GROUPE DE TRAVAIL

UMR 6085

GROUPE DE TRAVAIL

Programme de l'année 2000

5 janvier 2000   (Université de Cergy)
17 février 2000   (exceptionnellement ENS)
24 mars 2000   (Université de Rouen)
28 avril 2000   (Université de Rouen)
12 mai 2000   (Université de Rouen)

7 juin 2000   (Université de Cergy)
20 octobre 2000   (Université de Cergy)
24 novembre 2000   (Université de Rouen)
15 décembre   (Chevaleret)

5 janvier 2000 - Université de Cergy

14H00 Pablo A. Ferrari (Université de Sao Paulo)

POISSON APPROXIMATION FOR LARGE-CONTOURS IN LOW-TEMPERATURE ISING MODELS

We consider the contour representation of the infinite volume Ising model at low temperature. Fix a subset V of Z^d, and a (large) N such that calling G_N,V the set of contours of length at least N intersecting V, there are in average one contour in G_N,V under the infinite volume "plus" measure.
We find bounds on the total variation distance between the law of the contours of length at least N intersecting V under the "plus" measure and a Poisson process. The proof builds on the Chen-Stein method as presented by Arratia, Goldstein and Gordon. The control of the correlations is obtained through the loss-network space-time representation of contours due to Fernandez, Ferrari and Garcia.

17 février 2000 - École Normale Supérieure

Exceptionnellement cette réunion du groupe de travail aura lieu à
Ecole Normale Supérieure
45 rue d'Ulm, Paris 5ème
Département de Mathématiques, Salle W, 4ème étage.

14H00 Dima Ioffe (Haifa)

SOME PROBABILISTIC PROBLEMS OF WETTING

24 mars 2000 - Université de Rouen

10H45 Roberto Fernández (Université de Rouen)

CLUSTER EXPANSION AND NON-CLUSTER EXPANSION METHODS

Cluster expansions are routinely used in statistical mechanics to study systems of objects interacting by some local exclusion rule (eg. volume- or perimeter-exclusion). They are used, for instance, to deal with contour models, or the so-called polymer and animal models. They allow the estimation of logarithms or ratios of partition functions. Their existence implies the analyticity of the free-energy density (pressure) and, as a consequence, the existence and uniqueness of the infinite-volume measure and its analyticity with respect to relevant parameters (eg. temperature).
The technique, while extremely popular in rigorous statistical mechanics, has seldomly been used in probability theory. Dobrushin was the first to explicitly point out this fact, and he suggested two reasons:

the excessive technicism of the papers on cluster expansion, heavy on notation and with emphasis on delicate but auxiliary combinatorial facts
the indirect character of the method from the probabilistic point of view. Indeed, the main target of the technique is the free-energy density; the probability measures are almost a side remark.
Dobruhsin went on to propose a ``non-cluster-expansion'' method, based on an inductive approach, which leads to the same results without any (explicit) combinatorial complication. While his approach is conceptually simpler, it still falls a bit outside a probabilistic framework, as it resorts to complex analysis and it leads to analyticity results.
Recently, with P. Ferrari and N. Garcia, we have proposed a purely probabilistic approach where the measures are obtained as invariant measures of a process (loss-network, point process) which is absolutely continuous with respect to a marked Poisson process. The approach yields all the desired properties of the measures, except analyticity with respect to parameters. Moreover, its proven region of validity is larger than that so far determined for standard cluster expansions.
In this talk I will review the standard cluster-expansion machinery and Dobrushin inductive approach, before discussing in some detail our loss-network approach. I will comment on open questions, points to be clarified, and directions for improvement.

14H15 Tada-Hisa Funaki (Université de Tokyo)

EQUILIBRIUM FLUCTUATIONS FOR A RANDOM SURFACE ON THE WALL IN ONE DIMENSION

28 avril 2000 - Université de Rouen

10H45 Alan Sokal (NYU/Oxford)

POTTS MODELS, CHROMATIC POLYNOMIALS, AND ALL THAT

The q-state Potts model is a statistical-mechanical model that generalizes the well-known Ising model. It can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph (including its chromatic polynomial and its reliability polynomial). The complex zeros of the Potts-model partition function are of interest both to statistical mechanicians (in connection with the Lee-Yang picture of phase transitions) and to combinatorists.
I begin by giving an introduction to all these problems. I then sketch two recent (and as yet unpublished) results:

Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree (maximum number of nearest neighbors to any site).
Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1| < 1.
This talk is intended to be understandable to both mathematicians and physicists; no prior knowledge of either graph theory or statistical mechanics is required.

12 mai 2000 - Université de Rouen

10H45 Claudio Landim (Université de Rouen)

RELAXATION VERS L'EQUILIBRE DE SYSTEMES INFINIS DE PARTICULES CONSERVATIFS

7 juin 2000 - Université de Cergy

13H30 Ellen Saada (Université de Rouen)

DYNAMIQUES DE PILES DE SABLE EN VOLUME INFINI

Les modèles de piles de sable ont beaucoup été étudiés en volume fini. Dans ce travail en commun avec C. Maes, F. Redig et A. L. van Moffaert, nous construisons le modèle de piles de sable en volume infini,et regardons ses mesures invariantes. Nous considérons les cas de Z et d'un arbre infini.

15H30 Kirone Mallick (CEA)

SOLUTIONS EXACTES POUR LE MODELE D'EXCLUSION ASYMETRIQUE

Le Processus d'Exclusion Asymétrique (ASEP) est l'un des modèles les plus simples etudiés en physique statistique hors d'équilibre. Ce système est constitué d'un ensemble de particules qui, en présence d'un champ externe, diffusent sur un réseau avec une interaction de coeur dur. De nombreux processus de transport peuvent être décrits par un tel modèle ou l'une de ses variantes : la croissance désordonnée d'une surface par dépôt aléatoire de matière au dessus d'un substrat (équation de Kardar-Parisi-Zhang), les polymères dirigés en milieu aléatoire, la conduction par saut d'ions dans les matériaux super-ioniques ou encore le trafic routier...
Des solutions exactes ont été obtenues ces dernières années par une méthode originale, inspirée de la mécanique quantique et des systèmes intégrables : on représente les probabilités stationnaires du problème étudié comme traces d'opérateurs. L'état stationnaire est ainsi caractérisé par une algèbre à trace idoine.
Nous exposerons cette technique et montrerons comment elle s'applique sur des exemples simples.