The Seminar on Analysis and Probability, organized by the Faculty of Mathematics HSE. Speaker: Marielle Simon

Abstract.

I will present some recent results which have been obtained for the facilitated exclusion process, in one dimension. This stochastic lattice gas is subject to strong kinetic constraints which create a continuous phase transition to an absorbing state at a critical value of the particle density. If the microscopic dynamics is symmetric, its macroscopic behavior, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time. When the particle system is put in contact with reservoirs of particles (which can either destroy or inject particles at both boundaries), we observe an usual impact on the boundary values of the empirical density. The asymmetric case can also be fully treated (for the infinite particle system): in this case, the empirical density converges to the unique entropy solution to a hyperbolic Stefan problem. All these results rely, to various extent, on a mapping argument with a zero-range process, which completely fails in dimension higher than 1. I will finally discuss some open problems and questions, especially in dimension 2. Based on joint works with O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.

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