Gaultier Lambert -- Applications of the theory of Gaussian multiplicative chaos to random matrices

Abstract: Log-correlated fields are a class of stochastic processes which describe the fluctuations of some key observables in different probabilistic models in dimension 1 and 2 such as random tilings, or the characteristic polynomials of random matrices. Gaussian multiplicative chaos is a renormalization procedure which aims at defining the exponential of a Log-correlated field in the form of a family of random measures. These random measures can be thought of as describing the extreme values of the underlying field. In this talk, I will present some applications of this theory to study the logarithm of the characteristic polynomial of some random matrices. I will focus on the Ginibre ensemble and also mention some results for the Gaussian unitary ensemble and circular beta ensembles.