Multiplicative chaos and the characteristic polynomial of the CUE: the $L^1$-phase

Abstract

In this note we prove that suitable positive powers of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix converge in law, as the size of the matrix tends to infinity, to a Gaussian multiplicative chaos measure once correctly normalized. We prove this in the whole $L^1$- or subcritical phase of the chaos measure.

References in corpus (7)

• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function
• Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
• Toeplitz determinants with merging singularities
• On the asymptotics of a Toeplitz determinant with singularities
• Integrability of Liouville theory: proof of the DOZZ Formula
• The law of large numbers for the maximum of almost Gaussian log-correlated fields coming from random matrices

Cited by in corpus (3)

• Critical Gaussian multiplicative chaos: a review
• On the critical-subcritical moments of moments of random characteristic polynomials: a GMC perspective
• Tail universality of critical Gaussian multiplicative chaos