Level-spacing distributions and the Airy kernel. (English) Zbl 0789.35152

Summary: 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 \([\text{Ai}(x) \text{Ai}(y)- \text{Ai}'(x) \text{Ai}(y)]/ (x-y)\). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of PDE’s found by Jimbo, Miwa, Môri, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlevé 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.


35Q58 Other completely integrable PDE (MSC2000)
15B52 Random matrices (algebraic aspects)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text: DOI arXiv


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