Thesis
Reference
On Ornstein-Zernike theory and some applications
OTT, Sébastien
Abstract
L'objet de cette thèse est l'étude des corrélations entre fonctions locales dans des modèles de physique statistique classique, hors du point critique. L'outil principal est la théorie Ornstein-Zernike.
OTT, Sébastien. On Ornstein-Zernike theory and some applications. Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5345
DOI : 10.13097/archive-ouverte/unige:119877 URN : urn:nbn:ch:unige-1198772
Available at:
http://archive-ouverte.unige.ch/unige:119877
Disclaimer: layout of this document may differ from the published version.
UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES
Section de Mathématiques Professeur Y. Velenik
On Ornstein-Zernike Theory and some Applications
THÈSE
présentée à la Faculté des sciences de l'Université de Genève pour obtenir le grade de Docteur ès sciences, mention mathématiques
par
Sébastien OTT
de
Winterthur (ZH)
Thèse N◦ 5345
GENÈVE
Atelier d'impression ReproMail 2019
Contents
0.1 Résumé . . . 7
0.2 Abstract . . . 8
0.3 Remerciements . . . 9
0.4 Notations and Conventions . . . 10
1 Introduction 11 1.1 Random Fields on a Lattice . . . 11
1.1.1 Some Motivations . . . 11
1.1.2 A Few Examples . . . 12
1.1.3 Observables . . . 14
1.1.4 Correlations . . . 14
1.1.5 Graphical Representation . . . 15
1.2 Ornstein-Zernike Theory . . . 15
1.2.1 The Renewal Equation . . . 16
1.2.2 Separation of Masses . . . 16
1.2.3 The Modern Picture . . . 16
1.3 More on Correlations . . . 18
1.3.1 The High-Temperature Picture . . . 18
1.3.2 The Ambient Field Picture . . . 18
1.3.3 The Low-Temperature Picture . . . 19
1.3.4 Energy-Energy Correlations . . . 19
1.4 Inhomogeneities in Ising and Potts Models . . . 20
1.4.1 Interfaces in Dimension Two . . . 20
1.4.2 Correlations . . . 22
2 Models to be Considered and their Basic Properties 25 2.1 The Ising and Potts Models . . . 25
2.1.1 Ising Model . . . 25
2.1.2 Boundary Conditions and Spatial Markov Property . . 26
2.1.3 Correlation Inequalities for the Ising Model . . . 26
2.1.4 Potts Model . . . 27
2.1.5 Innite Volume Measures . . . 28
2.2 The Random Cluster Model . . . 28
2.2.1 Boundary Conditions . . . 29
2.2.2 Basic Properties . . . 29 2.2.3 Random Cluster Model on Zd and Percolation Transition 31
2.2.4 Weak Exponential Ratio Mixing when β < βc . . . 32
2.2.5 Edward-Sokal Coupling . . . 37
2.2.6 Planar Duality . . . 38
2.3 The Random Current Model . . . 39
2.3.1 Motivation: the Link with the Ising Model . . . 40
2.3.2 Percolation Associated with Currents . . . 40
2.3.3 Parity Functions Associated to Currents . . . 41
2.3.4 Random Current Associated to the Ising Model with a Positive Field . . . 41
2.3.5 Weak Exponential Ratio Mixing in the Random Current with a Positive Field . . . 42
2.4 The High-Temperature Expansion of the Ising Model . . . 44
2.4.1 The Expansion . . . 44
2.4.2 Path Extraction . . . 45
2.4.3 Link with Random Current . . . 47
3 Modern Ornstein-Zernike Theory for Percolation-Type Mod- els 49 3.1 The Problematic . . . 49
3.2 The Inverse Correlation Length . . . 51
3.2.1 Ising Spins: Decay Rate for Two Point Function Through GKS . . . 51
3.2.2 Decay Rate for Percolation Models with FKG . . . 52
3.2.3 More on the Ising Model: Existence Via Random Current 52 3.2.4 Transfer Matrix Representation of Truncated Two Point Function Using Reection Positivity . . . 54
3.2.5 Decay Rate for Truncated Correlations Through Reec- tion Positivity . . . 55
3.2.6 Comments and a Few Open Questions . . . 56
3.3 Convex Shapes Linked to the Inverse Correlation Length . . . 56
3.4 Percolation Models and Coarse Graining . . . 60
3.4.1 Assumptions on Percolation Models . . . 60
3.4.2 The Coarse Graining . . . 62
3.5 Lattice Embedded Trees and Tree Measures . . . 64
3.5.1 Control via Energy . . . 65
3.5.2 Surcharge of Trunks . . . 66
3.5.3 Cone-Points of Trunks . . . 68
3.5.4 Cone-Points of Trees . . . 70
3.5.5 Concluding Comments and Remarks . . . 72
3.6 Local Surgery . . . 72
3.7 Irreducible Decomposition and Factorization . . . 75
3.7.1 Irreducible Decomposition . . . 75
3.7.2 Factorization of Measure . . . 78
3.8 OZ Theory for Random Current Model: double random current, h >0 . . . 83
3.8.1 The Coarse-Graining . . . 84
3.8.2 Local Surgery . . . 90
3.8.3 Finishing the Proof . . . 91
3.9 Some Applications of the OZ Representation . . . 91
3.9.1 OZ Pre-Factor . . . 91
3.9.2 Analiticity of ξ in the Direction . . . 94
4 Application of OZ Theory I: Asymptotic of Even-Even Corre- lations in Sub-critical Ising Model 97 4.1 About the Problem . . . 97
4.2 Results . . . 100
4.2.1 Conventions for this Chapter . . . 101
4.3 Upper Bound . . . 102
4.4 Lower Bound . . . 105
5 Application of OZ Theory II: Potts Model with a Defect Line111 5.1 The Model . . . 111
5.2 About the Problem . . . 112
5.3 Summary of Results . . . 113
5.4 Basic Facts and Tools . . . 116
5.4.1 A Monotone Coupling . . . 117
5.4.2 A Second Monotone Coupling . . . 118
5.4.3 A Decoupling Inequality . . . 119
5.4.4 Existence of ξx0 and Fx0 . . . 119
5.4.5 Uniform Exponential Decay . . . 121
5.4.6 Continuity of ξx0 and Fx0 . . . 125
5.4.7 OZ Representation . . . 126
5.5 Upper Bound on Free Energy . . . 126
5.6 Lower Bound on Free Energy . . . 130
5.7 The Localized Regime Behaviour . . . 134
5.7.1 Coarse Graining Procedure . . . 134
5.7.2 Renewal on L . . . 136
5.7.3 Strict Monotonicity of ξx0 . . . 138
5.7.4 Ornstein-Zernike Correction: Pure Exponential Decay . 139 5.7.5 Analyticity of the Free Energy . . . 140
5.7.6 Interface localization . . . 143
5.8 The Delocalized Regime . . . 144
5.8.1 Upper Bound . . . 144
5.8.2 Lower Bound . . . 146
A ToolBox 149 B Random Walks Estimates 151 B.1 Walks to be Considered . . . 151
B.2 Local Limit Theorem . . . 152
B.3 Random walk estimates . . . 152 B.4 Walks with Non-Zero Endpoints . . . 157 B.5 Synchronization of Walks . . . 158
C Factorization of Measures 161
C.1 Setting, Notations and Denition . . . 161 C.2 The Memory Percolation Picture . . . 163 C.3 Decoupling of Random Sequences . . . 166 C.4 Application to Random Walks with Exponentially Decaying
Memory . . . 171
0.1 Résumé
Cette Thèse traite des corrélations dans certain modèles de spin sur réseau non-critiques. Plus précisément, le but est d'étudier le comportement n de certaines covariance dans les modèles d'Ising et de Potts. L'outil récurrent est la théorie d'Ornstein-Zernike originellement conçue pour étudier la fonc- tion à deux points tronquée dans des modèles de gaz dans un espace continu.
Le Chapitre 3 présente une construction de cette dernière dans le cadre de modèles de percolation sur réseau et du courant aléatoire, une représentation graphique encodant les corrélations du modèle d'Ising. Cette construction, qui est l'amélioration d'une approche développée entre 1998 et 2008, est l'objet autour duquel les preuves des autres chapitres gravitent.
Le Chapitre 2 présente les modèles qui seront l'objet de l'étude réalisée et en énumère les propriétés qui seront utilisées dans les chapitres suivants. En particulier, les propriétés de mélange de plusieurs modèles sont étudiées en détail.
Dans le Chapitre 4, nous considérerons les covariances paires-paires du modèle d'Ising à haute température. Ces covariances apparaissent comme une version généralisée des corrélations énergie-énergie et possèdent un comporte- ment très diérent de la fonction à deux point. Le traitement de ce problème passe par la représentation de ces covariances par de multiple connexions en in- teractions, le comportement de ces connexions est ensuite étudié via la théorie d'Ornstein-Zernike.
Finalement, le Chapitre 5 est une étude du comportement de la fonction à deux points dans le modèle de Potts face à l'introduction d'inhomogénéités, présentes sous forme de modication des constantes de couplage le long d'un axe de coordonnées. La présence d'un changement drastique de comportement selon la modication eectuée (liée à la transition de pinning pour les marches aléatoires) est prouvé en toute dimensions plus grande ou égale à deux. En dimensions deux et trois, une étude détaillée du comportement critique est ef- fectuée. La preuve est une formalisation de l'idée suivante: les covariances, à la fois dans le modèle homogène et dans le modèle inhomogène, sont représentées par des connexions dans le modèle de percolation FK. On étudie le change- ment de comportement dans les corrélations en voyant l'inhomogénéité comme un potentiel agissant sur les connexions du modèle homogène. Ces connexions dans un potentiel sont ensuite étudiées à l'aide de la théorie d'Ornstein-Zernike.
0.2 Abstract
This Thesis's topic is the study of correlations in some o-critical lattice spin models. To be more precise, the goal is the ne study of some covariances in the Ising and Potts model. The recurrent tool is the Ornstein-Zernike theory, originally designed to study the covariances between local densities in gas models in a continuous space. Chapter 3 constructs such a theory for some lattice percolation models and for the random current model, a graphical representation encoding the Ising model correlations. This construction, which is an improvement over an approach developed between 1998 and 2008, is a central tool in the proofs present in the remainder of the Thesis.
Chapter 2 presents the models under investigation and their properties. In particular, a detailed study of their mixing properties is made.
In Chapitre 4, we consider even-even covariances of the Ising model in the high temperature regime. These covariances appear as a more general version of the energy-energy correlations and exhibit a very dierent behaviour from the two point function. This problem treatment goes by representing these covariances by multiple interacting connections, these connections behaviour is then studied using Ornstein-Zernike theory.
Finally, Chapter 5 is a study of the two point function in the high tem- perature Potts model when introducing inhomogeneities, under the form of a modication of the coupling constants along a coordinate axis. The presence of a drastic change of behaviour depending on the magnitude of the change (linked to the pinning transition for random walks) is proved in all dimensions greater or equal to two. In dimensions two and three, a detailed study of the critical behaviour is done. The proof is a formalization of the following idea: covariances, in both the homogeneous and the inhomogeneous models, are represented by connections in the Random Cluster model. We study the correlations change of behaviour by seeing the inhomogeneity as a potential acting on the connections in the homogeneous model. These connections in a potential are then studied using Ornstein-Zernike theory.
0.3 Remerciements
Tout d'abord, je souhaiterais remercier Yvan Velenik pour m'avoir suivi durant ces quatre ans, les (nombreuses!) discussions que nous avons eu ont grande- ment contribué à ma compréhension actuelle des mathématiques et au plaisir que j'ai eu à faire cette thèse.
Ensuite, je tiens à remercier Hugo Duminil-Copin pour ses conseils et son dynamisme qui contribue de manière essentielle à l'ambiance du groupe de probabilité de Genève. Je le remercie également d'avoir accepté de faire partie de mon jury.
De même, je remercie Fabio Toninelli de me faire l'honneur d'être membre du jury.
Mathematics are more enjoyable when shared, I thus thank all people I had the pleasure to mathematically interact with during those four years. In particular, I would like to thank Dima Ioe for a very nice stay in Technion.
Les mathématiques c'est aussi beaucoup d'échecs, les échecs ça se passe mieux à plusieurs. Je remercie donc mes amis pour leur présence et leur soutien, en particulier Anthony Conway et Jeremy Dubout, amis et collègues depuis mes débuts universitaires, ainsi qu'Elise Raphael, Sandie Moody et Maxime Gagnebin.
Merci également aux amis non-mathématiciens: les grimpeurs, les Võ Sinh, les joueurs de go... et les autres!
Finalement, je remercie ma famille; en particulier mes parents, sans qui je ne serais pas là aujourd'hui... et par induction étend ces remerciements aux générations précédentes.
0.4 Notations and Conventions
The following conventions will be applied for this thesis:
• Constant c, C will be generic positive constants that can change from line to line. They never depend on parameters relevant to the analysis at hand.
• We will write on(1) a quantity that is negligible as n becomes large, i.e.
f =on(1) if limn→∞f = 0.
• To lighten notations, integer part will almost systematically be omitted.
When a point xwith non-integer coordinates is used as being inZd, it is meant that the considered point is the site in Zdclosest to x(with some arbitrary, xed, breaking of draws).
• Unless explicitly stated, the order ≤ on {f : Ω → R} is the point-wise partial order:
f ≤g if f(x)≤g(x) for all x∈Ω
• Weighted graphs G are the data of a set of vertices VG and a weighting J : VG×VG → R≥0. The set of edges induced is EG =
{i, j} ⊂ VG : Jij >0 .
• Two vertices i, j ∈ VG are adjacent if {i, j} is an edge (equivalently Jij > 0). We denote it i ∼ j. Summing over i ∼ j is understood as summing over unoriented edges.
• For Ga graph and H ⊂G, dene three notions of boundary forH:
∂extH =
v ∈VG\VH :∃v0 ∈VH,{v, v0} ∈EG ,
∂intH =
v ∈VH :∃v0 ∈VG\VH,{v, v0} ∈EG ,
∂edgeH =
{v, v0} ∈EG :v ∈VH, v0 ∈VG\VH .
• The rst person of the plural will be almost systematically used in the text even though there is only one author. A justication can be provided as follows. When someone is reading, there are at least two people involved in the text: the author and the reader, using of the we pronoun therefore feels appropriate...
Chapter 1 Introduction
The purpose of this introductory Chapter is to motivate and present the prob- lems which will be treated in the remainder of the Thesis as well as go over the results proven in the next Chapters. As each Chapter contains some in- troduction to the problematic it is concerned with, the aim is here directed to emphasize the link between the dierent problems and to provide the reader an intuition on the way the proofs are conducted. Rigorously introducing ev- ery quantity and object of interest feeling too heavy for an introduction, the choice of presentation is to heuristically introduce the problems and objects as well as the strategies of proof. Theorems will thus be only referred to and the claims will be stated in a loose way as Results that will be a reformula- tion of the Theorems in the informal language used in this rst Chapter. In that way, readers only interested by an overview of ideas can go through the introduction while the readers interested by what is actually proven can go in the concerned Chapter to nd denitions of the objects, Theorems and proofs of the result they are interested in.
Although it could have been reasonably included in this introductory Chap- ter, a discussion on the pressure or other thermodynamic quantities will be omitted as this Thesis is mostly concerned with correlations functions and the correlation structure of lattice models. Apologies are made to the reader this choice hurts!
1.1 Random Fields on a Lattice
1.1.1 Some Motivations
In order to study many physical phenomena, it is a natural procedure to search for models replicating the behaviour of the real world. Statistical me- chanic is concerned with nding microscopic models (modelling the behaviour of elementary constituents of the system -atoms, molecules, electrons...-) that display the same macroscopic (statistical) behaviour as the real world. Ex- amples contain
• hard spheres with attractive interactions, used to mimic the behaviour of uids such as the liquid/gas phase transition,
• lattice constrained models modelling the behaviour of electrons in a metal or crystal,
• percolations models (discrete and continuous) modelling erosion or cor- rosion...
Lattice models appear as both models of independent interest or as discretiza- tion of continuous systems that are often harder to study. A generic way to construct a lattice models (say on Zd) is to
• Choose the value that a given site in the lattice can take (e.g. {0,1} for a one species gas: 0 when no particle is close to the site, 1 when a particle is; {−1,+1} for a magnet giving the overall agreement of the magnetic moment of a particle with an a priori eld...), denoteS the set of possible values. The congurations we look at are thus Ω =SZd.
• Dene an energy function H : Ω → R (the Hamiltonian) giving the energy of a particular conguration.
• Nature favouring low energy state, one can then construct a probability measure by giving a weight e−βH(σ) to a conguration σ ∈Ω (the exact form of the weight being prescribed by entropic considerations) where β = 1/T is the inverse temperature. Let µ be the probability measure on Ωsuch that µ(σ)∝e−βH(σ).
• Now, the typical congurations of the system at inverse temperature β are the ones that have statistical properties happening with high prob- ability. A measurement made is a laboratory can be approximated by computing the average value of the measurement under µ.
1.1.2 A Few Examples
Let us give a few very classical models as illustration. We will come back to some of them later on.
Lattice Gas
In this model, the spin space is {0,1} (particle or no particle) and particles interact via a pair potential: for a conguration n ∈Ω ={0,1}Zd,
H(n) =− X
{i,j}⊂Zd
Kijninj− µ β
X
i∈Zd
ni,
where µ is the chemical potential and Kij ≥ 0 decays suciently fast as function of ki−jk. This model displays a liquid/gas transition: for β large
enough, the density of particles has a discontinuity as a function of the chemical potential (i.e., the system goes from a dilute to a dense phase abruptly). This model is closely related (in perfect correspondence) with the Ising model, which is our next example.
Ising Model
Here, the state space is{−1,+1}and the Hamiltonian is (forσ ∈ {−1,+1}Zd) H(σ) =− X
{i,j}⊂Zd
Jijσiσj −hX
i∈Zd
σi,
where h ∈ R is the magnetic eld and |Jij| is suppose to decay fast enough.
The case Jij ≥ 0 is the ferromagnetic Ising model (which is discussed in Section 2.1) while Jij ≤ 0 is the anti-ferromagnetic Ising model. As for the lattice gas, the ferromagnetic model displays a magnetic phase transition: for β small enough, the average magnetisation is discontinuous as a function of h. This model is used to study the behaviour of magnets in a eld acting in a xed direction.
Potts Model
The Potts model is a possible generalization of the Ising model (with h= 0):
the spin space is now {1, ..., q} (often called colours) for q ≥ 2 integer (the Ising model is q = 2). This model is discussed in more details in Section 2.1.
The Hamiltonian is
H(σ) = − X
{i,j}⊂Zd
Jij1σi=σj.
Again, one can consider the ferromagnetic or the anti-ferromagnetic version of the model (that display very dierent behaviour). As one varies the inverse temperature, the model goes from an equilibrated phase (the average density of each colour is 1/q) when β is small (smaller than a transition value βc) to an ordered phase (one of the colour has a larger density than the others) when β > βc.
Spin O(N) Model
This is another possible generalization of the Ising model where the spin can now take value in theN−1-dimensional sphere: S =SN−1 (N = 1 is the Ising model) and the Hamiltonian is
H(σ) =− X
{i,j}⊂Zd
Jijσi·σj −X
i∈Zd
h·σi, whereh∈RN and · is the scalar product.
Figure 1.1: The transition in the 4-states Potts model. Left: β = 0.4, right β = 1.2. The symmetry between phases is broken by forcing the boundary spins to be blue in both cases.
1.1.3 Observables
Study a system in a laboratory is to measure quantities of interest. The mathematical analogue of this procedure is to compute expectation value of functions of the congurations. The rst thing is to nd out what are the functions we could compute the expectation of, the second thing is to nd out what are the functions we want to compute the expectation of. The real study is to compute the expectations we can and to deduce something about the model... The rst point has a natural answer: (both with regard to has the function a well dened expectation value? and with regard to is it a function one could measure in a lab?) take the algebra of functions generated by the real-valued functions with nite support (support = minimal set of site whose value determines the value of the function, see Section 2.1). These functions contain the average density for the lattice gas, the average magnetisation for the Ising model. More generally, they contain the macroscopic observables one can hope to measure in a lab.
A rst layer of study of a model can be done through the study of ob- servables representative of the physical phenomenon we want to study.
1.1.4 Correlations
Studying the correlation structure of a model is another way to look at the model. Here the focus is not on how do a few observables behave but rather to study how do dierent regions of space interact. The question is thus not about the average value of a xed function but on how much information a measurement done in a region of space A gives information about a region B?. A rst way to look at this question is to study covariances between two local functions, where now the parameters of interest are not the functions but their supports and the distance between them: for ∆a xed nite set of sites
containing 0 and ∆x =x+ ∆ the translate of∆ byx, how does Cov(f, g) = µ(f g)−µ(f)µ(g)
decay as a function of x where f is any function supported on ∆ and g is any function supported on∆x? For many models at high temperature the be- haviour of covariances is exponentially decaying in kxk while critical models are characterized by power law decay of correlations. The focus of Chapter 3 will be the presentation of a procedure allowing to go beyond the exponen- tial decay in high temperature model and to nd the correction term to this exponential decay.
Correlations between two regions of space is only the rst level of the correlation structure study, further information is obtained by studying higher order truncated functions (called Ursell functions or cumulants). This will not surprise readers used to the study of correlated random variables family as cumulants are a fairly natural way to control dependencies. Givenn functions f1, ..., fn, the n-Ursell function of f1, ..., fn is dened as
Un(f1, ..., fn) = ∂n
∂tn...∂t1
log
µ ePni=1tifi
t1=...=tn=0.
A precise study of the behaviour of these higher order correlations remains restricted to a few models in restricted regimes of temperature. A reasonably complete study being missing even in the most studied case of the Ising model, we will not spend more time on these. When talking about correlations in the remainder of the Chapter, we mean covariances.
1.1.5 Graphical Representation
We are interested in the study of correlation functions. For us, a graphical representation ofCov(f, g) is a pair Ω, q where Ωis a set of connected graphs and q: Ω→R+ is a weighting such that:
Cov(f, g) = X
γ∈Ω γ∩suppf6=∅ γ∩suppg6=∅
q(γ), (1.1)
wheresuppf is the support of f. Some examples of these representations are:
the High-Temperature representation of the Ising model (see Section 2.4), the Random Cluster model associated to the Potts model (see Section 2.2), the Random Current representation of the Ising model (see Section 2.3), the BFS representation (covering many models including Ising, spinO(N),... see [18]).
1.2 Ornstein-Zernike Theory
In what follows, we will assume translation invariance of µ. The Ornstein- Zernike theory is a heuristic on the structure underlying the transfer of infor- mation in models. It originates in the works of Ornstein and Zernike [59, 72]
where they postulated a given structure and, under hypotheses believed to hold in many high-temperature models, they derived the precise asymptotics of the covariance between local densities in classical uid models. In the language of the lattice gas introduced previously, they (non rigorously) derived
Cov(n0, nx) = ψ(x/kxk)
kxk(d−1)/2e−kxkξ(x/kxk)
(1 +o(1)), (1.2) where ξ(s) =−limN→∞ 1
N log Cov(n0, nN s) for s ∈Sd−1 (the existence of this quantity will be discussed in Section 3.2). This theory is well suited to study covariances between one-point observables (functions with support being a singleton).
1.2.1 The Renewal Equation
The rst idea of OZ was that the correlation displayed a renewal structure.
We formulate it again in the language of the lattice gas. Write G(x, y) = Cov(nx, ny). They postulated the existence of a functionD such that
G(x, y) = X
z∈Zd
D(x, z)G(z, y). (1.3)
Under the right hypotheses on G, D, the equation can be studied and gives their prediction (1.2).
1.2.2 Separation of Masses
We present here briey the main hypotheses of OZ, called the separation of masses. It is simply the fact that
G(x, x+ks) =e−kξ(s)(1+ok(1)),
D(x, x+ks) = e−kc(s)(1+ok(1)), (1.4) with c(s) < ξ(s) for all s ∈ Sd−1 (i.e. G decays exponentially and D decays exponentially faster thanG). In words, it says that to vehicle information from x to y, one uses a linear (in kx−yk) number of jumps using D. One can then see G as the green function for a massive random walk with transitions D (this is not the way the equation is originally presented but this is the way we will see it in Chapter 3).
1.2.3 The Modern Picture
We start by a review (non-exhaustive) of rigorous works based on OZ idea (renewal equation + separation of masses). The rst rigorous result are due to Abraham and Kunz [4] and Paes-Leme [63] where they derived (1.2) for a class of gas models (lattice and continuous) in a perturbative regime (very
large temperature). Then, it was developed by Abraham, Chayes and Chayes in [3] for study the surface correlation (the probability that two distant points lie in the same surface) for models of random surfaces. Follows the work of Chayes and Chayes [29] where they treated self avoiding walk at all non-critical temperatures. The approach presented in Chapter 3 started with the work of Ioe [51] on self avoiding walk (reproducing the results of [29]), where the sep- aration of masses was proven non perturbatively (in the whole regime where the Green function of the self avoiding walk is decaying exponentially). He introduced some tilting of measure ideas that were subsequently used by Cam- panino and Ioe in [24] where the OZ asymptotic was proven for connections in sub-critical Bernoulli percolation. This last work introduced a new way to prove separation of masses. Building on this work, Campanino, Ioe and Velenik extended the result for the High Temperature expansion of the Ising model in [25], giving the rst non-perturbative rigorous derivation of the OZ asymptotic in a lattice spin system. The approach was then extended by the same authors to Potts model [27] via the derivation of the asymptotic of the connection probability for the Random Cluster model. An improvement of the construction done in [27] was realized in [62], transforming the renewal with memory that was obtained in [27] by a real renewal structure.
On the perturbative side, we can mention [22, 23] that derived the OZ asymptotic for nite connections in (very) super-critical Random Cluster model (the spirit is closer to the study of random surfaces but the arguments follow anyway the general OZ picture).
We will now concentrate on the approach developed in [51, 24, 25, 27, 62].
The goal is to construct a functionD satisfying (1.3) and (1.4) with the help of graphical representations. Moreover, to do it in the whole regime where G decays exponentially. For the discussion, we x a functionf with support{0} and we will look at the covariance between f and its translate by x, denoted fx. The idea is to use a suitable graphical representation of correlations (as in (1.1)) to get
Cov(f, fx) = X
γ30,x
q(γ) (1.5)
and to show that the graphs contributing to the sum are mainly assimilable to a line going from0toxin a directed fashion with sub-linear uctuation in the direction transverse tox/kxk. The next step is to show that these graphs can be decomposed in smaller pieces of morally bounded size such that a graphγ in (1.5) is a concatenation of small pieces: γ = γ1 ◦γ2...◦γm and such that the weight q(γ) = ˜q(γ1)...˜q(γm) (this gives the renewal equation (1.3)) where
˜
q has a decay rate strictly larger than q (which yields separation of masses (1.4)). In other words: P
γ3x,yq(γ) plays the role of G(x, y) (by denition) andP
γ1:x→yq(γ˜ 1)plays the role ofD(x, y). This is an overly simplied picture, but it conveys the idea.
We will call the data of a graphical representationΩ, q and a set of pieces with a weight( ˜Ω,q)˜ satisfying the above, an OZ theory for Cov(f, fx). Chap-
ter 3 is concerned with the realization of such a programme for a family of percolation models and a graphical representation of Cov(σ0, σx) in the Ising model with a eld.
Result 1. One can construct an OZ theory for:
• a class of percolation models, including the Random Cluster model,
• the eld-free connections in the Random Current model with a eld.
The corresponding Theorems are Theorems 3.4 and 3.6 in Chapter 3. In particular we can deduce (see Theorem 3.7)
Result 2. In the high temperature regime of the Potts model and in the positive eld regime of the ferromagnetic Ising model, truncated two-point functions have the asymptotic 1.2.
1.3 More on Correlations
In this Section we describe heuristically (and supercially) what is the phe- nomenon responsible for the exponential decay of covariances in dierent regimes of temperature and eld (we have the Ising model in mind, but the discussion should be similar for other models). We will base the discussion on the idea that there exists a graphical representation of the form (1.1) and we will consider the same type of covariances as in the previous Section (between one-point observables).
This short discussion can be seen as a way of illustrating what can be the diculties encountered while implementing the lastly described approach to OZ theory to other regimes than the high temperature one.
We will conclude the Section by discussing how OZ theory can be adapted to understand more complicated functions than the two-point function.
1.3.1 The High-Temperature Picture
In the High Temperature regime, the graph appearing in (1.1) themselves will cost to the system: the weight q will be small on large graph because of the intrinsic will of the system to be disordered. One can think of the information being transferred as a massive random walk. In this regime, the weight qoften has a simple structure: it is the cost of the graph itself (think q(γ) = e−c|γ|
with c large).
1.3.2 The Ambient Field Picture
This case is slightly more subtle: the decay now comes from the fact that the truncation removes the eect of the eld on the spins. Information transfer from site to site has to be done without interacting with the eld. qshould thus
look like the probability for a random walk path to survive in an (annealed) environment of traps (which displays the same behaviour as in the previous Section). The diculty is that now the decay does not come from the graph itself but from its interaction with something, which is often harder to deal with technically speaking.
1.3.3 The Low-Temperature Picture
This last regime is the hardest to study. The exponential decay comes from the fact that the truncation removes the eect of innitely far, the covariance is thus governed by a transfer of information that has to stay free from infor- mation coming from innitely far. At the level of the weightq, this means that the weight encompasses the interaction of the graph with everywhere present paths of informations coming from innity. The decay should thus come from the fact that we disconnect this information from the background, the graphs γ have to be thought of as graph plus disconnecting tube. Factoring out the eect of the background is a heavy task and this regime remains the least understood of the three cases mentioned here: it is only solved perturbatively in [17].
1.3.4 Energy-Energy Correlations
Energy-energy correlations appear naturally in most models with nite range two-body interactions: if the Hamiltonian is of the form
H(σ) = −X
{i,j}
JijV(σi, σj),
and {i, j},{u, v} are such that Jij6=0 and Juv 6= 0, energy-energy correlations between {i, j} and {u, v} measure the inuence of the energy produced by the pair {i, j} on the energy produced by the pair {u, v}. In what follow, we will restrict the discussion to the nearest-neighbour ferromagnetic Ising model (V(σi, σj) = σiσj) and to the Potts model (V(σi, σj) = 1σi=σj). Take e={0, i}an edge containing0andex ={x, i+x}its translate by x. We want to studyCov(V(e), V(ex)). For such correlations, one expects (and can prove in the Ising model, see [57, 58] for other models in very high temperature - perturbative- regimes) that the correlations are represented by pairs of graphs, i.e.:
Cov(V(e), V(ex)) = X
γ:γ30,x γ03i,i+x
q(γ)q(γ0)I(γ, γ0), (1.6) where I is an interaction term between γ and γ0. Depending on whether I is attractive (the weight is large ifγ∩γ0 is big) or repulsive, dierent behaviours are expected going from the usual OZ-type asymptotic to path interacting through hard core exclusion giving a very dierent behaviour (as is the case in
the Ising model). Chapter 4 is concerned with (a class of covariances containing the) energy-energy correlations for the nite range Ising model.
Result 3. In the high-temperature ferromagnetic Ising model, c−
ψd(kxk)e−2kxkξ(x/kxk)
≤Cov(V(e), V(ex))≤ c+
ψd(kxk)e−2kxkξ(x/kxk)
, where ξ is the decay rate of the truncated two-point function and
ψd(n) =
n2 d= 2
(nlog(n))2 d= 3
nd−1 d≥4
The corresponding Theorem is Theorem 4.2. The proof goes by nding graphical representations (1.6) as upper and lower bounds with I a hard core exclusion interaction. The sums over γ, γ0 are then handled using the OZ theory in each of the graphical representations.
1.4 Inhomogeneities in Ising and Potts Models
This last Section of the introduction is devoted to present the topic treated in Chapter 5. It concerns the reaction of the system to the introduction of inhomogeneities. We present here the general ideas employed in solving these kind of problems. For this discussion, we will have in mind the near- est neighbour homogeneous Potts model on Zd: σ ∈ {1, ..., q}Zd, H(σ) =
−P
{i,j}:ki−jk1=11σi=σj. The inhomogeneity will be modelled by modifying the coupling constant Jij from 1 to some value J along a given set of nearest- neighbour pairs E. In Chapter 5 we treat the case where E is the set of nearest neighbour pairs belonging to the rst coordinate axis, denoted L but the strategy should work for any geometry of E.
1.4.1 Interfaces in Dimension Two
The problem here is as follows: take the Potts model at low temperature (β > βc) in a two dimensional square box and enforce an interface between two colours through the system using boundary conditions (see Figures 1.3 and 1.2). Modify the coupling constant along a line as depicted in 1.2. The question is: how does the interface behaves as one changes the value ofJ? This problem was introduced by Abraham in [1, 2] and motivated many studies and the introduction of eective models (such as pinning of renewal processes, that are integrable models). See [39] for an early review and [44] for a recent account on the theory from the mathematical point of view.
The result we obtain on interfaces is
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1
2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
J
Figure 1.2: Modied Potts models with 1/2boundary conditions.
Figure 1.3: The dierence of behaviour between dierent regimes of J. Left:
J = 1, centre: J = 0.5, right: J = 25. β = 1.2 in all cases.
Result 4. When J < 1, the interface converges to a straight line under any polynomial scaling. Moreover, it has a density of contact points with this straight line.
See Theorem 5.4.
The idea is to use planar duality to treat the interface as a correlation path between two points (a graphical representation for a suitable correlation func- tion) and to use the OZ theory to treat the problem of the interface geometry.
As this will be the topic of the next Subsection, we continue this discussion there.
1.4.2 Correlations
Here the problem is closely related to the one of the previous Subsection. The frame is: take the Potts model at high temperature (β < βc) and modify the coupling constant along a set E. DenoteµJ the modied measure (the homo- geneous measure is denoted µ). The question is: how is CovJ(1σ0=1,1σx=1) modied by the change in the weights (from J = 1 to J 6= 1)? The basic idea is to use a graphical representation of correlation forµJ and one forµthat use the same collection of graphs but a dierent weighting:
CovJ(1σ0=1,1σx=1) = X
γ30,x
qJ(γ), Cov(1σ0=1,1σx=1) = X
γ30,x
q(γ).
Then simply write P
γ30,xq(γ)qq(γ)J(γ) and see qq(γ)J(γ) as a potential acting on the homogeneous model. The behaviour of the graphs under q appearing in the sum is handled via OZ theory and one studies the potential qq(γ)J(γ) by using properties of the graphical representation (in this case, the Random Cluster model). In the case where we modify the weight along a line and we look at correlations between points on the line, we nd that the rate of exponential decay is modied when taking J > Jc for some transition value ∞> Jc ≥ 1. Moreover we computeJcin dimension 2and3and study the critical behaviour in these cases.
Result 5. There is a ∞ > Jc ≥ 1 depending on the dimension such that for anyJ > Jcthe rate of exponential decay ofCovJ(1σ0=1,1σx=1)(where0, x∈ L) is strictly smaller than the one of Cov(1σ0=1,1σx=1) while it is unchanged for J ≤Jc. Moreover, Jc= 1 in dimensions 2 and 3.
The corresponding Theorems are Theorems 5.1 and 5.2.
Result 6. In dimension 2 and 3 we can compute the behaviour of the rate of exponential decay as J ↓Jc.
Figure 1.4: The dierence of behaviour between dierent regimes of J. Left:
J = 1, right: J = 25. β= 0.4in both cases.
See Theorem 5.3.
Finally, we obtain that while the rate is not modied for J ≤ Jc, the correction to exponential decay changes from the usual OZ asymptotics (1.2) whenJ <1 and d= 2,3.
Result 7. When J <1, c−
ψd(kxk)e−kxkξ(x/kxk)
≤CovJ(1σ0=1,1σx=1)≤ c+
ψd(kxk)e−kxkξ(x/kxk)
, where
ψd(n) =
n3/2 d= 2 nlog(n)2 d= 3 n(d−1)/2 d≥4 . For this last result, see Theorem 5.5.
Chapter 2
Models to be Considered and their Basic Properties
2.1 The Ising and Potts Models
2.1.1 Ising Model
The ferromagnetic Ising model on a nite graph G = (V,J) at inverse tem- perature β ≥ 0 and external elds h ∈ RV is the probability measure µG,β,h
on Ω = {−1,+1}V given by (σ ∈ Ω, we make the choice of including the inverse temperature in the Hamiltonian as it is technically more convenient;
sometimes,β will even be absorbed in the coupling constants Jij):
H(σ) =−β X
{i,j}⊂V
Jijσiσj −X
i∈V
hiσi, ZG,β,h =X
σ∈Ω
e−H(σ), µG,β,h(σ) = 1
ZG,β,h
e−H(σ), hfiG,β,h =X
σ∈Ω
µG,β,h(σ)f(σ).
When clear from the context, we will drop β and h from the notation.
2.1.2 Boundary Conditions and Spatial Markov Prop- erty
When the graph G is a subgraph of a (possibly innite) graph F, one can dene the model with boundary conditions η∈ {−1,1}VF by formally setting
HηG(σ) = −β X
{i,j}⊂VG
Jijσiσj −β X
i∈VG,j∈VF\G
Jijσiηj− X
i∈VG
hiσi, ZG,β,hη =X
σ∈Ω
e−HηG(σ), µηG,β,h(σ) = 1
ZG,β,hη e−HηG(σ), hfiηG,β,h =X
σ∈Ω
µηG,β,h(σ)f(σ).
When there exists R < ∞ such that dG(i, j) > R =⇒ Jij = 0, we say that the model has nite range, otherwise it has innite range. In the innite range case, one need some summability condition on J for the above to make sense.
In what follows we will mostly consider nite range models. Two boundary conditions play a particular role: the + boundary conditions η ≡+1 and the minus boundary condition η ≡ −1, associated measures are denoted µ+G and µ−G. The measure without boundary conditions will be denotedµG,µ0G orµfG.
2.1.3 Correlation Inequalities for the Ising Model
As we are interested in studying the properties of the probability measures µG,β,h asG, β andh vary, we want to study the behaviour of hfiG,β,h for large classes of functions f. A class of function (multi-points functions) plays a special role in this analysis: for any A⊂V dene
σA =Y
i∈A
σi.
They are of particular interest as they generate the functions with nite sup- port:
Lemma 2.1.1. Let ∆ be a nite set. Let f : {−1,1}∆ → R be a function.
Then, there exist real numbers ( ˜fA)A⊂∆ such that:
f(σ) = X
A⊂∆
fˆAσA, for all σ ∈ {−1,1}∆.
Proof. Start by noticing that for any B ⊂∆,σ,σ˜∈ {−1,1}∆ X
A⊂B
σAσ˜A=Y
i∈B
(σiσ˜i+ 1) = 2|B|1σi=˜σi,∀i∈B.
Then, for anyσ ∈ {−1,1}∆, f(σ) = X
σ∈{±1}˜ ∆
f(˜σ)1σi=˜σi,∀i∈∆
= 2−|∆| X
σ∈{±1}˜ ∆
f(˜σ)X
A⊂∆
σAσ˜A
= X
A⊂∆
σA2−|∆| X
˜σ∈{±1}∆
f(˜σ)˜σA,
and the claim is proved forfˆA= 2−|∆|P
σ∈{±1}∆f(σ)σA.
The Ising model enjoys a wealth of correlation inequalities. We recall here some of them that will be of use later on. The rst one is the Grif- thsKellySherman inequality:
Lemma 2.1.2 (GKS inequalities). For any weighted graph G = (V,J), non- negative elds h= (hi)i∈G and sets A, B ⊂V, the following hold:
hσAi∗G,β,h ≥0
hσAσBi∗G,β,h≥ hσAi∗G,β,hhσBi∗G,β,h, for ∗= 0,+.
Proof. It is a direct consequence of the Random Current expansion of the correlations function (see section 2.3) and of the Switching Lemma 2.3.2.
The second one is based on the notion of increasing functions: the set {−1,+1}V has a natural partial order: σ ≥ σ0 if σi ≥ σi0 for all i ∈ V. The functions f : {−1,+1}V → R thus inherit a natural notion of monotonicity:
f is said to be non-decreasing if σ ≥ σ0 =⇒ f(σ) ≥ f(σ0). The Fortuin- Kasteleyn-Ginibre inequality is the property of positive association between non-decreasing functions:
Lemma 2.1.3 (FKG inequality). For any weighted graphG= (V,J), any real eldsh, any boundary conditionsη and any two f, g non-decreasing functions,
hf giηG,β,h≥ hfiηG,β,hhgiηG,β,h. A proof of this inequality can be found in [41].
2.1.4 Potts Model
The Potts model is a natural way to generalize the Ising model to more than two symmetric colours. Given q∈N the ferromagnetic q-states Potts model
on a nite graph G⊂ F at inverse temperature β ≥ 0 with boundary condi- tions η ∈ {1,2, ..., q}VF is the probability measure µηG,q,β onΩ = {1,2, ..., q}VG given by, for σ∈Ω:
HηG(σ) =−β X
{i,j}⊂VG
Jijδσi,σj −β X
i∈VG,j∈VF\G
Jijδσi,ηj, ZG,q,βη =X
σ∈Ω
e−HηG(σ), µηG,q,β(σ) = 1
ZG,q,βη e−HηG(σ), hfiηG,q,β =X
σ∈Ω
µηG,q,β(σ)f(σ).
We will omit indices from notation when they are clear from the context. The q pure boundary conditions are of particular use, we denote the measures with boundary conditions η ≡ 1,2, ..., q by µ1G, ..., µqG. The Ising model at inverse temperature 2β and magnetic eld h ≡ 0 is equivalent to the 2-states Potts model at inverse temperature β.
2.1.5 Innite Volume Measures
As we will be interested in the behaviour of the model for very large systems, it is natural to try to dene the Potts model on innite graphs. Of particular interest to us will be the graph with vertex set Zd and weights J with nite range. Innite volume measures are probability measures µ on {1, ..., q}Zd satisfying the Dobrushin-Lanford-Ruelle equations: for any Λ nite subset of Zd and any event A Λ-measurable,
µ(A) = Z
µηΛ(A)dµ(η).
A consequence of Theorem 2.1 and of the Edward-Sokal coupling (see sec- tion 2.2) is that there is a unique innite volume measure at high temperature (β small).
2.2 The Random Cluster Model
Let G = (VG,J) be a nite graph. Write EG =
{i, j} ⊂ VG : Jij > 0 . The congurations of the Random Cluster Model (or FK-percolation) are subsets of EG. For a xed graph, the Random Cluster Models are a two parameters family: for q≥1, β ≥0, the probability of a conguration ω is given by
ΦG,q,β(ω) = 1 ZG,q,β
qκ(ω)Y
e∈ω
(eβJe −1),
where κ(ω) = |C(ω)|, C(ω) being the set of clusters of ω and ZG,q,β is the normalization constant. We will use the following notation for the weights:
xe = eβJe −1 and the Random Cluster Measure will also be denoted ΦG,q,x. When clear from the context, q and xor β will be omitted.
2.2.1 Boundary Conditions
Given a nite graph G, one can impose boundary condition on a subset I of VG by specifying a partition π of I and looking at the measure
ΦI,πG,q,β(ω) = 1
ZG,q,βI,π qκI,π(ω)Y
e∈ω
(eβJe−1),
where κI,π(ω) = |CI,π(ω)|, CI,π(ω) being the set of clusters of ω with edges added between any two sites of I sitting in the same class of π. A classical example is when G is a subgraph of another graph F and I =∂intVG. In this case, two boundary conditions are of particular interest: the one induced by the partition π = {∂intVG} and the one induced by π = {{v} : v ∈ ∂intVG};
the corresponding measures will be denotedΦwG,q,β and ΦfG,q,β respectively and be called the wired measure and the free measure. Boundary conditions will often be omitted from notation when clear from the context. Notice that the random cluster measure with a given boundary condition can be seen as a free random cluster measure on a modied graph where each class inπare merged into one vertex.
2.2.2 Basic Properties
Lemma 2.2.1 (FKG inequality). Letf, g be two non-decreasing functions on the subsets of EG (with respect to the inclusion). Then, for any I ⊂VG and π partition of I,
ΦI,πG (f g)≥ΦI,πG (f)ΦI,πG (g).
An event A is non-decreasing if the function 1A is. The FKG inequality implies straightforwardly that ΦwG and ΦfG are extremal for the FKG order amongst the Random Cluster measures on G ⊂ F. The FKG inequality is a direct consequence of the following lemma.
Lemma 2.2.2 (Increasing coupling). Let A be a non-decreasing event. Then for any I ⊂ VG and π partition of I, one can construct a coupling Ψ of ΦI,πG (· |A) and ΦI,πG such that, if (ω, η)∼Ψ, ω ≥η almost surely.
A proof can be found in [34] or in [49].
Remark 2.2.1. The previous Lemma has a version with B non-increasing: it then yields the existence of a coupling betweenΦI,πG (· |B)and ΦI,πG such that, if (ω, η)∼Ψ,ω ≤η almost surely.
Using Lemma 2.2.2 and its non-increasing version, one can obtain:
Lemma 2.2.3. LetAbe a non-decreasing event andB a non-increasing event.
Then for any I ⊂VG and π partition of I, one can construct a coupling Ψ of ΦI,πG (· |A) and ΦI,πG (· |B) such that, if (ω, η)∼Ψ, ω≥η almost surely.
Proof. LetΨ1be the coupling betweenΦI,πG (· |A)andΦI,πG given by Lemma 2.2.2 and let Ψ2 be the coupling between ΦI,πG (· |B) and ΦI,πG given by the non- decreasing version of Lemma 2.2.2. Let (ω,τ˜) ∼ Ψ1 and (η, τ0) ∼ Ψ2. We then construct a coupling of ω and η by gluing Ψ1 and Ψ2 along their second coordinate: construct (ω, τ, η)by
• Sampling τ ∼ΦI,πG .
• Sampling ω ∼Ψ1(· |τ˜=τ) and η∼Ψ2(· |τ0 =τ).
Denote Ψthe law of the triplet (ω, τ, η). Let ψ be any conguration, one has Ψ(ω=ψ) =X
a,b
Ψ(ω=ψ, τ =a, η=b)
=X
a,b
ΦI,πG (a)Ψ1(ω=ψ|τ˜=a)Ψ2(η=b|τ0 =a)
=X
a
Ψ1(˜τ =a)Ψ1(ω=ψ|τ˜=a)
= Ψ1(ω=ψ) = ΦI,πG (ψ|A).
Thus, ω ∼ ΦI,πG (· |A). In the same fashion, η ∼ ΦI,πG (· |B). Moreover, ω ≥ τ ≥ η implies ω ≥η, thus the restriction of Ψto (ω, η) is a coupling between ΦI,πG (· |A) and ΦI,πG (· |B) with ω ≥η.
Lemma 2.2.4 (Russo-type formula). Let A be an non-decreasing event. Then for any e∈EG, I ⊂VG and π partition of I,
d dxe
ΦI,πG (A)≥ 1
xe(1 +xe)ΦG(e∈PivA, ωe= 1).
Where PivA(ω) is the set of edges that are pivotal for A in ω (ie. whose addition or deletion will change the value of 1A ).
Proof. We drop I and π from the notation.
d dxe
ΦG(A) = 1 xe
ΦG(1Aωe)−ΦG(A)ΦG(ωe)
= 1 xe
ΦG(ωe)ΦG(1−ωe)
ΦG(A|ωe = 1)−ΦG(A|ωe = 0)
≥ 1
xe(1 +xe)ΦG(ωe)Ψ 1A(ω)−1A(η) ,
where we used nite energy and Ψ is the coupling of ΦG(· |ωe = 1) and ΦG(· |ωe = 0) given by Lemma 2.2.3 and (ω, η) ∼ Ψ. Then, as ω ≥ η a.s., A is non-decreasing and ωe = 1 and ηe= 0,
Ψ 1A(ω)−1A(η)
= Ψ 1A(ω)1Ac(η)
≥Ψ 1e∈PivA(ω)
= ΦG(e∈PivA|ωe = 1)
2.2.3 Random Cluster Model on Z
dand Percolation Tran- sition
Dening the Random Cluster Model on an innite graph can be done through DLR formalism (see for example [49] or [34]). In what follows, we will need only two innite volume measures, Φw and Φf on the innite graph (Zd,J) that are obtained as weak limits of the nite volume measures ΦwΛN,ΦfΛN on ΛN = [−N, N]d.
Lemma 2.2.5. There exist two innite volume measures,Φw and Φf, possibly equal, such that for any local event A and any innite volume measure Φ,
1. Φw(A) = limN→∞ΦwΛN(A), Φf(A) = limN→∞ΦfΛN(A), 2. Φf 4Φ4Φw,
3. Φw and Φf are ergodic and invariant under translation.
We refer to [49] and [34] for proof and general denition of innite volume measures. We are now in position to describe the percolation transition in the Random Cluster Model.
Theorem 2.1. On(Zd,J)(supposed connected) with J nite range and trans- lation invariant, d ≥ 2, for any q ≥ 1, there exists 0 < βc = βc(q, d,J) < ∞ such that:
• Φf(0↔ ∞)>0 for any β > βc.
• For any β < βc, there exists c >0 such that:
ΦwΛN(0↔∂intΛN)≤e−cN, for any N large enough.
This Theorem was rst proved for q = 1 in [56] and in [6] and for q = 2 in [7]. The general version stated here is the main result of [37]. A fairly easy consequence of Theorem 2.1 is that there exists a unique innite volume random cluster measure onZd when β < βc.
Lemma 2.2.6. Whenβ < βc, Φw= Φf, in particular there is only one innite volume Random Cluster Measure.
Proof. Let ∆ be a nite set of edges. For Λ = [−N, N]d containing ∆, let A = {∂intΛ ↔/ ∆}. On A, ΦwΛ restricted to ∆ is stochastically dominated by ΦfΛ restricted to ∆. As β < βc, by Theorem 2.1 and a union bound, ΦwΛ(A) ≥ 1−e−cN when N is large enough. Letting N to innity, one gets that Φw is dominated by Φf on ∆. As ∆ was an arbitrary nite set, they are equal.
2.2.4 Weak Exponential Ratio Mixing when β < β
cThis section is devoted to the proof of
Theorem 2.2. Let β < βc. Then, there exist c >0, R ≥ 0 such that for any two events A, B supported respectively on E, F with E at distance at least R from F,
log
Φ(A, B) Φ(A)Φ(B)
≤ X
e∈E,f∈F
e−cd(e,f).
A proof of this result rst appeared in [9] (with a proof working for a larger class of models). We employ here a slightly dierent strategy (even though using the same phenomena for decoupling) based on FKG inequality. The result will follow from a combination of mixing estimates for particular events.
For E a set of edges, dene
OE ={ωe = 1 ∀e∈E}, CE ={ωe= 0 ∀e∈E}, (E)K ={e:d(e, E)≤K}.
We start by a technical result (which is a semi-direct consequence of The- orem 2.1), it provides a useful estimate on connections occurring away from known lands. The strategy of the proof will be at the heart of many other arguments used in this thesis, it is thus worth exposing it in some detail.
Lemma 2.2.7. Let β < βc. There exist c >0 andK ≥0such that for any F set of edges, u, v ∈Zd with d({u, v}, F)≥K,
Φ(u←→∆c v| OF)≤e−cd(u,v), where ∆ = (F)K.
Proof. First, by Theorem 2.1, one can ndc > 0such thatΦwΛK(0↔∂intΛK)≤ e−cK for all K large enough. Dene B(i) =i+ Λ3K and for V ⊂Zd,B(V) = S
i∈V B(i). One then coarse grains the cluster ofu restricted to ∆cas follows.
Fix a total ordering on Zd and run
Algorithm 1: Coarse graining of the restriction to∆c of Cu. Set v0 =u, V ={v0}, E =∅, n = 1;
while A=
z ∈∂extB(V) :z ←→∆c (∂intΛK+z) 6=∅ do Set vn = minA;
Letv∗ be the smallest i∈V such thatvn∈∂extB(i);
Set en={v∗, vn};
UpdateV =V ∪ {vn},E =E∪ {en},n =n+ 1; endSet M =n−1;
returnV ={v0, ..., vM} and E ={e1, ..., eM};
