Alix Deleporte: Semiclassical analysis of free fermions

Abstract:

To each orthogonal projector of finite rank \(N\) on \(L^2(\mathbf{R}^d)\) is associated a point process on \(\mathbf{R}^d\) with \(N\) points, which gives the joint probability density of fermions that fill the image of the projector. The study of the statistical properties of these fermions, in the large \(N\) limit, is linked to semiclassical spectral theory problems, some of them well studied (the Weyl law gives a law of large numbers), some of them new. In particular, the behaviour of the variance is linked with the properties of commutators involving spectral projectors, which are not so well understood.