Séminaire de Probabilités commun ICJ/UMPA

Anomalous correlations in the symmetric simple exclusion process out of equilibrium

par M. Benoît Dagallier (University of Cambridge)

Europe/Paris
Description

For finite size Markov chains, the probability that a time-averaged observable take an anomalous value in the long time limit was quantified in a celebrated result by Donsker and Varadhan. In the study of interacting particle systems, one is interested not only in the large time, but also in the large system size limit. To each observable (for instance the density of particles or the two point correlations) corresponds a different scaling in terms of the system size. In the large size limit, one then has to focus on a specific observable, and could try to extract, from the Donsker-Varadhan functional, information pertaining to the scale of that observable. 
For reversible dynamics, the functional is explicit, and taking the large system size limit boils down to a computation. For non-reversible dynamics, i.e. out of equilibrium, the functional is not explicit. In this case, using the Donsker-Varadhan result as a starting point seems considerably more difficult. 

In a joint work with Thierry Bodineau, we study a paradigmatic example of out of equilirium interacting particle systems: the one-dimensional symmetric simple exclusion process connected with reservoirs of particles at different density. We focus on two point correlations and obtain the long time, large system size limits on the probability of observing anomalous correlations. This is done through quantitative, non-asymptotic estimates at the level of the dynamics. The key ingredient is a precise approximation of the dynamics and its invariant measure (not explicitly known), that is of independent interest. The quality of this approximation is controlled through relative entropy bounds, making use of recent results of Jara and Menezes.