Subcritical multiplicative chaos and the characteristic polynomial of the C$\beta$E

Abstract

The goal of this article is to expand on the relationship between random matrix and multiplicative chaos theories using the integrability properties of the circular beta-ensembles. We give a comprehensive proof of the multiplicative chaos convergence for the characteristic polynomial and eigenvalue counting function of the circular beta-ensembles throughout the subcritical phase, including negative powers. This generalizes recent results in the unitary case, [NSW20,BF22], to any beta>0 and for the eigenvalue counting field.