Year: 1998
Type: article
Abstract: In an earlier work we 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, however, 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 class of universality for the level spacing distribution when the spectrum of the source is adjusted to produce a vanishing gap in the density of the state. The problem is solved through coupled nonlinear differential equations, which turn out to form a Hamiltonian system. As a result we find that the level spacing probability $p(s)$ behaves like $\mathrm{exp}[\ensuremath{-}{\mathrm{Cs}}^{8/3}]$ for large spacing s; this is consistent with the asymptotic behavior $\mathrm{exp}[\ensuremath{-}{\mathrm{Cs}}^{2\ensuremath{\beta}+2}],$ whenever the density of state behaves near the edge as $\ensuremath{\rho}(\ensuremath{\lambda})\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{\beta}}.$
Authors: E. Brézin, S. Hikami
Institutions: École Normale Supérieure - PSL, The University of Tokyo
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2048981998