Diffusion in the space of complex Hermitian matrices - microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial

TL;DR

This paper studies how the averaged characteristic and inverse characteristic polynomials of Hermitian matrices evolve under a matrix-valued random walk, deriving diffusion-like equations valid for any matrix size and analyzing their asymptotics.

Contribution

It introduces diffusion equations for averaged polynomials of Hermitian matrices undergoing a random walk, providing integral solutions and asymptotic analysis for arbitrary matrix sizes.

Findings

Derived diffusion-like PDEs for averaged polynomials

Provided integral representations for solutions

Analyzed asymptotic behaviors for various initial conditions

Abstract

We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like, equations. These equations are valid for matrices of arbitrary size. Their solutions can be given an integral representation that allows for a simple study of their asymptotic behaviors for a broad range of initial conditions.