceEvia the appearance of n;pin the denition of hff;jin (4.46), and thusthe estimates it provides also depend on the choice of eigenspace for h. Thisis the source of the two distinct powers n;pfor the logarithms appearing inTheorems 1.6{1.7.Theorem 4.9. Let d = 4. Let n = 0 and p  1, or n  1 and p = 1; 2. LetCDand L be suciently large. Let h = h2 E, and choose  = n;pin(4.42) and (4.46). There exist M > 0 and  > 0 such that for ~g 2 (0; )and ~m22 I+, and with the domain D dened using any  > M , themapsR+: D(~s) ~I+( ~m2)! V(1)h; K+: D(~s) ~I+( ~m2)!W+(~s+) (4.60)dene (U;K) 7! (V+;K+) as in (4.8) and obeying (4.5), and satisfy theestimateskR+kVhM ~+~g3+; kK+kW+M ~+~g3+: (4.61)In addition, R+;K+are jointly continuous in all arguments m2; V;K.In particular, the bounds of (4.61) hold when ~m2= m22 Ij, and in thiscase ~+= j+1. Also, ~gjcan be replaced in estimates by gj, due to (4.19).This leads to the replacement of the right-hand sides of (4.61) by j+1gj+1,which itself can be replaced by jgj. Thus there is no need for distinctionbetween these various options.914.4. A single renormalisation group step including observablesMore can be said about R+, for which we have the exact formulas (4.9){(4.10). Since the distance between a and b is at least12Ljabby (3.10), whenj < jab, no scale j small set contains both points a and b, in addition(Cj+1)ab= 0 for such j, so abR+= 0. We write Rx+for the coupling con-stant corresponding to xin R+, and similarly for Rqx+. With this notation,Rqx+= 0 for j < jab. Similarly, since only constants are in the range of(a+ b) Loc(j+1)Xwhen j  jab, Rx+= 0 for j  jab. Recall that we writefjffi gjto mean that fj cgj, then by denition of the Vhnorm, the rstbound of (4.61) implies that, for (U;K) 2 Dj(~sj),jRx+j ffi `pj`1ff;jjg3j1j<jabffi jjg2j1j<jab; (4.62)jRqx+j ffi `2ff;jjg3jffi 2jabL2pjab22p(jjab)jgj1jjab: (4.63)As discussed below the statement of Proposition 3.12, the rst scale forwhich qptof (3.71) can be nonzero is qpt;jab+1. The indicator function in(4.63) shows that this remains true on a non-perturbative level.With observables, according to [24, (1.69)], the statement for the bulkow in (4.11) is accompanied by the statement that, for x = a or x = b,if xV = 0 and xK(X) = 0 for all X 2 P thenxR+= abR+= 0 and xK+(U) = abK+(U) = 0 for all U 2 P+.(4.64)Moreover, as discussed below [24, (1.69)], a;+is independent of each of b,bK, and abK, and a similar statement holds for b;+.Proof of Theorem 4.9 Theorem 4.9 is an adaptation of Theorems 1.10{1.11 in [24] to include more general observables. Its proof requires modica-tion to some aspects of [23,24], which focus specically on the case of p = 1and WSAW, to to handle arbitrary p  1 for WSAW, and p = 1; 2 for j'j4.These modications can be sorted into three categories:(i) Dierent choices of parameters and changes to stability estimates,which we already discussed in Section 4.3.2.(ii) Modication needed in one aspect of the renormalisation group map924.4. A single renormalisation group step including observables(4.8), which is discussed in Section 4.4.1.(iii) For n  2 and h 2 E, we use new ideas to prove that the full non-perturbative ow of the coupling constants remains in the space Vh.This is seen perturbatively in Proposition 3.12, and non-perturbativelyfrom the fact that R+maps into Vhin Theorem 4.9. The new ingre-dient is the requirement of h-factorisability in Denition 4.1, and thefact that this property is preserved by the renormalisation group map.We discuss this in Section Modification to [24, Map 6]For the analysis of Map 6 in [24, Section 6.2], we must estimate the incre-ments qa, qb, ta, tb, and u that arise in R+. The discussion of uprovided there holds without change here. There is a small modication tothe treatment of qa, qb, which we discuss rst, and ta, tbare new here.We use the notation of [24, Section 6.2].Let x = a; b. It suces to show that kqxffaffbkT0ffi 1, and for this wemay assume that j  jab. In this case, x= jab;xand xis not updatedby Q. By [11, Proposition 6.1], for m22 Ij, jCj;xyj  cL2(j1). From thiswe conclude that [wpab] ffi L2pjffi `2pj. Therefore,qx= p!ab[wpab] ffi ab`2pj: (4.65)Since V 2 Dj, we have jxj  CD~gj`pj`1ff;j. Therefore,kqxffaffbkT0= jqxjh2ff;jffi ~g2j(hff;j=`ff;j)2: (4.66)Since the right-hand side is ~g2jfor h = `, and is ~g2p=4jfor h = h, this issucient.Finally, txonly arises for n  1 and p = 2, which we assume in thefollowing sketch. It suces to show that jtxjhffffi 1. By (3.72),tx= tpt;x(V Q) tx= 1n1^x(e+ h)&^ ; (4.67)934.4. A single renormalisation group step including observableswhere&^ =C0;0(1 1j+1<jab2^w(1)) + 1j+1<jab^+[w(2)] + 1j+1jab[^w(2)];(4.68)with^x; ^ the relevant coupling constants of V Q. Thus,^x= x x;Qand ^ =   Q, with x;Qand Qfrom Q. As above, we have C00ffi `2jandjxj  CD~gj`2j`1ff;j. As in [24, (1.43)], we denej=8<:1=2j~gj(h = `)1=2j~g1=4j(h = h):(4.69)In the setting of Map 6, we have jx;Qjh2hffffi . The largest term on theright-hand side of (4.68) is the rst one, and its contribution to jtxjhffisbounded by a multiple ofj^xj`2jhff;jffi (~gj`2j`1ff;j+ jh2jh1ff;j)`2jhff;jffi ~g1=4j+ j~g1=2j; (4.70)for both h = ` or h = h (recall (4.45){(4.46)). This is sucient.4.4.2 Reduced symmetryAs discussed in Section 3.3.2, for n  2 the O(n) symmetry can be reducedby choice of h. To handle this, we replaced the denition of the spaceK in [24, Denition 1.7] by the adapted version in Denition 4.1. WithDenition 4.1, we can prove that if h 2 E, and if U 2 Vhand K 2 K(h)obey appropriate estimates, then under the renormalisation group map itis also the case that V+2 Vhand K+2 K+(h). This is the content of thefollowing proposition, in which we place more prominence than usual on hin the notation. Recall that the domain D depends on h directly throughthe denition of Vhin Section 3.2.3 and W depends on the eigenspace Ethrough the coecient jin hff;j(see (4.46)).Proposition 4.10. The renormalisation group map of [24, Section 1.8]obeys (R+;K+) : D(~s; h) ~I+( ~m2)! V(1)hW+(~s+; h).The following proposition gives the R+part of Proposition 4.10.944.4. A single renormalisation group step including observablesProposition 4.11. Let h 2 E. If (U;K) 2 D(~s) and m2 2 ~I+( ~m2), thenR+(U;K) 2 V(1)h.Proof of Proposition 4.11. Let h be in one of the eigenspaces Eof thematrices in M2(n) (and h = 1 if n = 0). The denition of R+is givenin (4.9){(4.10). It is already established in [24, Section 2.1] that ∅R+ 2∅V(1)h(for this h plays no role). Thus we concentrate on R+for  2fa; b; abg. Note that the superscript in V(1)hplays no role in these observablesubspaces.By assumption, U 2 V(0)h Vh, and by Proposition 3.12, Vpt: Vh!Vhpt= Vh(with the last equality due to h 2 E). Thus, by denitionof R+, it suces to show that the polynomial Q dened by (4.9) obeysQ 2 Vh. By denition of Q, to prove that Q 2 Vhit sucesto prove that LocX: Nh! Vh, because K(Y )I(Y; V )12 NhbyLemma 3.10 and because K(Y ) 2 Nhsince K 2 K(h) (recall Denition 4.1).This last requirement is provided by Lemma 3.9, and the proof is complete.We now complete the proof of Proposition 4.10, by proving its K+part.We extend the notion of h-factorisation in Denition 3.3 to maps F : Pj!N , as follows. We say that F is h-factorisable if F (X) 2 Nhfacfor allX 2 Pj. By Lemma 3.5, if F;G 2 K are h-factorisable, then F  G is h-factorisable as well since the O(n)-invariance of ∅F and ∅G is guaranteedby the denition of K in Denition 4.1.Proof of Proposition 4.10. By Proposition 4.11, R+(U;K) 2 V(1)h, so itremains to check that K+2 W+(~s+). This statement is provided by [24,Theorem 1.11], apart from the requirement that K+is h-factorisable, and,if p = 2, that K+is R(n)-invariant. To check that the map K+constructedin [24] is h-factorisable, we recall that the construction is a composition ofsix maps which produce K(1); : : : ;K(6)= K+. We examine these one byone and show that K(i)2 Nh-facimplies K(i+1)2 Nh-fac.1. According to its construction in [24, Lemma 4.2], K(1)is a polynomialin I, K, and J (see [24, (4.9)]). Since J is given by localised products954.4. A single renormalisation group step including observablesof I and K (see [24, (4.12){(4.13)]), it is h-factorisable, and hence sois K(1), by Lemma 3.5.2. By [24, Lemma 4.3], K(2)is a circle product of I(2)and K(1). Bothof these are h-factorisable, and hence so is K(2).3. The denition of K(3)is given in [24, (5.9)]. All of the quantities onthe right-hand side of [24, (5.9)] are h-factorisable, and hence so isK(3).4. According to its construction in [24, Lemma 5.8], K(4)is a polynomialin~Ipt, K(3), and hlead(see [24, (4.9)]). By [24, (5.18)], hleadis atruncated expectation of V 's, so it is h-factorisable by Lemma 3.6,and hence so is K(4).5. Map 5 replaces W (Vpt) by W (V+). Since both Vpt(V ) and V+areh-factorisable, so is K(5).6. The role of Map 6 is to perform summation by parts and to moveconstant elds out of the circle product. Only the second aspect isdierent in our present setting, in which [24, (6.24)] becomes replacedby((eI+pt) K(5))() = e()(I+pt (eK(5))(); (4.71)where(X) =Xx2XVpt;x(V Q)j'=0: (4.72)We have shown above that V Q and Vpt(V Q) are h-factorisable,and hence so is . It can then be seen from its denition in [24, (6.21)]that K(6)is h-factorisable.Since K+= K(6)by denition, this completes the proof of h-factorisability.If p = 2, the initial polynomial V0contains only even powers of ', so (I0;K0)is R(n)-invariant. It is straightforward to check that this property is pre-served by all of the above operations, so K+is also R(n)-invariant.964.5. Complete renormalisation group ow4.5 Complete renormalisation group flowGiven (m2; g0) 2 [0; )2, the initial conditions for the global existence of thebulk renormalisation group ow are given by∅U0 = Uc0= (g0; zc0(m2; g0); c0(m2; g0)); (4.73)and this gives rise via Theorem 4.2 to the sequence Uj(m2; g0). The initialcondition for the ow with observables is the pair (0; U0), with U02 V(0)hand 0as in (4.4), that is given by∅U0 = Uc0; a;0; b;02 f0; 1g; and 0= 0; (4.74)The next three propositions show that the ow with observables with theinitial condition (0; U0) exists for all j  N , and they state properties ofthat ow. The ow of x; qx; txdoes depend on the choice of the vectorh = h2 E, and on the choice of initial condition a;0; b;0, but we donot add labels to indicate this dependence. When a;0= 0 or b;0= 0, wedene the coalescence scale jabto be jab=1 rather than via (3.9), since inthis case at least one of the observable elds ffa; ffbis absent and its point aor b no longer plays a special role.Proposition 4.12. Let d = 4. Let n = 0 and p  1, or n  1 andp = 1; 2. Let h = h2 E, and choose  = n;pin (4.42) and (4.46). Let(0; U0) be given by (4.74), and let K0= 1∅. Let N 2 N and (m2; g0) 2[L2(N1); )(0; ). There exist (j; Uj;Kj) such that (Uj;Kj) 2 Djand(4.5) hold for 0  j  N . This choice is such that ∅Uj = Ucj(m2; g0).For x = a or x = b, if x;0= 0 then x;j= 0 for all 0  j  N , whereasif x;0= 1 thenx;j=8<:j11 +Pj1k=0rx;k(j  jab)x;jab1(j > jab);(4.75)974.5. Complete renormalisation group owwhere rx;k2 R obey, for some c > 0,jrx;kj  ckg2k: (4.76)Also, with M given by Theorem 4.9, for all j,kKjkWjMjg3j: (4.77)Proof. We rst observe that if x;0= 0 then x;j= 0 for all 0  j  N , dueto (3.70) and (4.64). We therefore assume that x;1= 1.The proof is by induction on j. We make the induction hypothesis:IHj: for all k  j, (Uk;Kk) 2 Dk, (4.75) and (4.77) hold with jreplaced by k; and (4.76) holds for all k < j.By direct verication, IH0holds since 1= 1 and kK0kW0= 0 by deni-tion. We now assume IHjand show that it implies IHj+1.We apply Theorem 4.9 with j= jin (4.46), where j= (gj=g0)n;pasin (4.42). By the induction hypothesis and (4.61), Kj+12 BWj(~j+1~gj+1)and satises (4.77). According to Theorem 4.2, the sequence Ucsatisesthe bounds required for ∅U in the denition of D and obeys (4.11){(4.12),so that ∅Uj = Ucjfor all j. Therefore, to verify (Uj+1;Kj+1) 2 Dj+1, itsuces to show that jx;j+1j < CD`pj+1`1ff;j+1.Let x denote a or b. Since Rj+1(Uj;Kj) 2 Vhby (4.60), the inclusion ofthe non-perturbative remainder Rxj+1in the ow of xgives, by (4.10) andProposition 3.12,x;j+1=8<:fjx;j+Rxj+1(j + 1 < jab)x;jab1(j + 1  jab):(4.78)The ow of xstops at the coalescence scale, so we restrict to j + 1 < jab.In this case, we insert (4.75) into (4.78) to obtainx;j+1= j0@1 +j1Xk=0rx;k1A+Rxj+1= j0@1 +jXk=0rx;k1A; (4.79)984.5. Complete renormalisation group owwith rx;j= (j)1Rxj+1. By (4.62), this givesjrx;jj = jjj1jRj+1j  c0(j)1jjg2j; for some c0> 0: (4.80)Then (4.76) follows since (j)1and jare comparable by Lemma 4.6 and(4.42).To complete the induction, it remains to prove that jx;j+1j`pj+1`ff;j+1<CD~gj+1. By (4.45){(4.46), it suces to prove thatjx;j+1j < CDj+1; (j + 1 < jab) (4.81)(the case of j + 1  jabthen also follows). We deduce (4.81) from (4.75){(4.76), the estimatePjk=0jrx;kj = O(g0) (by (4.14)), and Lemma 4.6, sincewe may assume that CD> 2.Proposition 4.13. Let d = 4. Let n = 0 and p  1, or n  1 andp = 1; 2. Let h = h2 E, and choose  = n;pin (4.42) and (4.46).Let (0; U0) be given by (4.74) with a;0= b;0= 1, and let K0= 1∅.Let N 2 N and (m2; g0) 2 [L2(N1); )  (0; ). Let a; b be such thatjab< jm. For j  N and x = a; b, the entry qx;jin jproduced byProposition 4.12 obeysqx;j= p!a;jabb;jabwpj;ab+j1Xi=jabRqxi; (4.82)with jRqxij ffi 2jabL2pjab22p(jjab)jgj1jjab.Proof. By (3.71) and (4.10),qx;j+1= qx;pt+Rqxj= p!a;jabb;jab[wpj;ab] +Rqxj: (4.83)For all j < jab, both qptandRqxjvanish, and summation of [wpj;ab] produces994.5. Complete renormalisation group owa telescoping sum, so thatqx;j=j1Xi=jabqx;i= p!a;jabb;jabwpj;ab+j1Xi=jabRqxi: (4.84)The desired bound on Rqxjis provided by (4.63), and the proof is complete.Proposition 4.14. For x = a; b and j  N , each of x;j; qx;jis inde-pendent of N , meaning that, e.g., the nite sequence fx;1; : : : ; x;Ngtakes the same values on the torus Nas on a larger tori N0withN0> N . Also, each of x;j; qx;jis dened as a continuous function of(m2; g0) 2 [0; )2, and a;jis independent of b;0, and b;jis independentof a;0.Proof. The proof is identical to the proof of [9, Proposition 4.3(ii)], whichprovides the (n; p) = (0; 1) version of the statement and extends withoutmodication to our more general context here. Note that by denition ofV+in (4.9), x;Nand qx;Nare constructed from KN1and IN1, so theyare independent of whether the torus has scale N , or a larger scale.100Chapter 5Infinite volume limit andproofs of main resultsWe now complete the proofs of our main results Theorems 1.6{1.8. As a rststep, in Section 5.1, we take the N ! 1 limit of the observable couplingcontants. The derivatives of ZNin Lemma 2.8 naturally lead us to studyderivatives of WNand KN, and estimates for these are given in Section 5.2.In Section 5.3, we identify the correlation functions of Theorem 1.6 in termsof the limiting values x;1of Lemma 5.1 and prove Theorem 1.6. Finally,in Section 5.4, we prove Theorems 1.7{ Inductive limit of observable flowProposition 4.14 permits the observable coupling constants to be dened asinnite sequences, not stopped at j = N , via an inductive limit N ! 1.Indeed, since x;j, qx;jare independent of N > j, we obtain sequencesdened for any given j 2 N0by choosing any N > j. For the case of initialcondition b;0= 0, we write a;jfor the inductive limit of the sequence a;j,and dene b;jsimilarly. By (4.75),x;j= j10@1 +j1Xk=0rx;k1Afor x = a; b and j 2 N0: (5.1)1015.1. Inductive limit of observable owBy Proposition 4.14 and by denition,x;j= x;j^(jab1)(5.2)for any choice of initial conditions a;0; b;02 f0; 1g. The following twolemmas analyse the sequences dened by inductive limits. The constantsvxin the rst lemma ostensibly depend on x, but they are shown below inProposition 5.4 to be independent of x = a; b. The function g1(m2) in itsstatement is given by Lemma 4.3.Lemma 5.1. Fix h 2 E and make the corresponding choice of  = .Let (m2; g0) 2 (0; )2. For x = a; b, there exist constants vx= 1+O(g0),such that for all j 2 N0,x;j= vxj1 +O(jgj): (5.3)The limit x;1(m2) = limj!1x;jexists, andx;1(m2) = vx g1(m2)g0!: (5.4)On the other hand, if a;0= b;0= 1 and if jm> jab, then, as jabj ! 1with N > jm> jab,x;jab= vx1bg0log ja bj(1 + E(p)ab): (5.5)Proof. Let rx=P1k=0rx;k. By (4.76) and (4.14), the sum converges and isO(g0), and in addition rxPj1k=0rx;k= O(jgj). Let ux= 1 + rx. Then,by (5.1),x;j= j11 + rx+O(jgj)= uxj1(1 +O(jgj)): (5.6)With (4.42) and Lemma 4.6, this implies that there exists 0= 1 + O(g0)such thatx;j= 0uxj(1 +O(jgj)): (5.7)1025.1. Inductive limit of observable owThis proves (5.3) with vx= 0ux, and then (5.4) follows immediately fromthe denition (4.42), Lemma 4.3, and (4.14).The proof of (5.5) follows similarly, using (4.75) and Lemma 4.7, withLemma 4.4 (and (4.20)) to bound the error term O(jabgjab).For m2 0, we writeGab(m2) = (Z4 +m2)1ab; Gab= Gab(0): (5.8)Lemma 5.2. Fix h 2 E, jhj = 1, and make the corresponding choice of = . Let (m2; g0) 2 [0; )2. For both x = a; b, the limit qx;1(m2; g0) =limj!1qx;j(m2; g0) exists, is continuous, and for a; b with jab< jm,obeysqx;1(m2) = p! a;jabb;jabGpab(m2) +O(gjab)(g0log ja bj)2Gpab: (5.9)As ja bj ! 1,qx;1(0) = p!vavb1bg0log ja bj2Gpab(1 + E(p)ab): (5.10)Proof. By Proposition 4.13 and the fact that limj!1wj;ab= Gab(m2) bydenition,limj!1qx;j= p!a;jabb;jabGpab(m2) +1Xi=jabRqxi: (5.11)The sum on the right-hand side converges uniformly in (m2; g0) by Propo-sition 4.13 and is therefore continuous by Proposition 4.14. By Proposi-tion 4.13 and the fact that jgj O(gjab) (see [12, Lemma 2.1(i)]), weobtain1Xi=jabjRqxij ffi 2jabL2pjab1Xi=jab22p(jjab)jgj= 2jabL2pjabO (gjab) : (5.12)Note that the exponential factor in the second sum is needed for convergence,which is not otherwise guaranteed by (4.14). Then (5.9) follows from (4.43)1035.2. Non-perturbative estimatesand (3.9), together with the fact that Gab=142ja bj2(1 +O(ja bj2))by (1.15). Finally, (5.10) follows from Lemma Non-perturbative estimatesThe following lemma allows us to control the non-perturbative quantitiesin the proofs of our main theorems. We write Dkffto mean no derivativefor k = 0, the derivative with respect to ffafor k = 1, and derivatives withrespect to ffaand ffbfor k = 2.Lemma 5.3. Let h = h 2 E, and let  = . For n = 0, p  1,the following estimates (all at zero eld) hold uniformly in g 2 (0; )and m22 [L2(N1); ). For initial conditions a;0= b;0= 1 and forl = 0; 1; 2,jDlffK0N()j ffi Ng3lN12p(Njab)+1ja bjp1(g log ja bj)l: (5.13)For initial conditions a;0= 1, b;0= 0, and for k = 0; 1; : : : ; p andl = 0; 1,jDkffiDlffK0N()j ffi Ng3lNLN(klp)(g0logm2)l; (5.14)jDkffiDlffW0N()j ffi Ng2lNLN(klp)(g0logm2)l: (5.15)The bounds (5.13){(5.15) also hold for n  1 and p = 1; 2, after changingK0Nto KNand making directional derivatives with respect to ' in thedirection of a constant eld 1.Proof. We give the proof for n = 0. The proof for n  1 involves only slightchanges in notation. By (4.19), ~gjand gjare interchangeable in estimates.Recall the denitions of the T0;j(`j) and j(`j) norms from [10, Sec-tion 6.3]. In (4.49), in the T0;j(`j) norm each occurrence of ff or ff producesthe weight`ff;j= `p01j^jab2p(jjab)+Lp(j^jab)~gj(5.16)1045.3. Proof of Theorem 1.6dened in (4.46). We apply [24, (1.62)] which uses this fact, together with(4.77), to see that for l = 0; 1; 2 the boundjDlffK0N(; 0; 0)j  `lff;NkKN()kT0;N(`N) `lff;NkKNkWNffi ljab2lp(Njab)+LlpjabNg3lN(5.17)holds uniformly in m22 [L2(N1); ). By (3.9), Ljabffi ja  bj1. Thelogarithmic behaviour of jabis given by (4.43), and (5.13) is proved.For any k  pN, l = 0; 1, F 2 N , and test functions Ji:  ! C(i = 1; : : : ; k), it follows from the denition of the T0;N(`N) norm thatjDkffiDlffF0(0; 0; J1; : : : ; Jp)j  `lff;NkFkT0;N(`N)kJ1kN(`N)   kJkkN(`N):(5.18)By denition, k1kN(`N)= `1N(as in [10, (8.55)]). As in (5.17), this givesjDkffiDlffK0N(; 0; 0; 1; : : : ; 1)j  `lff;NkKN()kT0;N(`N)k1kkN(`N) `lff;N`kNkKNkWN:(5.19)With the initial conditions assumed for (5.14){(5.15), we have jab=1. By(5.16), (4.45), (4.42), (4.25), and (4.23),`lff;N`kN= ~glNlN`0LNlpkffi glN(g0logm2)lLN(klp): (5.20)With (4.77), this proves (5.14). Finally, for the bound on WN, we recallfrom [23, Proposition 4.1] thatkWN()kT0;Nffi Ng2N; (5.21)and (5.15) then follows exactly as in (5.19).5.3 Proof of Theorem 1.6Let n  0. For small g; " > 0, set  = c+ ", and let (m2; g0; 0; z0) =( ~m2; ~g0; ~0; ~z0) be the functions of (g; ") given by Proposition 4.8. This1055.3. Proof of Theorem 1.6choice is consistent with the initial condition (4.74) that guarantees theexistence of the global ow with observables. By [10, (4.34)] for n = 0,and [13, (4.24)] for n  1, it provides the identity = (g; ) =1 + ~z0~m2=1 + z0m2: (5.22)Proposition 5.4. Let h = h 2 E and  = . Let n = 0 and p  1,or n  1 and p = 1; 2. For n  1, let H be a constant eld with valueH0. For (g; ") 2 (0; )2, and for x = a; b,1pS(p)(g; ) = p!x;1(n = 0) (5.23)1ph(';H)p;'pa hig;= p!(Hp0 h)x;1(n  1): (5.24)In particular, the innite volume limit on the left-hand side of (5.24)exists.Proof. We use initial conditions a;0= 1 and b;0= 0. We start with (4.6),but without setting the elds to zero, to getZN= eN(IN+KN) ; (5.25)where ZN; IN;KNdepend on (ffi;ffi) for n = 0, and on ' for n  1.We rst prove (5.23). In this case, N=12(qa;N+ qb;N)ffaffb. By (3.54)(since  is a single block at scale N),Z0N= eN(I0N+K0N) = eN(eV0N(1 +W0N) +K0N): (5.26)Since ∅eV0N= eU0Nand DffaeV0N= a;Nffipa,DffaZ0N= a;NffipaeU0N1 + ∅W0N+ eU0NDffaW0N+DffaK0N: (5.27)We dierentiate with respect toffi in direction 1, p times, and set ffi =ffi = 0to obtainDpffiDffaZ0N= p!a;N+DpffiDffaW0N+DpffiDffaK0N; (5.28)1065.3. Proof of Theorem 1.6where we used the facts that eU0N= 1 and W0;∅N= 0 when ffi =ffi = 0. ByLemma 5.3 and (2.75),1 + z0m2pS(p)N(g; ) = DpffiDffaZ0N= p!a;N+O NgN(g logm2)+!(5.29)(since = +for n = 0). We let N ! 1 in (5.29), using the facts thatNgN! 0 by (4.14), S(p)N! S(p)by Proposition 2.1, and a;N! a;1byLemma 5.1. We also use (5.22) to identify the factor pon the left-handside. This proves (5.23).To prove (5.24), we apply both Dp'Dffato the logarithm of the right-hand side of (5.25). Since Nis independent of ', in terms of the pressure(2.70) this givesDp'DffaPN= Dp'Dffalog (IN+KN) : (5.30)By denition, IN= eUN(1+WN), and we write IN+KN= eUN(1+WN+eUNKN). Since Dp'Dffa(UN) = p!a;N(Hp0 h),Dp'DffaPN= p!a;N(Hp0 h) +Dp'Dffalog(1 +WN+ eUNKN): (5.31)It is now an exercise in calculus to apply Lemma 5.3 (and the fact that UNlies in the domain DNof (4.48)) to conclude thatDp'Dffalog(1 +WN+ eUNKN) = ONgN(g logm2): (5.32)Then, by (2.73),1 + z0m2ph(';H)p;'pa hig;;N= p!a;N(Hp0 h) +ONgN(g logm2):(5.33)Again we use NgN! 0 and a;N! a;1to see that the limit as N !1 of the right-hand side exists and equals the right-hand side of (5.24).1075.3. Proof of Theorem 1.6Therefore, the limit of the left-hand side must also exist and soh(';H)p;'pa hig;= limN!1h(';H)p;'pa hig;;N(5.34)exists in the sense of (1.18). We complete the proof of (5.24) by appealingto (5.22).Corollary 5.5. The constant vxin (5.3) is, in fact, independent of x,and moreover, for p = 1 and for all n  0, x;1= 1.Proof. Since the left-hand sides of (5.23) and (5.24) are independent of x,x;1and vxmust also be independent of x.Let p = 1. By denition, S(1)is the susceptibility , so (5.23) yieldsx;1= 1 (as was proved in [9, Lemma 4.6]). For n  1, since (';H) =Px'x H0, we take H0= e^1, the rst standard basis vector. Using 'ix7!'ixsymmetry and (1.12),h(';H);'a hig;= limN!1Xx2ND'1x;'a hEg;;N= limN!1Xx2Nh1D'1x'1aEg;;N= (H0 h) :(5.35)Thus (5.24) simplies to x;1= 1.Proof of Theorem 1.6. (i) By (1.13) and (1.28), (5.22) givesm2 (1 + z0)A1g"(log "1)+as " # 0; (5.36)and hence logm2 log "1. Using (5.4) and Lemma 4.3, and since g0=g1 +O(g),1(m2) ~v(log "1); ~v=v(g0b)=1(gb)1 +O(g); (5.37)where, in view of Corollary 5.5, we have dropped the labels x on 1and1085.3. Proof of Theorem 1.6v. For n = 0, we use (5.23) (recall that += for n = 0) to obtain1pS(p)(g; ) = p!1(m2)  p!~v+(log "1)+: (5.38)This proves (1.35).(ii) Let n  1 and p = 2. Now we use (5.24) to obtain12D(';H)2;'2a hEg;= 2!(H20 h)x;1 (H20 h)2~v(log "1); (5.39)where H0is the constant value of the eld H, and H202 Rn is the vectorwhose components are the squares of the components of H0. For the choiceh = n1=2e+2 E+, we have '2a h = n1=2j'aj2and H20 h = n1=2jH0j2.We cancel the n1=2factor on both sides of (5.39) and obtain12D(';H)2;'2a hEg; jH0j22~v+(log "1)+: (5.40)We take H0= e^kto be the kthstandard basis vector, and then sum over k,to obtain12Xx;yD'x 'y; j'aj2Eg;=12Xx;ynXk=1D'kx'ky; j'aj2Eg;2n~v+(log "1)+:(5.41)This proves (1.36). Suppose now that n  2. By symmetry, (5.41) gives12Xx;yD'1x'1y; ('1a)2Eg;+ (n 1)12Xx;yD'1x'1y; ('2a)2Eg;2~v+(log "1)+:(5.42)In (5.39) we take H0= e^1and h = 21=2(1;1; 0; : : : ; 0) 2 E. Sinceh 2 E, now  = . We obtain12Xx;yD'1x'1y; ('1a)2Eg;12Xx;yD'1x'1y; ('2a)2Eg;2~v(log "1): (5.43)1095.4. Proof of Theorems 1.7{1.8Since < +, the combination of (5.42){(5.43) gives12Xx;yD'1x'1y; ('1a)2Eg;n 1n2~v(log "1); (5.44)12Xx;yD'1x'1y; ('2a)2Eg; 1n2~v(log "1); (5.45)which proves (1.37){(1.38).(iii) The asymptotic formula (1.39) follows from (5.37), and the proof iscomplete.5.4 Proof of Theorems 1.7–1.8Proof of Theorem 1.7. (i-ii) We denote the parameters (n; p) by super-scripts. The innite volume limit of the watermelon network can be com-puted as a limit using Proposition 2.1, and for n  1 we have dened thecritical innite volume limits of correlation functions as in (1.17). For n  1,let (Sc)ij=D('ia)p; ('jb)pEc(n)denote the matrix of critical correlation func-tions. According to (2.72) and (2.74) (we drop the notation for evaluationat zero as all elds are evaluated at zero here),1 + ~z0(g; 0)plim"#0limN!1D2ffaffbP(n;p)N=8<:W(p)ab(c(0)) (n = 0; p  1);h  Sch (n  1; p = 1; 2):(5.46)For the prefactor on the left-hand side, we have used Proposition 4.8 forexistence of the limit ~z0(g; ") ! ~z0(g; 0) as " # 0. It also follows fromTheorem 4.2 that ~z0= O(g).By (4.6),P(n;p)N= logZ(n;p)N= N+ log(1 +KN()); (5.47)with N=12(qa;N+ qb;N)ffaffb+ ta;Nffa+ tb;Nffb+ uNjj (if n = 0 then1105.4. Proof of Theorems 1.7{1.8ta;N= tb;N= uN= 0). Dierentiation givesD2ffaffbP(n;p)N=12(qa;N+ qb;N) +D2ffaffbKN1 + ∅KN(DffaKN) (DffbKN)(1 + ∅KN )2: (5.48)According to (5.13), the last two terms vanish in the N ! 1 limit. Wewrite vfor the common value of vaand vb(recall Corollary 5.5). ByLemma 5.2,lim"#0limN!1D2ffaffbP(n;p)N=12qa;1(0) + qb;1(0)= p!(v)21bg0log ja bj2Gpab1 + E(p)ab:(5.49)When we make the choice h = h+= n1=2e+2 E+and carry out therenormalisation group analysis, it is the exponent +that occurs in (5.49),and we conclude (1.41){(1.43) (for (1.42) we use += 0 when p = 1).(iii) Next we prove (1.44){(1.45). Let n  2 and p = 2. We make the twochoices h+= n1=2e+2 E+and h= 21=2(1;1; 0; : : : ; 0) 2 E, whichobey jhj = 1. By symmetry,D'2a h+;'2b h+Ec=D('1a)2; ('1b)2Ec+ (n 1)D('1a)2; ('2b)2Ec; (5.50)D'2a h;'2b hEc=D('1a)2; ('1b)2EcD('1a)2; ('2b)2Ec; (5.51)and hencenD('1a)2; ('1b)2Ec=D'2a h+;'2b h+Ec+ (n 1)D'2a h;'2b hEc;(5.52)nD('1a)2; ('2b)2Ec=D'2a h+;'2b h+EcD'2a h;'2b hEc: (5.53)The rst term on the right-hand sides has been computed already in theproof of (1.43). For the second term, we instead use h = h, and now obtain(5.49) with  = . This leads to (1.44){(1.45).(iv) The asymptotic formula (1.46) for the amplitudes A0n;p;follows directly,1115.4. Proof of Theorems 1.7{1.8using the amplitude1(2)2for Gabin (1.15) and (5.37).Proof of Theorem 1.8. We must show thatW(p)aa(c(0)) = Gpaa(p! +O(g)) (p  1); (5.54)D'1a;'1aEc(n)= Gaa(1 +O(g)) (n  1); (5.55)Dj'aj2; j'aj2Ec(n)= G2aa(2!n+O(g)) (n  1); (5.56)D('1a)2; ('2a)2Ec(n)= O(g) (n  2): (5.57)Now the coalescence scale is jaa= 0, and hence j= 1 for all j. Also, (4.46)now gives `ff;j= 2pj~gj, and (5.13) is replaced byjDkffKN()j ffi Ng3kN2kpN: (5.58)Minor changes to the proof of Lemma 5.2 show that for the case of a = bwe obtain12(qa;1(0) + qb;1(0)) = p!Gpaa+O(g0); (5.59)and using this in place of (5.49) leads to the desired results. In particular,the main terms cancel now in (5.53), leading to (5.57).112Chapter 6Conclusion6.1 SummaryIn this thesis, we set out to investigate the critical and near critical be-haviour of two 4-dimensional models: the n-component j'j4model and thecontinuous time weakly self-avoiding walk (WSAW). The formal connectionbetween self-avoiding walks and spin models goes back to [39], but we areable to interpret the WSAW as the n = 0 case of the j'j4in a mathematicallyrigorous manner using the integral representation discussed in Chapter 2,and we provide a unied treatment for all n  0.The distinction between dierent models, represented by dierent valuesof the parameter n, comes into our discussion in the form of symmetriesof the interaction polynomials as discussed in Chapter 3. For example,the case n = 0 of the WSAW is represented by the gauge invariance andsupersymmetry, the case n = 1 is characterised by discrete symmetry similarto the Ising model, while for n  2, the O(n) symmetry is continuous. Infact, Proposition 3.12 provides a lot of insight into the dependence of thequantities of interest on the symmetries of the initial condition V0from(2.64) and (2.66). It is especially interesting how a full and a reduced O(n)symmetry lead to two dierent powers of the logarithmic corrections inTheorems 1.6{1.7.The introduction of observables was another major theme in this work.1136.2. OutlookObservables were used in our setting in [9] to compute the decay of theWSAW two-point function. In this thesis, we extend the variety of observ-ables considered and also apply the formalism of observables to the j'j4model (this was not done in [13]). In fact, it is through the evolution ofthe observable terms of our interaction polynomials that the symmetries ofthe models determine the values of the logarithmic corrections. This factcan already be seen at a perturbative level in Proposition 3.12, but it isremarkable that the same separation is true for the full non-perturbativeows in Proposition 4.12. The notion of h-factorisability from Chapter 3was developed and used in Chapter 4 together with the results of [23,24] toprove this fact.With this unied approach in dimension d = 4, we derive the asymptoticdecay of several critical correlation functions, and also obtain results forlogarithmic corrections to scaling for correlations of elds, as the criticalpoint is approached. For the case n = 0 of the WSAW, we obtain the decayof the critical \watermelon" networks, consisting for xed p  1 of p weaklymutually- and self-avoiding walks joining two distant sites, at the criticalpoint. This extends the result for p = 1 obtained in [9]. For p  2, wealso determine the logarithmic corrections to scaling for \star networks"consisting of p weakly mutually- and self-avoiding walks which intersect atthe origin, as the critical point is approached. For the case of n  1 of thej'j4model, we prove jxj2decay for the critical two-point function for alln  1. This extends previous results for n = 1 due to [33, 37, 38], to alln  1. In [38], it was also shown that for n = 1 the critical correlationbetween '20and '2xdecays as jxj4(log jxj)2=3. We extend this to generaln  1.6.2 OutlookNetworks of WSAWs The factor jabj2pthat appears in the asymptoticformula for W(p)ab(c) is what would occur for p independent WSAWs (oreven simple random walks) joining a and b. The power 24p2of the loga-1146.2. Outlookrithmic correction for interacting walks can be explained as follows. Pairsof walks at each of the vertices a and b must join despite their penchant toavoid, so each pair gives rise to a penalty (log ja bj)1=4.We expect that if instead of the watermelon network we studied an arbi-trary network of p paths joining vertices v1; : : : ; vkappropriately divergingfrom one another, the exponent of the logarithmic factor log jvi vjj wouldbe related to the number of pairs of walks joining at each vertex, analogousto the behaviour of amplitudes (constants) in d > 5 (see [45] and [31]). Theanalysis of arbitrary networks requires a more accurate perturbation theoryfor the observable elds, but seems to be within the reach of our method.Magnetisation. A very important critical exponent in the jffij4theory isthe exponent , describing the magnetisation. If one applies a constantuniform magnetic eld h in the direction of some xed unit vector e 2Rn, then the magnetisation is M(; h) = hffi0 ei;h. At criticality and asjhj ! 0, M has a power-law behaviour M(c; h)  jhj1=similarly to (1.9).This behaviour is modied by a logarithmic correction for d = 4, wherethe conjecture is the mean-eld value  = 3 together with a (log jhj)1=3correction [14,34].It would be very interesting to investigate whether methods similar tothose used in [13] to study the susceptibility could produce asymptoticsof the magnetisation in the vicinity of the critical point. The divergentparameters could be controlled by the mass term in the renormalised theory,but stability estimates would have to be improved compared to those from[23] that we currently use.Scaling limit to massless GFF. In [13] is is shown that in the subcrit-ical regime (for  > c), the scaling limit of the j'j4eld under standardcentral limit theorem rescaling is white noise with intensity given by thesusceptibility. It is also shown there that when the critical point is suitablyapproached, the scaling limit is a multiple of a massive Gaussian free eld(GFF) on the continuum torus. The convergence to the free eld requires1156.2. Outlookanomalous scaling and the mass parameter m2of the GFFm2is connectedto the exact way the eld j'j4is rescaled. Extending the result of [13] tothe case of m2= 0 remains an open problem.116Bibliography[1] A. Abdesselam, A. Chandra, and G. Guadagni. Rigorous quantumeld theory functional integrals over the p-adics I: Anomalousdimensions. Preprint, (2013).[2] M. Aizenman. Geometric analysis of '4elds and Ising models, PartsI and II. Commun. Math. Phys., 86:1{48, (1982).[3] M. Aizenman. The intersection of Brownian paths as a case study of arenormalization group method for quantum eld theory. Commun.Math. Phys., 97:91{110, (1985).[4] M. Aizenman, H. Duminil-Copin, and V. Sidoravicius. 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