Level Spacing of Random Matrices in an External Source

TL;DR

This paper investigates the level spacing distribution of Gaussian random matrices with an external source, revealing a new universality class characterized by a specific exponential decay in spacing probability.

Contribution

It introduces a novel universality class for level spacing in random matrices with external sources, solved via coupled nonlinear differential equations forming a Hamiltonian system.

Findings

Level spacing probability behaves like exp(-C s^{8/3}) for large s.

New universality class identified when the source spectrum creates a vanishing gap.

Solution involves coupled nonlinear differential equations forming a Hamiltonian system.

Abstract

In an earlier work we had considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials, or on the Coulomb gas representation, are not available. Nevertheless the n-point correlation functions are still given in terms of the determinant of a kernel, known through an explicit integral representation. This kernel is no longer symmetric though and is not readily accessible to standard methods. In particular finding the level spacing probability is always a delicate problem in Fredholm theory, and we have to reconsider the problem within our model. We find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state. The problem is solved through coupled non-linear differential equations, which turn out to form a Hamiltonian system. As a result we find that the level spacing probability $p(s)$ behaves like $\exp[ - C s^{{8\over{3}}}]$ for large spacing $s$; this is consistent with the asymptotic behavior $\exp[ - C s^{2 \beta + 2}]$, whenever the density of state behaves near the edge as $\rho(\lambda)\sim \lambda^{\beta}$.