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 25 mai

11h00 - 11h45 : Ben GRAHAM - *Metastability in the dilute Ising model.*

11h45 - 12h30 : Frank REDIG - *Random walk in dynamic random environment.*

12h30 - 14h00 : Déjeuner

14h00 - 16h00 : Fabio TONINELLI - *Mini-cours 1*

16h00 - 16h15 : Pause

16h15 - 17h00 : Sana LOUHICHI - *Towards a motion by mean curvature under
Glauber dynamics at zero temperature.*

17h00 - 17h45 : Amine ASSELAH -

*Fluctuations pour un nuage de points aléatoires.*

## Jeudi 26 mai

09h15 - 11h15 : Fabio TONINELLI - *Mini-cours 2*

11h15 - 11h30 : Pause

11h30 - 12h15 : Hubert LACOIN - *Approximate Lifshitz law for the mixing time of the
zero-temperature stochastic Ising model with + boundary conditions in any dimension.*

12h15 - 13h45 : Déjeuner

13h45 - 14h15 : Christophe POQUET -

*Noise induced transitions for active rotator models.*

14h15 - 15h00 : Krishnamurthi RAVISHANKAR -

*Euler hydrodynamics for attractive particle*

systems in random environment

systems in random environment

15h00 - 15h15 : Pause

15h15 - 15h45 : Eric LUÇON -

*Quenched limits and fluctuations in the Kuramoto*

synchronization model.

synchronization model.

15h45 - 16h30 : Christophe BAHADORAN -

*Stationary states for a two-species asymmetric*

exclusion process.

exclusion process.

## Mini-cours

Fabio TONINELLI (ENS, Lyon)
Low temperature Ising dynamics in 2 and 3 dimensions.
I will discuss the Glauber dynamics for the Ising model at low temperature.
A long-standing open problem is to prove that a bubble of "-" phase in a sea
of "+" phase evolves according to a mean-curvature type equation. A less
ambitious but still hard problem is to prove that the disappearence time of
the bubble is of order $L^2$, where $L$ is its initial radius.
I will report recent progress on these issues. Specifically I will discuss:

-the zero-temperature 3D Ising dynamics with either "+" or "Dobrushin"
boundary conditions; I will show how to obtain "almost optimal" bounds on the
mixing time (of order $L^2$ up to logarithmic corrections)

-the low-temperature ($\beta>\beta_c$) 2D dynamics with "+" boundary conditions.
I will show how a recursive procedure allows to get "almost polynomial" bounds on the mixing time.

(based on a series of works in collaboration with: P. Caputo, E. Lubetzky,
F. Martinelli, F. Simenhaus, A. Sly).

## Conférences

Amine ASSELAH (Paris 12)
Fluctuations pour un nuage de points aléatoires.
Nous obtenons des bornes sur les fluctuations
d'un nuage de points aléatoires généré par agrégation
limite par diffusion interne.

C'est un travail commun avec A.Gaudilliere.

Christophe BAHADORAN (Clermont Ferrand) Stationary states for a two-species asymmetric exclusion process. I will describe some work in progress, whose purpose is to derive the stationary density profile of a model first studied on the whole space by Fritz & Toth (CMP 04). There are particles of opposite sign with exclusion within each type and between types, moving in opposite directions with rate 1, exchanging with rate 2. There is a two-parameter family of product invariant measures, and the hydrodynamic equation is a 2x2 hyperbolic system of conservation laws known as Leroux's system. Here we consider a version of this model coupled to boundary reservoirs. The expected phase diagram can be computed explicitely. It is reminiscent of TASEP but has a more complex structure due to interplay between Riemann invariants of Leroux's system.

Ben GRAHAM (Warwick, Royaume-Uni)
Metastability in the dilute Ising model.
Consider Glauber dynamic for the Ising model on the hypercubic lattice
with a positive magnetic field.
Starting from the minus configuration, the system initially settles
into a metastable state with negative magnetization.
Slowly the system relaxes to a stable state with positive magnetization.
Schonmann and Shlosman showed that in the two dimensional case the
relaxation time is a simple function of the energy required to create
a critical Wulff droplet.

The dilute Ising model is obtained from the regular Ising model by
deleting a fraction of the edges of the underlying graph.
This produces a catalyst effect. Even an arbitrarily small dilution
can dramatically reduce the relaxation time.

Joint work with Thierry Bodineau and Marc Wouts.

Hubert LACOIN (Rome 3, Italie) Approximate Lifshitz law for the mixing time of the zero-temperature stochastic Ising model with + boundary conditions in any dimension. It has been noticed that below the critical temperature the mixing properties of the Stochastic Ising model are strongly dependent on boundary condition. Indeed if one considers Heat-Bath Dynamics for Ising in the cube of side $L$, the mixing time is exponential in $L$ whereas it is believed that for all + boundary condition, it behaves like $L^2$ (conjecture called "Lifshitz law"). What we present here is a new step toward the verification of the conjecture, showing that in all dimension, the mixing time with + boundary condition is $O(L^2 \log L^c)$ for some appropriate $c$ in any dimention. This generalizes a recent result by Caputo, Martinelli, Simenhaus and F.L. Toninelli (who proved it in two and three dimension). We will present the key results obtained by Caputo et al. for the three dimensional model and explain how they can be used to obtained the result for dimension greater than three.

Sana LOUHICHI
(LJK IMAG Université de Grenoble)
Towards a motion by mean curvature under Glauber dynamics at zero temperature.

Joint work with R. Cerf.
We consider the 2D stochastic Ising model evolving according to
the Glauber dynamics at zero temperature. We compute the initial
drift for droplets which are suitable approximations
of smooth domains. A specific spatial average of the derivative at time 0 of the
volume variation of a droplet close to a boundary point is equal
to its curvature multiplied by a direction dependent coefficient.
We compute the explicit value of this coefficient.

Eric LUÇON (Université Paris 6) Quenched limits and fluctuations in the Kuramoto synchronization model. The Kuramoto model was proposed to describe synchronization phenomena for large populations of oscillators in a random environment. Various applications can be found in biological or physical contexts (collective behavior of insects, synchronization of cells, circuits...). It is a mean field system of interacting diffusions on the circle, in the presence of disorder. The purpose is to understand the behavior of the empirical measure of the oscillators, in the large scale limit, for a fixed realization of the disorder (quenched model). One can show the convergence of this empirical measure to the deterministic solution of a McKean-Vlasov PDE (self-averaging phenomenon). Synchronization reads in terms of the existence of non-trivial stationary solutions of this equation, and their stability. Secondly, one can prove a quenched Central Limit Theorem corresponding to this convergence which shows that self-averaging no longer holds at the scale of fluctuations.

Christophe POQUET (Université Paris 7) Noise induced transitions for active rotator models. The long time behaviour of a system of many interacting stochastic dynamical systems is typically difficult to predict on the base of the behaviour of one isolated system. We tackle this issue for mean field active rotators by exploiting the closeness of these models with the Kuramoto synchronisation model. We will see that the normally hyperbolic structure of the manifold of stationary solutions of the Kuramoto model allows us to apply a perturbation method that yields qualitative and quantitative estimates on active rotators.

Krishnamurthi RAVISHANKAR (New Paltz, USA) Euler hydrodynamics for attractive particle systems in random environment We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on Z in random ergodic environment. Our result is a strong law of large numbers and applies to a large class of particle systems in random environments.

Frank REDIG (Nijmegen, Pays-Bas) Random walk in dynamic random environment. We consider a random walk driven by an environment that converges sufficiently fast to its unique stationary measure (e.g. high-temperature spin-flip dynamics). We prove that the environment process (EP), i.e., the enviroment seen from the walker has a unique invariant measure. For the position of the walker, as well as for additive functionals of the EP we prove law of large numbers and central limit theorem. The main ingredients are coupling and martingale approximation. This is joint work with F. Vollering (Leiden).