Speaker:
David García-Zelada, Sorbonne Université
Abstract:
We shall consider the classical Boltzmann-Gibbs measure on the space of random normal matrices given by \(\exp[-\mathrm{Tr}(\kappa_n V(M))] \mathrm d M\) for \(\kappa_n \sim n\), where \(n\) is the dimension of the matrix. It is well-known that the empirical measures associated to the eigenvalues of such random matrices converge to some probability measure \(\mu_V\) that depends on \(V\). The question in this talk will be about the behavior of the eigenvalues outside the support of \(\mu_V\), i.e., are there outliers? An affirmative answer will be given in the case where \(V\) is the potential generated by \(\mu_V\). Moreover, I will present a description of a family of universal determinantal point processes arising in the limit. We will see that the language where the answer seems very natural is the one of Hermitian line bundles. So, I will explain how the classification of flat Hermitian line bundles over the analyzed region gives us the classification of possible limit point processes. This is joint work with Raphaël Butez, Alon Nishry and Aron Wennman arXiv:2104.03959.