Correlations in spin models and delocalization of random surfaces

Thesis

Reference

Correlations in spin models and delocalization of random surfaces

GAGNEBIN, Maxime Henri

Abstract

L'objectif de cette thèse est l'étude de plusieurs modèles de physique statistique, plus particulièrement de modèles de spins sur réseaux. Nous nous concentrons sur les corrélations entre les spins et sur la croissance ou décroissance de cette corrélation lorsque la distance entre les spins varie. Après une introduction définissant les modèles et les outils nécessaires à leur étude, chaque chapitre est basé sur un article publié ou en cours de publication et peut être lu indépendamment.

GAGNEBIN, Maxime Henri. Correlations in spin models and delocalization of random surfaces. Thèse de doctorat : Univ. Genève, 2017, no. Sc. 5113

DOI : 10.13097/archive-ouverte/unige:97005 URN : urn:nbn:ch:unige-970057

Available at:

http://archive-ouverte.unige.ch/unige:97005

Disclaimer: layout of this document may differ from the published version.

Section de Math´ematiques Professeur Yvan Velenik

Correlations in Spin Models and Delocalization of Random Surfaces

TH`ESE

Pr´esent´ee `a la Facult´e des Sciences de l’Universit´e de Gen`eve pour obtenir le grade de Docteur `es Sciences, mention Math´ematiques

par

Maxime GAGNEBIN de

Renan (BE)

Th`ese N^o5113

GEN`EVE

Atelier d’impression ReproMail 2017

mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems.

George Pólya

L’objectif de cette thèse est l’étude de plusieurs modèles de physique statistique, plus par- ticulièrement de modèles de spins sur réseaux. Nous nous concentrons sur les corrélations entre les spins et sur la croissance ou décroissance de cette corrélation lorsque la distance entre les spins varie. Après une introduction définissant les modèles et les outils nécessaires à leur étude, chaque chapitre est basé sur un article publié ou en cours de publication et peut être lu indépendamment.

Le chapitre 2 porte sur l’étude des corrélations dans un modèle de spins continus, le modèle O(N),N ≥2. Ces modèles exhibent une invariance sous les rotations locales. Il en résulte que la corrélation entre deux spins tend vers zéro avec la distance entre les deux spins. Calculer la vitesse asymptotique de cette perte de corrélation est une question encore partiellement ouverte. Dans ce chapitre, nous étendons un résultat de McBryan et Spencer permettant de majorer la corrélation par une fonction à décroissance algébrique. Notre résultat permet de traiter tous les potentiels continus, même dans le cas où l’interaction a une portée infinie.

Le deuxième résultat porte sur la délocalisation des surfaces aléatoire. Ce résultat est un résultat d’universalité prouvant une partie d’un théorème central limite deux dimen- sionnel. Nous étudions l’ensemble des fonctionsφ:Z² →Rmuni d’une mesure dépendant des gradients ∇φ. Proprement normalisé, ce modèle converge vers un objet limite, le champ libre gaussien. Cette propriété des surfaces aléatoires n’est prouvée que pour une classe restreinte de potentiel. Dans ce chapitre, nous nous intéressons à la variance de la surface en un point, comprendre cette variance est nécessaire pour renormaliser le modèle correctement. Nous prouvons une borne inférieure sur cette variance et cette borne est valable pour une classe très large d’interaction. Seule une condition d’intégrabilité est im- posée à l’interaction et cette condition se rapproche d’une condition nécessaire à l’existence du modèle.

Notre but dans les chapitres suivants est de prouver que la transition de phase du modèle de percolation FK de paramètreq est de premier ordre pourq >4. Ce résultat était connu pour q suffisamment grand (q≥27), mais la question restait ouverte pour de plus petites valeurs deq. Il est connu que pourq≤4la transition de phase est de deuxième ordre. Les chapitres 4 et 5 traitent du modèle de six sommets. Ce modèle est utilisé car il est lié au modèle de percolation FK, et comprendre le modèle de six sommets permet de récupérer des informations sur la percolation FK. Dans le chapitre 4, nous examinons l’ansatz de Bethe. Cette méthode réduit la recherche des vecteurs propres et des valeurs propres de la matrice de transfert du modèle à la recherche d’une solution d’un système d’équation.

Cette méthode est décrite dans la littérature, cet article nous permet de présenter une preuve rigoureuse et auto-contenue.

Le dernier chapitre utilise l’ansatz de Bethe et explique comment trouver une solution au système d’équation donné. Cette solution est ensuite analysée pour en déduire une

FK.

The objective of this thesis is the study of several models of statistical physics, more particularly of spin models on lattices. We focus on the correlations between the spins and on the growth or decay of this correlation as the distance between the spins varies. After an introduction defining the models and the tools necessary to their study, each chapter is based on an published article (or in preparation) and can be read independently.

Chapter2deals with the study of the correlations in a continuous spin model, theO(N) model for N ≥2. These models exhibit an invariance under local rotations. As a result, the correlation between two spins tends to zero with the distance between the two spins.

Computing the asymptotics of this loss of correlation is still a partially open question. In this chapter, we extend a result of McBryan and Spencer bounding the decay of correlation by an algebraic function. Our result allows us to treat any continuous potential, even in the case when the interaction has infinite range.

The second result concerns the delocalization of random surfaces. This result a type of universality result proving part of a central limit theorem for two dimensional objects.

We study functions φ:Z²→Requipped with a measure only depending on the gradients

∇φ. This model, properly normalized, converges to a limit object, the Gaussian free field.

This property of random surfaces is proved only for a restricted class of potential. In this chapter, we are interested in the variance of the surface at a point, to understand this variance is necessary to renormalize the model correctly. We prove a lower bound on this variance and this bound is valid for a very wide class of potentials. Only an integrability condition is imposed on the interaction and this condition is close to necessary condition for the existence of the model.

Our aim in the following chapters is to prove that the phase transition in the FK percolation model with parameter q is first order for q > 4. This result was known for q sufficiently large (q ≥ 27), but the question remained open for smaller values of q. It is known that for q ≤ 4 the phase transition is continuous. Chapters 4 and 5 deal with the six-vertex model. This model is used because it is related to FK percolation, and understanding the six-vertex model allows us to recover information on FK percolation. In chapter 4, we examine the Bethte ansatz. This method reduces the search for eigenvectors and eigenvalues of the transfer matrix to the search of a solution of a system of equations.

This method is described in the literature, this article allows us to present a rigorous and self-contained proof. The last chapter uses the Bethe ansatz and explain how to find a solution to the given system of equations. This solution is then analyzed to deduce an exact expression for the various eigenvalues of the transfer matrix. We then explain how these eigenvalues dictate the behavior of FK percolation.

Je souhaite remercier plusieurs personnes qui ont contribué à l’aboutissement de cette thèse.

Tout d’abord, merci à Yvan Velenik. Il a été un directeur de thèse toujours disponible et m’a dirigé vers un domaine des maths passionnant. La physique statistique est pleine de questions naturelles et dont la résolution est souvent ingénieuse.

Merci beaucoup à Vincent Beffara, David Cimasoni et Senya Shlosman de me faire l’honneur de constituer mon jury.

Je remercie tous mes coauteurs avec qui nous avons passé de nombreuses sessions de réflexion, soit à la rue du lièvre, au CIB ou lors d’une conférence. Hugo Duminil-Copin, Ioan Manolescu, Matan Harel, Vincent Tassion, Piotr Miłoś, Ron Peled, you have given me a sense of what it is to be an active researcher and it has been a privilege to have worked with you.

Merci à toute les personnes qui ont contribué à rendre ces années agréables, par des discussions mathématiques ou par d’autres divertissement. Mes co-bureau: Jhih-Huang et Sébastien. Mes collègues, camarades ou amis mathématiciens: Aglaia, Anthony, Caterina, les deuxième, Elise, Jérémy, Lida, Loren, Mounir, Paul-Henry, Sacha, Sandy, Talia, le Thugstier, le Tictora, les Vignouds, Xavier. Mes amis du collège: Alice, Anne-Laure, Chris, Dario, Kaya, Rachel.

Je remercie Jeanne Guenot qui a partagé ma vie à Vernier pendant ces années de thèse.

Enfin, un grand merci à ma famille pour leur soutien et leurs encouragements.

Résumé i

Summary iii

Remerciements iv

1 Introduction 1

1.1 Statistical Physics . . . 1

1.1.1 Notations and Conventions . . . 3

1.2 Electric Networks and Random Walks . . . 4

1.3 Spin Systems . . . 8

1.3.1 Theq-Potts Model . . . 8

1.3.2 TheO(N) Model . . . 13

1.3.3 The Random Height Model . . . 21

2 Upper bound on the decay of correlations in a general class of O(N)- symmetric models 27 2.1 Assumptions and results . . . 27

2.1.1 Proof of Theorem 2.1.1 . . . 30

2.1.2 Proof of Theorem 2.1.5 . . . 40

2.1.3 Percolation estimates . . . 41

2.1.4 Some technical estimates . . . 43

3 Continuous Height Functions 45 3.1 Introduction . . . 45

3.1.1 Surfaces models . . . 45

3.1.2 Rotator models . . . 46

3.1.3 Organization of the paper . . . 46

3.2 Notation and outline of the proof . . . 47

3.3 Preprocessing . . . 48

3.3.1 High-temperature expansion . . . 53

3.4 The addition algorithm . . . 55

3.5 Model on annulus . . . 56

3.5.1 Determining good configurations . . . 59

3.6 Proof of the main results . . . 62

3.6.1 Rotator model . . . 62

3.6.2 Surface model . . . 63

3.7 Proofs of the properties of the addition algorithm . . . 64

3.7.1 Model with Ω =S¹ . . . 64

3.7.2 Description of the addition algorithm . . . 66

3.7.3 Increments and Lipschitz property . . . 67

3.7.4 Bijectivity . . . 69

3.7.5 The shifts produced by the algorithm . . . 73

3.7.6 Jacobian definition . . . 74

3.7.7 Properties ofT⁻ . . . 77

3.7.8 The geometric average of the Jacobians . . . 77

3.7.9 Periodicity properties . . . 78

4 The Bethe ansatz for the six-vertex and XXZ models: an exposition 79 4.1 Introduction . . . 79

4.2 Definitions and statements of main theorems . . . 80

4.2.1 The six-vertex model and its transfer matrix . . . 80

4.2.2 Statement of the Bethe ansatz . . . 82

4.2.3 Comments on Theorem 4.2.2 . . . 83

4.2.4 The XXZ model . . . 84

4.3 Proof of the Bethe ansatz (Theorem 4.2.2) . . . 85

4.3.1 Relations satisfied by the coefficientsAσ . . . 86

4.3.2 Toroidal boundary conditions . . . 87

4.3.3 Encoding with words . . . 88

4.3.4 Proof of Theorem 4.2.2 when no entry is zero . . . 90

4.3.5 Proof of Theorem 4.2.2 when one entry is zero . . . 91

4.4 The six-vertex transfer matrix . . . 97

4.5 The XXZ Model . . . 98

5 Discontinuity of the phase transition for the planar random-cluster and Potts models with q >4 103 5.1 Introduction . . . 103

5.1.1 Motivation . . . 103

5.1.2 Results for the Potts model . . . 104

5.1.3 Results for the random-cluster model . . . 105

5.1.4 Results for the six-vertex model . . . 107

5.1.5 Organization of the paper . . . 109

5.2 Study of the Bethe Equation . . . 110

5.2.1 The continuous Bethe and Offset Equations . . . 110

5.2.2 The discrete Bethe Equation . . . 112

5.2.3 The asymptotic behaviour of the solutions to the Bethe Equation . . 118

5.3 Proofs of the theorems . . . 120

5.3.1 Perron-Frobenius eigenvalues of six-vertex model via Bethe Ansatz . 120 5.3.2 From the Bethe Equation to the six-vertex model: proof of Theo- rem 5.1.3 . . . 126

5.3.3 From the six-vertex to the random-cluster model: proof of Theo- rem 5.1.2 . . . 133

5.3.4 From the random-cluster to the Potts model: proof of Theorem 5.1.1 142 5.4 Fourier computations . . . 143

Introduction

1.1 Statistical Physics

Statistical physics is a part of mathematics based on the following idea:

macroscopic phenomena can be deduced from microscopic behavior.

Matter is composed of a very high number of very small particles. By observing a limited number of particles, one can deduce their interaction and intrinsic behavior. It is then a matter of deduction to say how the material as a whole will act. The most well known example might be the behavior of water. Water molecules are always the same, but a bucket of water is very different depending on the temperature.

The main feature of a model of statistical physics is the Hamiltonian, also called the energy. Given a configuration ω of the model, this function will return its total en- ergy H(ω) ∈ (−∞,+∞]. Physical reasonings imply that the weight we associate to a configuration should be exponentially small in the energy

w(ω) =e^−βH(ω)

We allow the Hamiltonian to take the value+∞, this just means that a configuration with such energy has weight0(and hence will almost surely never happen). The parameterβ is the inverse of the temperature; in classical physics literature we often findβ= 1/kT where k is the planck constant and T the temperature. For a function f on the configurations, we study the integral of f against our measure

[f] :=

Z

Ω

f(ω)w(ω)dω or [f] := X

Ω

f(ω)w(ω).

The normalizing constant is called the Partition function and it has been long studied in physics. It will be noted with the letter Z,

Z := [1] = Z

Ω

w(ω)dω or in the discrete case Z:= [1] =X

Ω

w(ω).

The probability of an event A is therefore given by the ratio between the mass ofA and the total mass,P(A) = [A]/Z.

Phase Transition Most models of statistical physics have one or more parameters.

Those can be the temperature, the electrical interaction or the chemical interaction between the particles. As the parameters of the system change, the global behavior may or may not change. When the temperature of a body of water goes from 10^◦C to 12^◦Cwe see little change. However, when the temperature of water goes from −1^◦C to 1^◦C then we see the solid water becoming liquid. It is important to note that the temperature was moved continuously, but that the state of the water changed suddenly, we say that the system underwent a phase transition. The value of the parameter at which such transition occur is called the critical value. It is possible that a system undergoes more than one phase transition and therefore has more than one critical value. This is the case for water that goes from solid to liquid at 0^◦Cand then from liquid to gaz at 100^◦C.

The natural world is filled with examples of phase transition. The change from solid to liquid to gaz is the most easy to observe. But many other ones exist. Ferromagnetic objects can have permanent magnetic properties at low temperature, and lose them at high temperature. In this case, the critical value is called theCurie point. Some alloys are superconductive at very low temperature, but not at higher temperature. We can also see the phenomenon of phase transition outside of physics, for example traffic can also undergo a phase transition, when adding only few more cars causes the formation of a traffic jam.

Because they are so key to the behavior of materials, phase transitions have been heavily studied in physics. Finding good models to be able to prove mathematically the existence of phase transition is difficult. Some phase transition, like the paramagnetic-ferromagnetic, have good models describing them, in this case theIsing model. Other phase transition, like the solid to liquid one, still aren’t well modeled.

Example 1.1.1. If the observable in some model isf(x) =P

k≥0x^k, then we have a phase transition at xc= 1.

Example 1.1.2. A more physically relevant example is given by the random walk on Z. Let (X_k)_k≥0, X₀ = 1 be a random walk with transition probability P(X_k−X_k−1 = 1) = 1−P(X_k−X_k−1 =−1) =p. We have the following change in behavior

• the random walk will visit 0 almost surely ifp≤1/2,

• there is a positive chance that the walk never visits0 if p >1/2.

Modeling water is beyond the reach of current mathematics. In the models considered below we make a few simplifications. First, we suppose that the particles of the material are constraint to lie on a given lattice (for example on a square grid). Second, we suppose that each particle only has one attribute that needs to be understood. The most well known example of such model is the Ising model: particles are placed on a grid and they carry electric charge (either+1or−1). The way a group of particle will interact with each other has to be specified (and finding which interactions best approximate the physical world is challenging). Models can be made as complex and intricate as one wants, there have been studies on models not assuming the above simplifications. As can be expected, it is a difficult task to represent the physical world and the more elaborate the model, the less one can say about it.

Phase Transition in the Ising Model The Ising model favors configurations where adjacent spin are equal (we say the model isferromagnetic), and this favoring will depend on the inverse temperatureβ.

• For largeβ, the model favors a lot equal adjacent spins,

• For small β, the model favors equal adjacent spins only a little.

Consider a large area totally surrounded by +1 spins, because of ferromagnetism the average spin in the area will be positive. But what should happen if the area becomes larger? It turns out that the answer to the above depends on “how much does the system

+ + + + + +

+

+ + +

+ +

+ + + + +

+ + + + +

+ + + +

+ + + + + +

+ + + + +

+ + + + +

+ +

+ + + +

+ +

+ + + + +

+ + +

+ + + + + +

+ + + +

+ +

+ + + + + +

+ + + + + + + + +

+ + + + + + +

+ + + + + +

+ + + + + +

Figure 1: In the figures above, we see that the graph becoming larger and the mesh of the lattice becoming smaller. Those two phenomena have the same effect on the behavior of the system.

favor adjacent spin to be equal”. We say that the model undergoes a phase transition at β =β_c.

Theorem 1.1.3. When β is small the average of spins will go to zero as the lattice gets bigger, but when β is large the average of the spins will stay bounded away from zero. A formal statement is given by Theorem 1.3.1 where the Ising model is the case q= 2.

In the following sections, we will define rigorously the models of study and we will properly state the quantity that we try to evaluate.

1.1.1 Notations and Conventions

− A graphG is defined by a set of vertices denotedV(G) and a set of edges denoted E(G)⊂V(G)². Graphs will be undirected,(x, y)∈E(G)if and only if(y, x)∈E(G), we’ll abbreviate this asx∼y.

− For a subset of vertices Λ⊂ G, we defineEΛ := {(x, y) ∈E(G) : {x, y} ∩Λ 6=∅} the set of edges intersecting Λ. For a sequence of subsets of vertices(Vn) and V we sayV_n⇑V if V_n→V and the sequence(V_n) is increasing.

− The boundary of a subsetW ⊂V is denoted by∂W :={w ∈W : there existsv /∈ W withv∼w}.

− T^d_Lwill be the torus of sizeL,V(T^d_L) :={0, . . . , L−1}^dand two vertices are adjacent, (v, w) ∈E(T), if their differences is 1 modulo L in one coordinates and zero in the

others. The reciprocal torus is T^?L:={2π

L(x₁, . . . , x_d) : x_i= 0, . . . , L−1}.

− We will use capital letters for fixed parameters of the model, such as the interaction Je, resistanceRe, base lattice Λ, state space Ω. Variables will be denoted by lower case letters, like the currenti_e, a random spin σ or the voltageν.

− For a pointx∈Z² we define(x1, x2) =xthe two coordinates of x and write 0= (0,0, . . .)or in general,n= (n, n, . . .)

where the dimension of the vector will be clear from the context.

− For an application a:V → R we use the gradient notation ∇_u,va=av−au, or for an oriented edge e= (u, v) we write ∇_ea =∇_u,va. The notation ∇a to denote the matrix of the gradients ofais also common, but we will not use it.

− The function1{·} is the indicator function.

1.2 Electric Networks and Random Walks

Before introducing the main objects of the thesis: spin systems, it is useful to introduce the random walk and some of its features. This section is inspired from Doyle and Snell [15] and Lawler and Limic [51].

The Random Walk. For a discrete spaceV the random walk onV is a random process S :N→V such that

P(S_n+1=v|S_n=u) =p_u,v, wherep_u,v≥0andP

vp_u,v = 1for allu∈V; we’ll denoteP= (p_u,v)_u,v. The notation for the measure on the set of random walks starting at siteu will bePu and expectation with respect to this measure will be noted Eu. The random walk can be defined and studied on many spaces. In this introduction, we will restrict ourselves to symmetric transitive walks (pu,v =pv,u =p0,v−u) on lattices with independent increments. The most common such lattices are thed-dimensional lattice,Z^d, or the discrete torus,TL. The random walk exhibits many interesting properties and we will show a few of them below. Central to the study of the random walk is Green’s function.

Definition 1. Green’s generating functionis defined asG(x, y;ξ) :=P

n≥0ξⁿPx(S_n= y). This sum is always well defined for|ξ|<1. When Green’s generating function converges for ξ= 1 we say that the walk is transientand we define Green’s functions as

G(x, y) :=G(x, y; 1) =X

n≥0

Px(Sn=y), G(x) :=G(0, x; 1) =X

n≥0

P0(S_n=x).

In the case where the above sum diverges, we say that the random walk is recurrent.

Of course, if the graph is finite, the random walk will be recurrent. The above definition of Green’s function G(x, y) is always divergent, however on a finite regular lattice, it is clear that in average the random walk will spend an equal amount of time on each site.

This quantity can be subtracted and this allows for the definition of a renormalized Green’s function on finite graphs,

Definition 2. On the d-dimensional torus of size L, T^d_L, we set GL(x, y) :=X

n≥0

Px(Sn=y)− 1 L^d

.

When the walk is transient, Green’s function can be rewritten by considering the first step of the walk

G(x, y) =1_{x=y}+X

n≥1

Px(Sn=y)

=1_{x=y}+X

n≥1

X

z

px,zPx(Sn−1 =y|S1 =z)

=1_{x=y}+X

z

px,z

X

n≥0

Pz(Sn=y)

=1_{x=y}+ (PG)(x, y).

This shows that (I −P)G =I where we recognize the Laplacian of the lattice ∆ = P−I. In the case of a finite graph, the same computation yields(I−P)G=I−1/L^d. One of the tools used to analyse the random walk and Green’s function is thecharac- teristic function of its first step,

φ(θ) :=E0

h e^iθS1i

. (1.2.1)

Proposition 1.2.1. Consider a random walk on a regular graph with independent and identically distributed increments. For |ξ|<1, we have

G(x;ξ) = 1 (2π)^d

Z

[−π,π]^d

e^−ixθ 1−ξφ(θ)dθ.

For a transient random walk, the above holds for ξ= 1, G(x) = 1

(2π)^d Z

[−π,π]^d

e^−ixθ

1−φ(θ)dθ (1.2.2)

Proposition 1.2.2. In the case of the torus T^d_L a similar equation holds, GL(x, y) = 1

L^d X

k∈T^?_L\{0}

e^ik(x−y)

1−φ(k). (1.2.3)

Proof. It is classical in group theory that 1{S_n=y}= X

k∈T^?L

e^ik(Sn−y) L^d ,

which allows to rewritePx(Sn=y).

GL(x, y) =X

n≥0

Px(Sn=y)− 1 L^d

=X

n≥0



Ex



 X

k∈T^?_L

e^ik(Sn−y) L^d



− 1 L^d



.

The termk= 0 will cancel with the1/L^d and we are left with G_L(x, y) = 1

L^d X

n≥0

X

k∈T^?L\{0}

Exe^ik(Sn−y).

This can be simplified asS_nis a sum of nrandom variable which all have the same law as S1. Moreover, a random walk starting fromxhas the same law as a random walk starting from0 and shifted byx. Hence

GL(x, y) = 1 L^d

X

n≥0

X

k∈T^?L\{0}

E0e^ik(Sn−y+x)

= 1 L^d

X

k∈T^?_L\{0}

e^ik(x−y) 1−φ(k).

Proposition 1.2.3. The simple random walk onZ^dis recurrent ford= 1,2and transient for d≥3.

The proof of this proposition is quite straightforward as soon we calculate the charac- teristic function for the simple random walk. For example, here are the asymptotics of a few characteristic functions asθ→0and the consequence on the recurrence of the random walk.

1. Nearest neighboor px,y =1_{|x−y|=1}/(2d) has1−φ(θ)∼C|θ|². The random walk is transient ford≥3and recurrent for d= 1,2.

2. Ifd= 2 and p_x,y = _kx−yk¹ s 1

then

1−φ(θ)∼C







|θ|^s−2 if s <4,

|θ|²log_|θ|¹ if s= 4,

|θ|² if s >4.

(1.2.4)

The random walk will be recurrent ifs≥4 and transient if s <4.

Another interesting quantity is the first time that the random walk visits a site v∈V, τv := min{n≥0 : Sn=v}. In general, let

τ_A:= min{n≥0 : S_n∈A}.

Define, for v₁6=v₂,

f(u) :=Pu(τ_v1 ≤τ_v2),

the probability of reaching a site v₁ before a sitev₂ when starting from u. This function has some trivial properties, f(v₁) = 1, f(v₂) = 0. On other sites, the following holds by summing over the possibilities for the first step

f(u) =Pu(τv1 ≤τv2) =X

v∼u

P(τv1 ≤τv2|S₁ =v)Pu(S₁ =v)

=X

v∼u

f(v)pu,v.

This property can be rewritten as (P−I)f(v) = 0 for all sites v 6= v1, v2. Functions satisfying this property are said to be harmonic or that such an f satisfies a Dirichlet problem,







f(v₁) = 1, f(v₂) = 0,

∆f(v) = 0 for v6=v1, v2.

(1.2.5)

Electric Network A seemingly unrelated subject is the amount of electrical current flowing through a system of wires and resistances. Denote the voltage at a pointxbyν(x) and the current fromxtoybyix,y. Suppose we ground one of the nodes (ν(v1) = 0) of the network and impose a voltage of1 at another node (ν(v₂) = 1). What is then the voltage at other nodes of the system? This question can be answered by knowing the resistance between each node and applying Kirchhoff’s and Ohm’s Laws:

1. the current through a wire is proportional to the difference of voltage, and inversely proportional to the resistance,

ix,y = ν(x)−ν(y) Rx,y

.

2. The amount of current flowing into each node must be equal to the current flowing out,

X

y

i_x,y = 0 for allx. (1.2.6)

Theorem 1.2.4. If one imposes a voltage difference on two vertices of the lattice V, then the induced voltage on the rest ofV is a harmonic function.

This is the same property as equation (1.2.5). It is straightforward to prove that there exists only one harmonic function satisfying those boundary condition. This implies that the voltages represent the probability of a random walk to reach v₁ before reaching v₂. The transition probability for this walker are given by

p_u,v = Ju,v

P

wJ_u,w, whereJu,v := 1/Ru,v is the conductance betweenu and v.

When a current flows through a circuit, it dissipates energy. For a function i:E →R we define the energy dissipated by i(also called theDirichlet energy) as

E= 1 2

X

e∈E

i²_eR_e.

As for most natural phenomena, an electrical network tries to dissipate as little energy as possible. This is the case when we derive a current in the above fashion,

Theorem 1.2.5. The current obtained by Ohm’s Laws minimizes the energy dissipation.

When looking at the network, we can also forget the fact that it is composed of many resistances. We can see some current entering at v1 and the same amount exiting at v2. We can now compute the effective resistanceof the network betweenv₁ andv₂ by

R(v₁, v₂) = ν(v1)−ν(v2) iv1,v2

.

This allows us to deduce two composition rules to calculate the effective resistance of more than one resistance – when the resistance are in series or in parallel. The behavior of a random walk can also be described by the effective resistance,

Theorem 1.2.6. The random walk on the infinite graphGis recurrent if and only if there exists a increasing sequence of sets Vn ⊂ V(G) such that 0 ∈ V₀, limnVn = V and the effective resistance between 0 andV_n^c diverges to +∞.

1.3 Spin Systems

The models examined in this thesis all fall in the family of spin systems. The ingredients of such models are

• A base graph Λ: this is the underlying graph. We will usually choose the lattice Z² or a subset of Z².

• The single-spin spaceΩwith an a priori measureµ0 on Ω. This defines the spins living on Λ. The configurations will be functions S : Λ→ Ω on which we consider the product measured^Λµ₀ :=×_i∈Λµ₀.

• Theboundary conditionsS¯ defined on parts ofΛ or onΛ^cwhen Λ is a subset of a larger graph. A usual example isΛ⊂Z² and S¯ is a function S¯: Λ^c→Ω.

• TheHamiltonianH, this function serves to weight the configurations. The measure that we study will be of the form

dµ(S) = 1

Z exp{−βH(S)}d^Λµ₀(S), Z =

Z

ΩΛ

exp{−βH(S)}d^Λµ₀(S).

Z is called thepartition function and serves to normalize the measure.

We will now define exactly these quantities for the models we study, and discuss some important results and open questions. The following section are in correspondance with the next chapters. Section 1.3.1 introduces the notions studied in Chapter 5. Section 1.3.2 introduces the O(N) Model which is studied in chapters 2 and as a side result in 3. Sec- tion 1.3.3 introduces random height functions which are the subject of Chapter 3.

1.3.1 Theq-Potts Model

The q-color Potts model (q ∈ N^∗) is obtained by Ω := {1, . . . , q} with a priori uniform measure on Ω. The set of configurations on the base Λ b Z² is Ω^Λ and the spin at u

will be σu. The boundary conditions we’ll consider will be i∈ {0, . . . , q} and for a given configuration σ∈Ω^Λ we set

H_Λⁱ(σ) :=− X

u,v∈Λ u∼v

1{σu=σv}− X

u∈∂Λ

1{σu=i}.

Notice that in the casei= 0the boundary term is always zero, this is called thefree bound- ary condition. The Hamiltonian gives rise to a measure on configurations at temperature 1/β with boundary condition i,

µⁱ_Λ,β(σ) := exp{−βH_Λⁱ(σ)}

Z_Λ,βⁱ The main question is to understand

µⁱ_β = lim

Λ⇑Z²

µⁱ_Λ,β.

In particular, we are interested in computing µⁱ_β(σ₀ = i) when i 6= 0. As in the Ising model, we have a phase transition,

Theorem 1.3.1 (Beffara and Duminil-Copin [5]). Define β_c= log(1 +√

q) then :

• For β < β_c, the measures µⁱ_β are all equal. We have µⁱ_β(σ₀ = j) = 1/q for j ∈ {1, . . . , q}.

• For β > β_c, the measures µⁱ_β are all different. We have µ^j_β(σ₀ = j) > 1/q and µ⁰_β(σ₀ =j) = 1/q for j∈ {1, . . . , q}.

This kind of phase transition is called an order/disorder phase transition. Indeed, whenβ passesβ_c we change from a regime where the spins can take any value with equal probability to a regime where a majority of spins take the same value.

Order of the Phase Transition. One can then ask what happens at β =β_c ? It was recently proved in [21]

Theorem 1.3.2. For 1≤q≤4, the measures µⁱ_β

c are all equal.

In the case q > 4 the situation should be different and the measures µⁱ_βc should be different. This was known for large values of q as first proved by Kotecký and Shlosman in [49] (or see [73] for a proof whenq≥47). In Chapter 5 we prove that this is true for all q >4.

Correlations. Another interesting question in understandingµⁱ_β is the behavior of the correlations

µⁱ_β(σ0 =σx).

The phase transition seen above can also be found in the correlations, Theorem 1.3.3 (Beffara and Duminil-Copin [5]). Define βc= log(1 +√

q) then :

• For β < βc, there existsc >0 such that

µⁱ_β(σ₀ =σx)−¹_q

≤e^−ckxk.

• For β > β_c, there existsc >0 such that

µⁱ_β(σ₀=σ_x)−¹_q

> c (independently of x).

The correlations of the Potts model can be represented by a percolation process in which the quantity |µⁱ_β(σu = σv)−1/q| is the probability that u and v are in the same cluster. This is called the random-cluster model (or Fortuin-Kasteleyn percolation) and it allows to take non integerq.

Fortuin-Kasteleyn percolation

FK-percolation (also called the random cluster model) was introduced by Fortuin and Kasteleyn in the 1970s. On a given subset Λ ⊂ Z² we consider configurations of states of edges. Each edge can be either open (ω(e) = 1) or closed (ω(e) = 0)), hence we have Ω ={0,1}^E. The boundary condition is the state of the edges outside of Λ. We will study two main boundary conditions,

• all outside edges are closed, this will be called free boundary condition and the measure denoted byφ⁰_Λ,p,q,

• all outside edges are open, this will be called wired boundary condition with the measure denoted byφ¹_Λ,p,q.

A cluster is a connected component of ω and we denote by k_i(ω) the number of clusters under boundary conditioni= 0,1. When counting the number of clusters, it is important to consider the boundary condition; indeed under wired boundary conditions, all open edges touching the boundary will be in the same cluster. We will endow the percolation

Figure 2: On the left we only have one cluster under wired boundary conditions, whereas on the right we have six clusters.

configurations with the following probability measure, φⁱ_Λ,p,q(ω) := 1

Zp^o(ω)(1−p)^c(ω)q^ki(ω) (1.3.1) where o(ω) is the number of open edges, and c(ω) the number of closed edges. In the limit Λ ⇑ Z² we have that those two measures converge to two measures φ⁰_p,q and φ¹_p,q. Understanding those two measures is central to the study of the FK model.

Let {0 ↔ ∞} be the event that the cluster of 0 is infinite. It has been proven in [5]

that this model exhibits a phase transition at p_c=p_c(q) =

√q 1+√

q in the following sense

• for p < pc we haveφ¹_p,q(0↔ ∞) =φ⁰_p,q(0↔ ∞) = 0,

• for p > p_c we haveφ¹_p,q(0↔ ∞) =φ⁰_p,q(0↔ ∞)>0.

Edwards-Sokal Coupling. There is a direct way to construct a Potts configuration with boundary condition 1from a configuration of random cluster model and vice versa:

1. Take ω a configuration of edges distributed according to φ¹_Λ,p,q. Color the cluster connected to∂Λ in color1. For each other cluster, choose a coloring uniformly (and independently) in {1, . . . , q} (every spin in one cluster takes the same value).

The spin configuration obtained this way has the same distribution as µ¹_Λ,β with q colors and β=−log(1−p).

2. Take σ a Potts configuration with q colors distributed according to µ¹_Λ,β. Indepen- dently, for each edgee= (u, v) set

• Ifσ_u6=σ_v, thenω(e) = 0.

• Ifσu=σv,then set ω(e) =

(1with probability p, 0with probabitilty 1−p.

The edge configuration obtained this way is distributed according to φ¹_Λ,p,q with p= 1−e^−β.

Using the above procedure, it is clear that for p= 1−e^−β we have φ¹_Λ,p,q(0↔x) =µ¹_Λ,β(σ₀ =σx)−1

q.

Order of the Phase Transition. For the Potts model, the order of the phase tran- sition changed at q = 4. Working with the random cluster allows us to ask wether this change happens exactly at q = 4? This is indeed the case and Theorem 1.3.2 proved in [21] remains true:

Theorem 1.3.4 ([21]). For 1≤q≤4 (q ∈R), we have φ⁰_pc,q =φ¹_pc,q. In Chapter 5 we prove the following

Theorem 1.3.5. For all q >4 we haveφ⁰_pc,q 6=φ¹_pc,q.

This result was first proved by Kotecký and Shlosman [49] in the case of very large q.

Further works were able to prove the result for q > 26 (see [73]). Our proof is based on the study of a third model: the 6-vertex model.

The 6-Vertex Model In this model the set Ωis given by the six choices:

We restrict configurations to having coherent arrows. For example if ωv = 1, then a site u just right from v will have ω_u ∈ {4,5}. The model can also be seen as arrow configurations where each vertex must have two incoming and two outgoing arrows (this

restriction is called the ice rule). In each configuration, the number of sites of type k (k= 1, . . . ,6) is calledn_k. The Hamiltonian is given by

H :=α(n₁+n₂) +β(n₃+n₄) +γ(n₅+n₆),

and we usually write the weight of a configuration directly, by setting a:=e^−α,b :=e^−β and c:=e^−γ,

w(ω) =aⁿ¹⁺ⁿ²·bⁿ³⁺ⁿ⁴ ·cⁿ⁵⁺ⁿ⁶. (1.3.2) If the configuration ω does not satisfy the ice rule, thenw(ω) = 0. The partition function is therefore given by

Z_6V(a, b, c) =X w(ω).

We will study the six-vertex model on TN,M the torus of sizeN ×M. The six vertex model could be defined on any subsetΛof Z² and the boundary condition is the direction of arrows with one endpoint inside Λand one endpoint outside Λ.

This model is said to be exactly solvable, this means that we can find an exact expression for the free energy.

Theorem 1.3.6 (Baxter [3] and chapter 5). Fix a=b= 1 andc >0. Define λ >0 such that cosh(λ) = ^c2₂⁻². Then the free energy is given by

f(c) = lim

N,M→∞

1

N M log (Z_6V(1,1, c)) = λ 2 +

∞

X

k=1

e^−kλtanh(kλ)

k .

This theorem follows from a detailed analysis of the partition function. As explain in Chapter 4, the partition function can be expressed as the trace of a matrix. The eigenvalues and eigenvectors of this matrix are found via the Bethe ansatz.

The correspondance between the 6-vertex model and FK percolation is detailed in Sec- tion 5.3.3. Like in the correspondance between random cluster and Potts model, one can obtain a 6-vertex configuration from a configuration of random cluster. This correspon- dance is classical. By carefully tracking the weight of each configuration, one finds:

Proposition 1.3.7. Let q >4 , a=b= 1 and c=p 2 +√

q. ForN, M divisible by 4 we have

X

ω∈ΩRC

w_RC(ω) 2

√q l0(ω)

q^−s(ω)= q^{N M/4} (1 +√

q)^{N M}Z_6V(N, M).

where we used

• wRC(ω) =p^o(ω)c (1−pc)^c(ω)q^k(ω) similar to equation 1.3.1,

• l₀(ω) is the number of non-retractible cluster (ie: cluster winding around the torus),

• s(ω) is the indicator of the configuration being a net:

s(ω) =

(1 ifω has a cluster that winds around the torus in both directions, 0 otherwise.

In chapter 5, we use the above proposition and the results on the six-vertex model to prove our theorems on FK-percolation.

1.3.2 TheO(N)Model

We choose Ω := S^N−1 = {S ∈ R^N : kSk2 = 1} with a priori measure being the Haar measure on S^N−1. The base graph will be Λb Z². In general, the Hamiltonian is of the form

H_Λ(S) :=−X

A⊂Z² A∩Λ6=∅

VA(S),

whereV_A depends only on the spins in A. This kind of interaction is too general, and we will restrict ourselves to two-body interaction of the following form

H_Λ(S) :=− X

u,v∈Z² {u,v}∩Λ6=∅

J_u,vV(S_u·S_v). (1.3.3)

The coupling constants are assumed to satisfyJu,v =Ju−v =Jv−u ≥0andP

x∈Z²Jx <∞;

we shall actually assume, without loss of generality, that P

u∈Z²Ju = 1. If the coupling constant satisfy

Jx = 0, for |x|>1,

we say that the model isnearest neighbor. Notice that the interaction V is taken to be invariant under simultaneous rotation of its two arguments; in other words, thatV(S_u, S_v) depends only on the scalar product Su·Sv.

The corresponding finite-volume Gibbs measure in Λ, with boundary condition S¯ and at inverse temperatureβ is then defined by

µ^S_Λ;β^¯ (d^ΛS) =







1

Z_Λ;β^S¯ e^−βHΛ(S)d^ΛS if Sy = ¯Sy for ally6∈Λ,

0 otherwise,

(1.3.4) where we used the notation d^ΛS = Q

x∈ΛdSx, with dSx denoting the Haar measure on S^N−1. The expectation with respect to this measure will be denoted h·i with indices indicated by the context.

Example 1.3.8. An important example of O(N) model is given by the choice H_Λ(S) :=− X

u,v∈Z² {u,v}∩Λ6=∅

1_{u∼v}Su·Sv.

This model is called theXY-model(or rotator model) in the casen= 2and the (classical) Heisenberg model in the case n = 3. The case n = 1 is the Ising model which is arguably the most famous statistical model.

The set of infinite volume Gibbs states arising by those Hamiltonian can be defined using classical theory (see for example [27, 35]). The results in this thesis will however all be obtained in finite volume, we will therefore only work in the finite volume setting.

Definition 3. A measureµ isrotation invariantif for every local function f and angle θ∈S^N−1 we have

hfi_µ=hf θiµ.

Where f θ(ω) =f(ω◦θ), ◦ is the group action on S^N−1 . More precisely, as we work in finite volume we will prove

|hfi_Λ− hf θi_Λ|=o_Λ(1), where the o_Λ(1) goes to zero asΛ⇑Z².

Phase Transition. The different (nearest neighbor) O(N)model exhibit a wide range of behavior:

• ForN = 1, the Ising model. This is the same as the Potts model withq = 2and the model undergoes a breaking of symmetry atβ_c.

• ForN = 2, the XY-model, we don’t have breaking of symmetry: at all temperature lim_Λ⇑Z²hS0i_Λ,β= 0. However, the correlations show a more subtle phase transition:

– Forβ large, there existsc_β >0such that we have hS₀Sxi_β ≥ kxk^−cβ. – Forβ small, there existscβ >0such thathS0Sxi_β ≤e^−cβkxk.

This is called a Berezinsky-Kosterlitz-Thouless phase transition.

• For N ≥ 3, the Heisenberg model, it was conjectured by Polyakov that the model has exponential decay at all temperatures: hS₀Sxi_β ≤ e^−cβkxk. This is one of the major open questions in statistical physics.

The absence of symmetry breaking for theO(N)model withN ≥2is called a Mermin- Wagner type results since the classical work of Mermin and Wagner [56] in which they proved rotation invariance for the XY-model. Numerous works have since been devoted to strengthening the claims and weakening the hypotheses. The following result proved in [39] is the strongest statement to date

Theorem 1.3.9 ([39]). Suppose that

• The random walk (S_n) (corresponding to the coupling constants) is recurrent.

Then all Gibbs states corresponding to the Hamiltonian (1.3.3) are rotation invariant.

The recurrence assumption is known to be optimal in general, as there are versions of the two-dimensional O(N) model for which rotation invariance is spontaneously broken at sufficiently low temperatures, whenever the random walk is transient (see page 19 for more details). The absence of any smoothness assumption onV was the main contribution of [39], such assumptions having played a crucial role in earlier approaches (see, for example, [14, 60, 30, 46, 8, 40]). We review the first such idea from page 15 onwards.

A particular consequence of the rotation-invariance is the fact that spin-spin correlations hS0·Sxi vanish askxk∞ → ∞. A natural problem is then to quantify the speed of decay of these quantities.

In the high temperature regime (β small) we have that the model exhibits exponential decay of correlations, we prove this below (see page 19). It is still open to show that the XY-model has exponential decay of correlations for all β < βc (the βc from the BKT phase transition). It is also open to show that forN ≥3 exponential decay occurs at any temperature.

We can however find other general bounds on the correlations. The first class of systems that have been studied had finite-range (usually nearest-neighbor) interactions. An upper bound on the decay of the form hS₀·Sxi ≤ (logkxk∞)^−c, c > 0, was derived by Fisher and Jasnow [24] for the model with V(S_u, S_v) = S_u·S_v. Their result was then extended by McBryan and Spencer [54], who obtained an algebraically decaying upper bound of the

formhS₀·Sxi ≤ kxk^−c/β∞ , which is best possible in general (this result is proved on page 17).

Indeed, Fröhlich and Spencer have proved an algebraically decaying lower bound of that type for the two-dimensionalXY-model (O(2)model) at low temperatures [32]. Building on [14], Shlosman managed to obtain upper bounds of the same type for a much larger class of interactions [69], under some smoothness assumption on V. More recently, Ioffe, Shlosman and Velenik showed how to dispense with the smoothness assumption, extending Shlosman’s result to very general interactionsV [39].

The first results for models with infinite-range interactions provided an upper bound à la Fisher-Jasnow [8, 40] for models with Jx such that the corresponding random walk is recurrent. An algebraically decaying upper bound was obtained by Shlosman [69] for a general class of models (with a smoothness assumption on V) in the case of exponentially decaying coupling constants. Algebraically decaying upper bounds were obtained forO(N) models by Messager, Miracle-Sole and Ruiz [57], when the coupling constants satisfyJx≤ Ckxk^−α_∞ withα >4. The results proved in Chapter 2 can be stated as

Theorem 1.3.10. The Gibbs state corresponding to the Hamiltonian (1.3.3) are rotation invariant if the interaction V is continuous (without a smoothness assumption) and if the coupling constant satisfy J_x≤Ckxk^−α_∞ with α >4.

First Ideas and Results for C² Potentials

We begin our study by a well known case. Consider the rotator model with a potential V ∈ C² and on a square part of the lattice, Λn = {−n, . . . , n}² ⊂ Z² with boundary condition Ψ : Z² → S¹. The associated space is Ωn := Ω_Λn and the associated measure will be µ_n and its meanh·i^Ψ_n. We will prove that the measureµ_n is rotation invariant.

The main idea of the proof is the following (it is a recurring one in this thesis), let A be the support of f.

If one can findτ = (τ_x)_x∈Λn such that

• τx=θ, for x∈A,

• τx= 0, for x∈∂Λn,

• |Hn(S+τ) +Hn(S−τ)−2Hn(S)| ≤on(1)uniformly inS.

thenlim_nhfi_n= lim_nhf θi_n.

The rest of this section is devoted to make this idea rigorous and to explain how in the case of theXY-model we can constructτ. Let’s start with the latter.

Choice of τ for theXY-model. ChooseM such thatA⊂ΛM and taken≥M, for x∈Λn define

τ_x=





 θ

1−τ(kxk+ 1) τ(n+ 1)

if x∈Λn\ΛM,

θ if x∈Λ_M,

whereτ(m) =

m−M

X

i=1

1

i. (1.3.5) This function has the property that τ|_ΛM = θ and τ|_∂Λn = 0. Let’s compare the energy of rotated configurations S−τ and S +τ. In the second equality, we use the fact that

V ∈C².

H(S+τ) +H(S−τ) =−X

x∼y

V(Sx+τx−Sy−τy)−X

x∼y

V(Sx−τx−Sy+τy)

=−2X

x∼y

V(Sx−Sy) +1

2V⁰⁰(ζ)(τx−τy)²

= 2H(S)−V⁰⁰(ζ)X

x∼y

(τx−τy)²

(1.3.6)

Using the given form ofτ, this last sum can be computed X

x∼y

(τ_x−τ_y)²= θ² τ(n−M + 1)²

n

X

i=M+1

8i 1

i−M 2

≤ Clog(n)

log(n+ 1)² →0. (1.3.7) We have just seen that theτ defined by (1.3.5) has the good properties stated above. Let’s now show how invariance under rotations follows. First let’s define a new measure on Ω_n

dµ^τ(S) = dµ(S−τ).

It is clear (by a change of variable) that hf θi=hfi^τ, therefore our result will follow if we show that the two measuresµand µ^τ are similar.

Relative Entropy. A good way to control if two measures are close (or similar) is via their relative entropy. For two measures µ, ν we define

h(µ, ν) :=

(h^dµ_dν log^dµ_dνi_ν, if µν,

∞ otherwise,

where ^dµ_dν is the Radon-Nikodym derivative of µ with respect to ν. The relative entropy has many useful properties:

1. Positivity: h(µ, ν)≥0 with equality if and only ifµ=ν.

2. Convexity: the map(µ, ν)7→h(µ, ν) is convex.

3. Pinsker’s inequality: for any functionf with kfk∞≤1 we have

|hfi_µ− hfi_ν| ≤p

2h(µ, ν).

4. Exponential inequality: for two measuresµ, νsuch thatνµνand a measurable eventA such thatν(A)>0we have

µ(A)≥exp

−h(ν, µ) +e⁻¹ ν(A)

ν(A).

In the case of the rotator model, the Radon-Nykodym derivative of the measuresµandµ^τ is easy to evalutate from the definition (1.3.4),

dµ

dµ^τ(S) = exp{H(S−τ)−H(S)}.

This allows us to calculate the relative entropy

h(µ, µ^τ) =hexp{H(S−τ)−H(S)}(H(S−τ)−H(S))i^τ

= 1 Z

Z

exp{H(S−τ)−H(S)}(H(S−τ)−H(S))e^−H(S−τ)dS_Λ

=hH(S−τ)−H(S)i.

Pfister remarked in [60] that it is easier to calculate h(µ, µ^τ) +h(µ, µ^−τ). Indeed one can see that this sum ish−2H(S) +H(S+τ) +H(S−τ)iand we have calculated this quantity in equation (1.3.6). Hence, we see thath(µ, µ^τ) +h(µ, µ^−τ) =CP

v∼w(τ_v−τ_w)². Which by Pinsker’s inequality implies that

|hfi − hf θi| ≤ kfk∞

r 2CX

v∼w(τ_v−τ_w)². The quantity ¹₂P

v∼w(∇_v,wτ)² is the Dirichlet energy of τ and has been analyzed in Sec- tion 1.2 (it’s the energy dissipated by seeingτ as electrical current). Using the calculation from equation (1.3.7) we see that we will have

|hfi − hf θi| ≤ kfk∞

s

2C logn

(logn+ 1)² →0 asn→ ∞.

McBryan-Spencer

McBryan and Spencer introduced the method shown below in their paper [54]. The method is similar to the spin wave method, but applying a complex shift to the spin rather that a real rotation. Rather than having S∈S¹, we will denote the spin by its angleS∈[0,2π).

The potential will be a function of the difference S_v −S_w taken modulo 2π. The proof below works for any graph and the associated Hamiltonian

H =−X

e∈E

J_ecos(∇_eS).

We begin by rewriting the correlation as an exponential.

|hcos(S0−Sx)i| ≤ |hcos(S0−Sx) + i sin(S0−Sx)i|=

1 Z

Z

e^{i(S0−Sx)−H(S)}dS . Now, thanks to the periodicity and analyticity of the functioni(S₀−S_x)−H(S), applying the (inhomogeneous) complex rotationSz →Sz+ iaz leaves the integral unchanged as long asaz= 0 for all z6∈Λ. Therefore,

1 Z

Z

e^{i(S0−Sx)−H(S)}dS =

1 Z

Z

e^{i(S0+ia0−Sx−iax)−H(S+ia)}dS

≤e^ax−a0 1 Z

Z

expnX

e∈E

Jecos(∇eS) cosh(∇ea)o dS,

where we used the identity cos(x+ia) = cos(x) cosh(a)−isin(x) sinh(a). We now recon- struct the original Gibbs measure by adding and subtractingH(S):

e^ax−a0 1 Z

Z

expnX

e∈E

Jecos(∇eS) cosh(∇ea)o dS

=e^ax−a0 1 Z

Z

expnX

e∈E

Jecos(∇eS)× cosh(∇ea)−1

−H(S)o dS

=e^ax−a0

expnX

e∈E

Jecos(∇eS) cosh(∇ea)−1o

≤expn

a_x−a₀+X

e∈E

J_e cosh(∇_ea)−1o

. (1.3.8)

We have proved that for alla: Λ→R we have,

|hcos(S0−Su)i| ≤exp (

ax−a0+X

e∈E

Je cosh(∇ea)−1 )

. (1.3.9)

This last quantity is deterministic and does not depend on the measure anymore. It is now left to optimize on ato find the best upper bound. This will be done using the theory of electrical networks.

Algebraic Decay in Transitive Graphs In equation (1.3.9) we can restrict ourselves to functions asuch that ∇_ea≤1 for all e∈E. In this case, the coshcan be bounded by a quadratic function

cosh(∇ea)−1≤(∇ea)² and we recognize P

e∈EJe(∇ea)² as the energy dissipated bya. This quantity is optimal when a is the flow given by Kirchhoff’s laws. If the differences of the effective resistance are bounded by1, the following theorem is a direct consequence of equation (1.3.9) (to the best of the author’s knowledge, this result is not present in the literature)

Theorem 1.3.11. For any regular graph such that the gradients of the effective resistance are bounded by 1, we have that

|hcos(S_v−S_w)i| ≤exp{−R(v, w)

4 }

Proof. Setbv = 1and bw= 0 and letb be defined by Kirchhoff’s law for all other vertices.

Set thena=−bR(v, w)/2. Therefore

|hcos(S_v−S_w)i| ≤exp (

a_v−a_w+X

e∈E

J_e cosh(∇_ea)−1 )

≤exp (

−R(v, w)/2 +R(z, w)² 4

X

e∈E

Je(∇eb)² )

≤exp{−R(v, w)/4}

This shows that having control on the effective resistance of your graph will bound the correlations of the XY-model. For example, if the graph is recurrent (R(v, w) → ∞ as

|v−w| → ∞) then the correlations will vanish. This Theorem holds for a large class of regular graph, including forZ².