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Level-Spacing Distributions and the Airy Kernel (1994)
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| Venue: | COMMUNICATIONS IN MATHEMATICAL PHYSICS |
| Citations: | 429 - 23 self |
BibTeX
@MISC{Tracy94level-spacingdistributions,
author = {Craig A. Tracy and Harold Widom},
title = {Level-Spacing Distributions and the Airy Kernel},
year = {1994}
}
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Abstract
Scaling level-spacing distribution functions in the "bulk of the spectrum" in random matrix models of N x N hermitian matrices and then going to the limit N — » oo 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 [Ai(x) Ai(y) — Ai (x) 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, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painleve 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.
Keyphrases
airy kernel level-spacing distribution fredholm determinant differential operator sine kernel integrable system painleve transcendent level-spacing distribution function airy kernel ai random matrix model hermitian matrix scaling limit sine kernel sin single interval spectrum quot following property
