Asymptotics of spacing distributions 50 years later

Summary

In 1962 Dyson used a physically based, macroscopic argument to deduce the first two terms of the large spacing asymptotic expansion of the gap probability for the bulk state of random matrix ensembles with symmetry parameter β. In the ensuing years, the question of asymptotic expansions of spacing distributions in random matrix theory has shown itself to have a rich mathematical content. As well as presenting the main known formulas, we give an account of the mathematical methods used for their proofs, and provide some new formulas. We also provide a high precision numerical computation of one of the spacing probabilities to illustrate the accuracy of the corresponding asymptotics.

1. Introduction

Random matrices were introduced in physics by Wigner in the 1950s; see [Porter 1965]. Wigner's original hypothesis was that the statistical properties of energy levels of complex nuclei could be reproduced by considering an ensemble of systems rather than a single system in which all interactions are completely described. This allowed for an entirely mathematical approach where statistical properties of the spectrum of an ensemble of random matrices were considered. But coming from physics, the aim was to use mathematics to compute experimentally measurable statistical quantities, and to compare against the data.

One viewpoint on a real spectrum from a random matrix is as a point process on the real line. As such, perhaps the most natural statistical characterization is that of the distribution of the eigenvalue spacing. This choice of statistic becomes even more compelling when one considers that in many cases of interest, eigenvalue spectra can be “unfolded”. This means that unlike many statistical mechanical systems, the density is not an independent control variable, but rather fixes the length scale only. Unfolding then is scaling the eigenvalues in the bulk of the spectrum so that the mean density is unity. It is indeed the bulk spacing distribution for the Gaussian orthogonal ensemble of real symmetric matrices— albeit in an approximate form known as the Wigner surmise (see, e.g., [Mehta 1991])—which was compared against the empirical spacing distribution for the energy level of highly excited nuclei (again, see [Mehta 1991], and references therein).