In this talk, we will provide a brief history of some recent developments in the study of random walks in dynamic random environments. Then, we will present a new result that establishes the Central Limit Theorem (CLT) for a family of environments that mix rapidly but not uniformly. We will give an outline of the proof, which follows a very simple idea: showing that the random walk cannot escape from a set of renewal traps. We will conclude the seminar by indicating some interesting directions for the continuation of this study.
This presentation is based on joint work with Julien Allasia, Rangel Baldasso, and Oriane Blondel
In this talk we will discuss some recent developments in random walks in dynamic random environments, when the environment is given by a realization of a particle system as the SEP or ZRP.
We use the framework of Quantitative Hydrodynamics to derive a CLT around its hydrodynamic limit for an interacting particle system inspired by integrate-and-fire neuron models. The hydrodynamic limit of this model was originally derived by Chariker, De Masi, Lebowitz and Presutti, and as an important intermediate step we show that this convergence holds at optimal L^2-speed.
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. Based on joint works with O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.
