Speaker
Francois, Université de Lille
Abstract
We prove the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure with weights assigned to each cycle length. The proof relies on singularity analysis of generating functions, together with the convergence of traces to explicit random variables expressed via a Poisson family. The limit function is the exponential of a Poisson series which appeared in the case of uniform permutation matrices. It is the Poisson analog of the Gaussian multiplicative chaos, related to the limit of characteristic polynomials for other matrix models such as Circular Ensembles, i.i.d. matrices, and Gaussian elliptic matrices.