Gaultier Lambert: Gaussian and non-Gaussian Limit theorems for some determinantal systems of particles

First, I will review the central limit theorem for linear statistics of the sine point process from Random Matrix theory. Then, I will explain how the proof can be generalized to a class of determinantal measures in one dimension which interpolate between Poisson and Random Matrix statistics. An example of such a process comes from considering a grand canonical ensemble of free fermions in a quadratic well at positive temperature. For this model, we obtain different limit theorems for linear statistics depending on the density of the process and the temperature. In particular, in a critical regime, we can observe some non-Gaussian limits. This is a joint work with K. Johansson.