Level Spacing of Random Matrices in an External Source

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 $\rho(\lambda)\sim \lambda^{\beta}$.