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 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 how to obtain universality of local correlations for these models. Then, 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. Finally, if time permits, I will also mention some work in progress concerning counting statistics for free fermions, the asymptotics of entanglement entropy and a conjecture of H. Widom.