Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory

TL;DR

This paper investigates the universality properties of a constrained random matrix model, revealing non-universality macroscopically but confirming universality microscopically at the scale of mean level spacing.

Contribution

It extends the understanding of universality in constrained random matrix models by analyzing both macroscopic non-universality and microscopic universality, providing explicit formulas and relations.

Findings

Macroscopic two-point resolvent is non-universal for RTE.

Microscopic correlations follow the universal sine-law.

Microscopic universality holds despite macroscopic non-universality.

Abstract

A random matrix model with a sigma-model like constraint, the restricted trace ensemble (RTE), is solved in the large-n limit. In the macroscopic limit the smooth connected two-point resolvent G(z,w) is found to be non-universal, extending previous results from monomial to arbitrary polynomial potentials. Using loop equation techniques we give a closed though non-universal expression for G(z,w), which extends recursively to all higher k-point resolvents. These findings are in contrast to the usual unconstrained one-matrix model. However, in the microscopic large-n limit, which probes only correlations at distance of the mean level spacing, we are able to show that the constraint does not modify the universal sine-law. In the case of monomial potentials $V(M)=M^{2p}$, we provide a relation valid for finite-n between the k-point correlation function of the RTE and the unconstrained model. In the microscopic large-n limit they coincide which proves the microscopic universality of RTEs.