Le but de ces rencontres est de présenter des résultats récents et de discuter des questions nouvelles et ouvertes sur les systèmes de particules et la mécanique statistique.
Mercredi 26 mai
11h00 - 11h45 : Federico CAMIA - Geometric Representation of the 2D Critical Ising Magnetization Field.
11h45 - 12h30 : Lorenzo ZAMBOTTI - Large deviations for the transfer of heat in collisional dynamics.
12h30 - 14h00 : Déjeuner
14h00 - 16h00 : Marton BALASZ - Mini-cours 1
16h00 - 16h15 : Pause
16h15 - 17h00 : Vivien LECOMTE - Dynamical phase transitions and kinetically constrained models.
17h00 - 17h45 : Milton JARA - Universality of KPZ equation.
Jeudi 27 mai
09h30 - 11h30 : Marton BALASZ - Mini-cours 2
11h30 - 11h45 : Pause
11h45 - 12h30 : Giambattista GIACOMIN - On the Kuramoto synchronization model.
12h30 - 14h00 : Déjeuner
14h00 - 14h45 : Thierry GOBRON - From McLane's to Kasteleyn's Theorem:
A graph theory approach to Pfaffian representation of Ising partition function.
14h45 - 15h30 : Marc WOUTS - Transition de localisation/délocalisation pour un polymère avec charges attractives.
15h30 - 15h45 : Pause
15h45 - 16h30 : Krishnamurthi RAVISHANKAR - Marking the Brownian web and Applications.
Mini-cours
Marton BALASZ (Budapest, Hongrie)
Scaling of current fluctuations for a class of asymmetric interacting particle systems.
Timo Seppäläinen showed in the 2008 minicourse how to prove t2/3-scaling of current fluctuations through the characteristics in the asymmetric simple exclusion process (ASEP). This time I will show the same argument valid for a class of systems that contains the ASEP, some zero range models, and more (no knowledge of the 2008 minicourse will be assumed). The treatment will be, within the class, as model-independent as possible, and I will explicitly point out a crucial step which limits the applicability of the method in that class.
The lectures are based on joint work with Júlia Komjáthy and Timo Seppäläinen.
Conférences
Federico CAMIA (Amsterdam, Pays-Bas) Geometric Representation of the 2D Critical Ising Magnetization Field. The two-dimensional Ising model has played a fundamental role in the theory of phase transitions and is one of the most studied models of statistical mechanics. In this talk, I will focus on the scaling limit of the lattice magnetization at the critical point, and discuss a geometric representation based on an ensemble of finite measures of fractal support characterized by the property of transforming in a particularly simple way under conformal maps. Such a representation seems useful in rigorously proving the expected conformal covariance properties of the magnetization field. (Based on Joint work with C.M. Newman and with C. Garban and Newman.)
Giambattista GIACOMIN (Paris 7) On the Kuramoto synchronization model. Synchronization (of cells, of complex organisms, of circuits,...) phenomena play a central role in a variety of fields. Several synchronization models have been proposed and studied and the Kuramoto model has emerged as the simplest non-trivial one. It is a mean field disordered system of interacting diffusions on a circle that reduces, in absence of disorder, to the classical XY spin model. We aim at giving an introduction to the Kuramoto model from a statistical mechanics viewpoint and to present some results on the non-local parabolic equation that describes the large scale limit of the system.
Thierry GOBRON (Cergy)
From McLane's to Kasteleyn's Theorem: A graph theory approach to Pfaffian representation of Ising partition function.
A classical theorem due to Kasteleyn (1961) states that the partition function of an Ising model on an arbitrary planar graph can be represented as the Pfaffian of a skew-symmetric adjacency matrix associated to the graph. This results both embodies the free fermionic nature of any planar Ising model and gives an effective way of computing its partition functions in closed form. An extension of this result to non planar models expresses the partition function as a sum of Pfaffians which number is related to the genus of the oriented surface on which the graph can be embedded.
In graph theory, McLane's theorem (1937) gives a characterization of planarity as a property of the cycle space of a graph and recently, Diestel et al. (2009) extended this approach to embeddings in arbitrary surfaces.
Here we show that McLane's approach naturally leads to Kasteleyn's results: McLane characterization of planar graphs is just what is needed to turn an Ising partition function into a Pfaffian. Its extension leads to a substantial improvement in non planar cases: it avoids the use of topological Arf invariants, allowing the embedding surface to be non-orientable. It also permits to set the problem on a larger algebraic setting, giving rise to new Pfaffian representations.
Milton JARA (Paris Dauphine)
Universality of KPZ equation.
We introduce a new concept, which we name the Second-order Boltzmann-Gibbs Principle. As an application of this principle, we prove that under minimal assumptions, fluctuations of one-dimensional, weakly asymmetric conservative systems are governed by the so-called KPZ equation. In particular, no combinatorial arguments or couplings are used in the proof. KPZ equation is well-known for being ill-posed. Even a suitable notion of solution is missing in the literature. As a consequence of our results, we formulate a self-consistent notion of solution of the KPZ equation through a martingale problem. In particular, we make sense of the nonlinear term of the equation and we prove that the Cole-Hopf solution constructed by Bertini and Giacomin is a weak solution in our sense. As a corollary, we obtain upper bounds on the variance of various functionals of the KPZ process, complementing recent results by Balazs, Quastel and Seppalainen.
Joint work with Patricia Gonçalves.
Vivien LECOMTE (Genève, Suisse)
Dynamical phase transitions and kinetically constrained models.
Among models aimed at describing slow dynamical properties of glassy phenomena, kinetically constrained models are one of the less sophisticated to formulate. We focus on Fredrickson–Andersen-like models, which are equilibrium lattice gases with trivial static properties but subtle dynamical ones, that we tackle through large deviations theory.
In the large size limit, we show that the large deviation function of some time-integrated observables displays a non-analyticity, meaning that the dynamics takes place at a first-order coexistence line between active and inactive dynamical phases. We formulate a mean-field approach, based on a Landau-like dynamical free energy, which provides a quantitative description of the coexistence of these two phases, in particular through finite-size scaling.
K. RAVISHANKAR (New Paltz, USA) Marking the Brownian web and Applications. In this talk I will discuss recent results (and some ongoing work) obtained in collaboration with C.M. Newman and E. Schertzer. I will start with a brief introduction to the discrete web of coalescing simple random walks and its continuum diffusive limit, the Brownian web (BW). After indicating how the continuum limit of the noisy voter model (Glauber dynamics of nonzero temperature stochastic Ising model) is obtained using the marking of (0,2) (or bulk nucleation) points of the BW, the remainder of the talk will focus on (1,2) points of the BW which correspond to the boundary nucleation points of the stochastic Potts model. I will then describe the marking procedure for the (1,2) points and indicate how it can be used to construct the Brownian net and then I will describe how these marking (along with the markings of (0,2) points) can be used to obtain the continuum limit of stochastic Potts model. If time permits I will discuss the marking construction of dynamical Brownian web.
Marc WOUTS (Paris 13) Transition de localisation/délocalisation pour un polymère avec charges attractives. Il s'agit d'un travail en collaboration avec D. Khoshnevisan et Y. Hu. Nos principaux résultats sont les suivants : à haute température, la trajectoire du polymère (renormalisée) converge vers le mouvement brownien, alors qu'à basse température, le diamètre du polymère est de l'ordre de log N, où N est le nombre de monomères.
Lorenzo ZAMBOTTI (Paris 6)
Large deviations for the transfer of heat in collisional dynamics.
(Joint work with Raphael Lefevere and Mauro Mariani).
We consider two models for the transport heat in systems described by local collisional dynamics. The dynamics consist of tracer particles moving through an array of hot scatterers describing the effect of heat baths at fixed temperatures. When the set of temperatures is fixed by the condition that in average, no energy is exchanged between the scatterers and the system, two behaviors may occur. When the tracer particles are allowed to travel freely through the whole array of scatterers, the temperature profile is linear. If the particles are locked in between scatterers, the temperature profile becomes nonlinear. In both cases, Fourier law holds. We are particularly interested in the Donsker-Varadhan large deviations functional, which turn out to be surprisingly difficult to analyze.