$\mathsf{GL}_N(\mathbb{C})$ Brownian motion and stochastic PDE on entire functions

TL;DR

This paper constructs the universal scaling limit of singular values of Brownian motion on $ ext{GL}_N( ext{C})$, showing they satisfy an infinite SDE system and a stochastic PDE with multiplicative noise, connecting to universal random matrix limits.

Contribution

It introduces a new universal limit for singular values of matrix Brownian motion, linking stochastic PDEs, SDE systems, and random matrix universality classes.

Findings

Limiting paths solve an infinite SDE with log-interaction.

Rescaled characteristic polynomial evolves via a stochastic PDE with multiplicative noise.

Special initial conditions recover universal limits related to random matrix products.

Abstract

We construct the full edge scaling limit of the singular values of Brownian motion on the general linear group $\mathsf{GL}_N(\mathbb{C})$ starting from general conditions. We show that the limiting paths solve an infinite system of SDE with log-interaction and have a Gibbs resampling property with exponential Brownian bridges. Moreover, we show that the evolution of the limiting rescaled reverse characteristic polynomial solves a stochastic partial differential equation with a non-linear multiplicative noise and linear drift. From a special initial condition the resulting line ensemble coincides, in logarithmic coordinates, with a line ensemble constructed by Ahn which arises as a universal scaling limit of singular values of products of random matrices. We prove some analogous results on the evolution of limiting characteristic polynomials for two models whose stationary measures are given by the Hua-Pickrell and Bessel stochastic zeta functions respectively.