Szegö limit theorems and central limit for random unitary matrices
In this introductory talk, I will present the classical link between determinants of large Toeplitz matrices and the statistics of eigenvalues of random unitary matrices. In particular, I will present a classical result by Kac, named "strong Szegö limit theorem", about a two-term asymptotics for these determinants, and recast it into a central limit theorem. This is a prime example of the usage of large frequency analysis in the analysis of determinantal point processes.
Semiclassical analysis of free fermions
To each orthogonal projector of finite rank N on $L^2(R^d)$ is associated a point process on $R^d$ 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.
In this talk, I will present my work in collaboration with Gaultier Lambert (KTH) on this topic.
