Multiplicative chaos measures from thick points of log-correlated fields

TL;DR

This paper constructs multiplicative chaos measures from thick points of log-correlated fields, applying the method to the characteristic polynomial of Haar random matrices, confirming a conjecture on fluctuations.

Contribution

It introduces a new method to build chaos measures from extreme level sets of log-correlated fields, applicable across the subcritical phase.

Findings

Established asymptotics for the characteristic polynomial of Haar random matrices.

Proved the conjecture of Fyodorov and Keating on fluctuations of thick points.

Extended the construction of chaos measures to the entire subcritical phase.

Abstract

We prove that multiplicative chaos measures can be constructed from extreme level sets or thick points of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish these estimates hold for the logarithm of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix (CUE), using known asymptotics for Toeplitz determinant with (merging) Fisher-Hartwig singularities. Hence, this proves a conjecture of Fyodorov and Keating concerning the fluctuations of the volume of thick points of the CUE characteristic polynomial.