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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×N hermitian matrices and then going to the limit N 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 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.

MSC:
35Q58Other completely integrable PDE (MSC2000)
15A52Random matrices (MSC2000)
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
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