Level-Spacing Distributions and the Airy Kernel

TL;DR

This paper explores the properties of the Airy kernel, which arises in the edge scaling limit of random matrix spectra, deriving integrable systems, Fredholm determinants, and asymptotic formulas similar to those known for the sine kernel.

Contribution

It extends known properties of the sine kernel to the Airy kernel, including integrable PDEs, Painlevé representations, and commuting differential operators.

Findings

Derived the Jimbo-Miwa-Mori-Sato PDE system for the Airy kernel.

Expressed the Fredholm determinant in terms of Painlevé transcendents.

Established a commuting differential operator for the Airy kernel.

Abstract

Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\times N$ hermitian matrices and then going to the limit $N\to\infty$, leads to the Fredholm determinant of the sine kernel $\sin\pi(x-y)/\pi (x-y)$. Similarly a scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x) {\rm Ai}(y)]/(x-y)$. In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\'e} transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general $n$, of the probability that an interval contains precisely $n$ eigenvalues.