The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The $L^2$-phase

Christian Webb

Published 2014-10-03Version 1

We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, the absolute value of the characteristic polynomial (and suitable powers of it) converges in law to a Gaussian multiplicative chaos measure (for complex powers the convergence is to a random distribution).