# Random Hermitian Matrices and Gaussian Multiplicative Chaos

## TL;DR

This paper demonstrates that normalized powers of the characteristic polynomial of certain random Hermitian matrices converge to Gaussian multiplicative chaos measures in the $L^2$-phase, using advanced asymptotic analysis techniques.

## Contribution

It introduces a general Fisher-Hartwig formula for Hankel determinants in one-cut regular unitary invariant ensembles, linking random matrix theory with Gaussian multiplicative chaos.

## Key findings

- Convergence of normalized characteristic polynomial powers to Gaussian multiplicative chaos.
- Development of a Fisher-Hartwig formula for Hankel determinants with singularities.
- Application of Riemann-Hilbert methods to establish asymptotics.

## Abstract

We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called $L^2$-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher-Hartwig singularities. Using Riemann-Hilbert methods, we prove a rather general Fisher-Hartwig formula for one-cut regular unitary invariant ensembles.