Glassy Random Matrix Models

TL;DR

This paper investigates random matrix models with multiple solutions at the same phase space point, revealing their spectral gaps, differences in free energy and correlations, and potential links to structural glasses.

Contribution

It provides analytical and numerical evidence for multiple solutions in glassy random matrix models, including their spectral properties and correlation functions, and explores their non-perturbative aspects.

Findings

Multiple solutions exist at the same phase space point.

Spectral gaps and eigenvalue density differences are characterized.

Two-point correlation functions reveal a new universality class.

Abstract

This paper discusses Random Matrix Models which exhibit the unusual phenomena of having multiple solutions at the same point in phase space. These matrix models have gaps in their spectrum or density of eigenvalues. The free energy and certain correlation functions of these models show differences for the different solutions. Here I present evidence for the presence of multiple solutions both analytically and numerically. As an example I discuss the double well matrix model with potential $V(M)= -{\mu \over 2}M^2+{g \over 4}M^4$ where $M$ is a random $N\times N$ matrix (the $M^4$ matrix model) as well as the Gaussian Penner model with $V(M)={\mu\over 2}M^2-t \ln M$. First I study what these multiple solutions are in the large $N$ limit using the recurrence coefficient of the orthogonal polynomials. Second I discuss these solutions at the non-perturbative level to bring out some differences between the multiple solutions. I also present the two-point density-density correlation functions which further characterizes these models in a new university class. A motivation for this work is that variants of these models have been conjectured to be models of certain structural glasses in the high temperature phase.