Level-Spacing Distributions and the Airy Kernel

TL;DR

This paper explores the properties of the Airy kernel in random matrix theory, drawing analogies to the sine kernel, and discusses its integrability, determinant expressions, and applications to eigenvalue probability calculations.

Contribution

It introduces the Airy kernel's properties analogous to the sine kernel, including integrability, Painlevé transcendent expressions, and commuting differential operators.

Findings

Airy kernel shares properties with the sine kernel.

Fredholm determinant expressed via Painlevé transcendent.

Differential operator aids in eigenvalue probability asymptotics.

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 double scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x) {\rm Ai}(y)]/(x-y)$. We announce analogies 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.