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Conformal invariance of lattice models
 in: Probability and Statistical Physics in Two and More Dimensions, in: Clay Math. Proc
, 2012
"... Abstract. These lecture notes provide an (almost) selfcontained account on conformal invariance of the planar critical Ising and FKIsing models. They present the theory of discrete holomorphic functions and its applications to planar statistical physics (more precisely to the convergence of fermi ..."
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Abstract. These lecture notes provide an (almost) selfcontained account on conformal invariance of the planar critical Ising and FKIsing models. They present the theory of discrete holomorphic functions and its applications to planar statistical physics (more precisely to the convergence of fermionic observables). Convergence to SLE is discussed briefly. Many open questions are included. Contents
Parafermionic observables and their applications to planar statistical physics models
, 2013
"... This volume is based on the PhD thesis of the author. Through the examples of the selfavoiding walk, the randomcluster model, the Ising model and others, the book explores in details two important techniques: 1. Discrete holomorphicity and parafermionic observables, which have been used in the pa ..."
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Cited by 7 (2 self)
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This volume is based on the PhD thesis of the author. Through the examples of the selfavoiding walk, the randomcluster model, the Ising model and others, the book explores in details two important techniques: 1. Discrete holomorphicity and parafermionic observables, which have been used in the past few years to study planar models of statistical physics (in particular their conformal invariance), such as randomcluster models and loop O(n)models. 2. The RussoSeymourWelsh theory for percolationtype models with dependence. This technique was initially available for Bernoulli percolation only. Recently, it has been extended to models with dependence, thus opening the way to a deeper study of their critical regime. The book is organized as follows. The first part provides a general introduction to planar statistical physics, as well as a first example of the parafermionic observable and its application to the computation of the connective constant for the selfavoiding walk on the hexagonal lattice. The second part deals with the family of randomcluster models. It studies the RussoSeymourWelsh theory of crossing probabilities for these models. As an application, the critical point of the randomcluster model is computed on the square lattice. Then, the parafermionic observable is introduced and two of its applications are described in detail. This part contains a chapter describing basic properties of the randomcluster model. The third part is devoted to the Ising model and its randomcluster representation, the FKIsing model. After a first chapter gathering the basic properties of the Ising model, the theory of sholomorphic functions as well as Smirnov and ChelkakSmirnov’s proofs of conformal invariance (for these two models) are presented. Conformal invariance paves the way to a better understanding of the critical phase and the two next chapters are devoted to the study of the geometry of the critical phase, as well as the relation between the critical and nearcritical phases. The last part presents possible directions of future research by describing other models and several open questions.
Phase Transition in Randomcluster and O(n)models
, 2011
"... Cette thèse traite des phénomènes critiques deux dimensionels. Plus précisément, nous étudions des modèles planaires de physique statistique qui exhibent une transition de phase, c’estàdire un changement brusque de leurs propriétés macroscopiques. L’étude se concentre sur deux familles de modèles: ..."
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Cited by 3 (2 self)
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Cette thèse traite des phénomènes critiques deux dimensionels. Plus précisément, nous étudions des modèles planaires de physique statistique qui exhibent une transition de phase, c’estàdire un changement brusque de leurs propriétés macroscopiques. L’étude se concentre sur deux familles de modèles: la FKpercolation et les modèles de boucles dénommés modèles O(n). Ces modèles englobent deux cas particuliers fondamentamentaux que sont le modèle d’Ising et les marches autoévitantes. Cette thèse est à l’interface entre la physique statistique, les combinatoires et les probabilitiés.
Graphical Representations for Ising and Potts Models in General External Fields
, 2015
"... This work is concerned with the theory of graphical representation for the Ising and Potts models over general lattices with nontranslation invariant external field. We explicitly describe in terms of the randomcluster representation the distribution function and, consequently, the expected valu ..."
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This work is concerned with the theory of graphical representation for the Ising and Potts models over general lattices with nontranslation invariant external field. We explicitly describe in terms of the randomcluster representation the distribution function and, consequently, the expected value of a single spin for the Ising and qstate Potts models with general external fields. We also consider the Gibbs states for the EdwardsSokal representation of the Potts model with nontranslation invariant magnetic field and prove a version of the FKG inequality for the so called general randomcluster model (GRC model) with free and wired boundary conditions in the nontranslation invariant case. Adding the amenability hypothesis on the lattice, we obtain the uniqueness of the infinite connected component and the almost sure quasilocality of the Gibbs measures for the GRC model with such
Parsimonious Description of Generalized Gibbs Measures: Decimation of the 2dIsing Model
, 2014
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Examples of DLR states which are not weak limits of finite volume Gibbs measures with deterministic boundary conditions
, 2014
"... We prove that the mixture 12 (µ ± + µ∓) of two reflectionsymmetric Dobrushin states of the 3dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finitevolume measures with deterministic boundary conditions. Furthermore, we discuss what is known about the stru ..."
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We prove that the mixture 12 (µ ± + µ∓) of two reflectionsymmetric Dobrushin states of the 3dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finitevolume measures with deterministic boundary conditions. Furthermore, we discuss what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and give a few conjectures. 1
On the Gibbs states of the noncritical Potts model on Z2
, 2012
"... We prove that all Gibbs states of the qstate nearest neighbor Potts model on Z^2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundar ..."
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We prove that all Gibbs states of the qstate nearest neighbor Potts model on Z^2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finitevolume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature.