Level-Spacing Distributions and the Airy Kernel
Craig A. Tracy, Harold Widom
TL;DR
This work establishes a parallel between the edge-level spacing statistics governed by the Airy kernel and the classical sine-kernel results in random matrix theory. It derives a fully integrable system of partial differential equations for the Airy kernel's Fredholm determinant, obtains a Painlev\'e II representation for single-interval determinants, and proves the existence of a commuting differential operator that yields detailed asymptotics for the eigenvalue-count probabilities $E(n;s)$. By combining Painlev\'e II analysis with the commuting-operator framework, the authors obtain precise large-$s$ asymptotics for the edge statistics, including explicit constants and expansions for $q_n(s)$, $r(n;s)$, and $\lambda_i$, as well as turning-point behavior of the associated eigenfunctions. The results deepen the connection between edge universality in random matrices and integrable systems, providing practical asymptotics for edge eigenvalue distributions and their densities with potential implications for related stochastic and spectral problems.
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 $\sinπ(x-y)/π(x-y)$. 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{ô}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{é} 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.