Universal shocks in random matrix theory

TL;DR

This paper connects universal kernels in random matrix theory to shock formation in fluid dynamics, revealing a new perspective on universality through Burgers equation analysis of eigenvalue spectra.

Contribution

It introduces a novel link between random matrix universality and shock phenomena in fluid equations derived from Dyson's random walks, focusing on the GUE case.

Findings

Orthogonal polynomials evolve via a viscous Burgers equation.

Spectral edges correspond to shocks in the Burgers equation.

Provides a new perspective on universality in random matrices.

Abstract

We link the appearance of universal kernels in random matrix ensembles to the phenomenon of shock formation in some fluid dynamical equations. Such equations are derived from Dyson's random walks after a proper rescaling of the time. In the case of the Gaussian Unitary Ensemble, on which we focus in this letter, we show that the orthogonal polynomials, and their Cauchy transforms, evolve according to a viscid Burgers equation with an effective "spectral viscosity" $\nu_s=1/2N$, where $N$ is the size of the matrices. We relate the edge of the spectrum of eigenvalues to the shock that naturally appears in the Burgers equation for appropriate initial conditions, thereby obtaining a new perspective on universality.