Dynamical invariance for random matrices

TL;DR

This paper introduces a dynamical generating function for a Coulomb gas model that satisfies a broad class of geometric constraints, linking random matrix dynamics with algebraic invariance principles.

Contribution

It develops a new dynamical generating function framework for Coulomb gas dynamics that incorporates Schr"odinger-Virasoro algebra invariance, extending understanding of matrix eigenvalue evolution.

Findings

The generating function satisfies a large class of geometric constraints.

The invariance under Schr"odinger-Virasoro algebra is established.

Connections to Virasoro and loop constraints in random matrix theory.

Abstract

We consider a general Langevin dynamics for the one-dimensional N-particle Coulomb gas with confining potential $V$ at temperature $\beta$. These dynamics describe for $\beta=2$ the time evolution of the eigenvalues of $N\times N$ random Hermitian matrices. The equilibrium partition function -- equal to the normalization constant of the Laughlin wave function in fractional quantum Hall effect -- is known to satisfy an infinite number of constraints called Virasoro or loop constraints. We introduce here a dynamical generating function on the space of random trajectories which satisfies a large class of constraints of geometric origin. We focus in this article on a subclass induced by the invariance under the Schr\"odinger-Virasoro algebra.