Non-intersecting Brownian Motions and Gaussian Multiplicative Chaos

TL;DR

This paper extends static Fisher-Hartwig asymptotics to a dynamical setting for the Ornstein-Uhlenbeck process, linking non-intersecting Brownian motions with Gaussian multiplicative chaos and advancing the understanding of random matrix theory and quantum gravity.

Contribution

It introduces a dynamical Fisher-Hartwig asymptotic framework for the Ornstein-Uhlenbeck process, connecting random matrix theory with Liouville quantum gravity measures.

Findings

Convergence of fractional powers of characteristic polynomial to Gaussian multiplicative chaos.

Leading order of log-characteristic polynomial and bulk rigidity results.

Extension of single-time convergence results to a dynamical setting.

Abstract

We obtain Fisher-Hartwig asymptotics with root and jump type singularities in space-time under the law of the stationary Hermitian Ornstein-Uhlenbeck process, which serve as a dynamical generalization of earlier static results obtained by Riemann-Hilbert methods. This extends previous asymptotics by [Krasovsky 2007], [Its, Krasovsky 2008], and [Charlier 2019]. As a consequence, fractional powers of the absolute value of the characteristic polynomial of this process (and the exponential eigenvalues counting process) converge to a two dimensional Gaussian multiplicative chaos measure on an infinite strip in the subcritical phase. The dynamical Fisher-Hartwig asymptotics also provide the leading order of the log-characteristic polynomial, together with optimal bulk rigidity for non-intersecting Brownian motions. These results offer (i) the second connection between random matrix theory and Liouville quantum gravity measures after [Bourgade, Falconet 2025], by proving a dynamical generalization of the single-time convergence to the GMC from [Berestycki, Webb, Wong 2018], (ii) a dynamical extension of the maximum of the log-characteristic polynomial [Lambert, Paquette 2019] and the optimal rigidity [Claeys, Fahs, Lambert, Webb 2021].