24 et 25 mai 2012 - Université de Rouen.

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.

## Jeudi 24 mai

11h00 - 12h00 : Stefan Grosskinsky - Condensation in zero-range processes and related models. (Partie I)
12h00 - 13h30 : Déjeuner
13h30 - 14h30 : Stefan Grosskinsky - Condensation in zero-range processes and related models. (Partie II)
14h30 - 15h10 : Alexandre Gaudillière - TBA.
15h10 - 15h30 : Pause
15h30 - 17h30 : Michalis Loulakis - Large Deviations and Subexponential Random Variables with Applications to Condensating Zero Range Processes.

## Vendredi 25 mai

09h00 - 11h00 : Claudio Landim - Metastability of reversible condensed zero range processes on a finite set.
11h00 - 11h20 : Pause
11h20 - 12h00 : Krishnamurthi Ravishankar - Ergodicity and Percolation for Variants of One-dimensional Voter Models.
12h00 - 13h30 : Déjeuner
13h30 - 15h30 : Ines Armendariz - Scaling limit of the condensate dynamics in the zero-range process.
15h30 - 15h50 : Pause
15h50 - 16h10 : Marios Stamatakis - Variational Characterization of Generalized Relative Entropy Functionals and Static Large Devations for the Empirical Embeddings of the Zero Range Process at Equilibrium without Full Exponential Moments.
16h10 - 16h50 : Milton Jara - The formation of the condensate on a metastable zero-range process.

## Mini-cours

Ines Armendariz (Buenos Aires) Scaling limit of the condensate dynamics in the zero-range process. We consider the zero-range process on the one dimensional torus with L sites and N particles, in the supercritical regime when N/L exceeds the critical density. It is known that in the stationary state, as L goes to infinity, the excess particles accumulate at a randomly chosen position, forming the condensate. We now show that, at the right scale, this condensates moves according to a Lévy process on the rescaled torus, with rates determined by the jump distribution of the original zero range process. For instance, for nearest neighbour probabilities, the limiting Lévy process will have rates inversely proportional to the jump length.
Joint work with Stefan Grosskinsky and Michalis Loulakis.

Stefan Grosskinsky (Warwick) Condensation in zero-range processes and related models. Zero-range processes or more general mass transport models can exhibit a condensation transition, where a finite fraction of all particles condenses on a single lattice site if the total density exceeds a critical value. This phenomenon can result from spatial inhomogeneities, an effective attraction between the particles and also from size-dependence in the jump rates. We give an introduction to the most basic results for the stationary measures to characterize the condensation transition, and describe also connections to the classical framework of the equivalence of ensembles in statistical mechanics. Zero-range processes will be the main example, but we will also mention other models such as inclusion processes or models with continuous state space such as the Brownian energy process.
This includes joint work with Herbert Spohn, Gunter Schuetz, Paul Chleboun, Frank Redig and Kiamars Vafayi.

Claudio Landim (Rouen et Rio de Janeiro) Metastability of reversible condensed zero range processes on a finite set. Let $r: S\times S\to \mathbb R_+$ be the jump rates of an irreducible random walk on a finite set $S$, reversible with respect to some probability measure $m$. For $\alpha >1$, let $g: \mathbb N\to \mathbb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = (k/k-1)^\alpha$, $k\ge 2$.
Consider a zero range process on $S$ in which a particle jumps from a site $x$, occupied by $k$ particles, to a site $y$ at rate $g(k) r(x,y).$ Let $N$ stand for the total number of particles. In the stationary state, as $N\uparrow\infty$, all particles but a finite number accumulate on one single site. We show in this article that in the time scale $N^{1+\alpha}$ the site which concentrates almost all particles evolves as a random walk on $S$ whose transition rates are proportional to the capacities of the underlying random walk.

Michalis Loulakis (National Technical University of Athens & ACMAC Heraklion) Large Deviations and Subexponential Random Variables with Applications to Condensating Zero Range Processes. We will prove a Gibbs Conditioning Principle analogue for subexponential random variables, and we use this result in the context of Zero Range Processes to explore the bulk fluctuations and the fluctuations of the size of the condensate in equilibrium, as well as the onset of condensation as we move from subcritical to supercritical densities.
Joint work with Ines Armendariz and Stefan Grosskinsky.

## Conférences

Alexandre Gaudillière (Université de Provence, Marseille) TBA
TBA

Milton Jara (Rio de Janeiro) The formation of the condensate on a metastable zero-range process. Let us consider a zero-range process with decreasing rates on a finite graph. Our setup is the same considered by Beltrán-Landim. Let N be a scaling parameter, which will be sent to infinity, and let us put initially N particles at each site. We prove that in a diffusive time scaling, the normalized number of particles on each site converges to a system of diffusions that shares some similarities with Bessel processes of negative dimension. In particular, each time a site is emptied, it remains empty (on the observed time window). The correlation among the diffusions on each site are given in terms of harmonic properties of the subjacent random walk on the graph.
Work in progress, joint with J. Beltrán.

Krishnamurthi Ravishankar (New Paltz, USA) Ergodicity and Percolation for Variants of One-dimensional Voter Models. We study variants of one-dimensional $q$-color voter models in discrete time. In addition to the usual voter model transitions in which a color is chosen from the left or right neighbor of a site there are two types of noisy transitions. One is bulk nucleation where a new random color is chosen. The other is boundary nucleation where a random color is chosen only if the two neighbors have distinct colors. We prove under a variety of conditions on $q$ and the magnitudes of the two noise parameters that the system is ergodic, i.e., there is convergence to a unique invariant distribution. The methods are percolation-based using the graphical structure of the model which consists of coalescing random walks combined with branching (boundary nucleation) and dying (bulk nucleation).
Joint work with Y. Mohylevskyy and C.M. Newman.

Marios Stamatakis (University of Crete, Greece) Variational Characterization of Generalized Relative Entropy Functionals and Static Large Devations for the Empirical Embeddings of the Zero Range Process at Equilibrium without Full Exponential Moments. It is well known that the empirical embeddings of the Zero Range Process satisfy the static large deviations principle at any equilibrium of full exponential moments. We prove a variational characterization of generalized relative entropy functionals allowing any lower semicontinuous convex function in place of u → u log u, in order to generalize the static large deviations principle for the empirical embeddings of the Zero Range Process in the absence of full exponential moments and in particular in the case of finite critical density.