Non-Equilibrium Random Matrix Theory : Transition Probabilities

TL;DR

This paper develops an analytic approach to compute transition probabilities between Gaussian matrices with specified eigenvalues, revealing how initial state memory diminishes over time in large systems.

Contribution

It introduces a novel method for calculating transition probabilities in non-equilibrium random matrix dynamics using Coulomb gas analogy.

Findings

Transition probabilities depend on a universal linear potential in large N limit.

Memory of initial eigenvalues persists in a linear potential form.

Transition probabilities converge to static ensemble values as time progresses.

Abstract

In this letter we present an analytic method for calculating the transition probability between two random Gaussian matrices with given eigenvalue spectra in the context of Dyson Brownian motion. We show that in the Coulomb gas language, in large $N$ limit, memory of the initial state is preserved in the form of a universal linear potential acting on the eigenvalues. We compute the likelihood of any given transition as a function of time, showing that as memory of the initial state is lost, transition probabilities converge to those of the static ensemble.