Gaultier Lambert: Statistical properties of free fermions

Abstract

The ground state of a free Fermi gas is a classical example of determinantal processes whose correlation kernel is associated with a Schrödinger operator on \(\mathbb{R}^n\). This observation is due to Macchi (1975) and determinantal processes have been intensively studied since then. In this talk, I will report on recent results that we obtained jointly with Alix Deleporte (Université Paris-Saclay) and Sylvain Chabredier describing the fluctuations of these determinantal processes using techniques from semiclassical analysis.

I will first explain the model and how to obtain universality of local correlations for these models. Then, if time permits, I will present a central limit theorem valid for linear statistics of generic one-dimensional Fermi gas, as well as an analogous result for the \(n\)-dimensional.