MARK KAC SEMINAR

In the second lecture I will discuss some interface problems which arise from physics. The context is the Fourier-Fick law where a stationary current flows in a system driven by the boundaries which are put in contact with suitable reservoirs. This is a much studied problem but only recently there have been results in models with phase transition, (DM,Olla,Presutti, arxiv.org/abs/1812.05799, 2018). I will present these models where the stationary non equilibrium distribution is known explicitly and exploit this to compare the stationary fluctuations of the interface in this case where a non zero current is present, with those at thermal equilibrium.

The Sine process is a fascinating determinantal point process. It appears as the bulk limit of some particle systems in various contexts (random matrices, zeros of L-funtions, growth models etc.) Its universality properties have drawn a lot of attention. From the point of view of statistical physics, it models the microscopic behavior of a one-dimensional log-gas at inverse temperature beta equals to two. More recently, Valko and Virag introduced a family of point processes as the bulk limit of Gaussian beta ensembles, for any positive beta. As soon as beta is different from two, much less is known. In a work with David Dereudre, Adrien Hardy (Université de Lille) and Thomas Leblé (Courant Institute, New York), we use tools from classical statistical mechanics based on Dobrushin-Lanford-Ruelle equations to better understand the Sine beta process and in particular show that it is number-rigid.