Orateur
Description
We study the coefficients of the characteristic polynomial of
unitary matrices drawn from the Circular Beta Ensemble. When the inverse
temperature parameter beta is strictly larger than 4, we obtain a new
class of limiting distributions that arise when both the order of the
coefficient and the dimension of the matrix goes to infinity. For beta
equal to 2, we solve an open problem of Diaconis and Gamburd by showing
that the middle coefficient tends to zero in probability when the
dimension goes to infinity. We introduce a new stochastic object
associated to the coefficients of the characteristic polynomial, which
we call Holomorphic Multiplicative Chaos (HMC). Viewing the HMC as a
random distribution, we prove a sharp result about its regularity in an
appropriate Sobolev space. Our proofs expose and exploit several novel
connections with other areas, including random permutations, Tauberian
theorems and combinatorics.
