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

TL;DR

This paper establishes the convergence of multiplicative chaos associated with the characteristic polynomial and eigenvalue counting function of circular beta-ensembles for all positive beta, extending previous results from the unitary case.

Contribution

It provides a comprehensive proof of subcritical multiplicative chaos convergence for circular beta-ensembles, generalizing prior unitary case results to all beta > 0.

Findings

Proves multiplicative chaos convergence for circular beta-ensembles

Extends results to all beta > 0, including negative powers

Connects random matrix theory with multiplicative chaos in a comprehensive way

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.