Multiplicative chaos measures from thick points of log-correlated fields

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.