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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 $$N\times N$$ 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.

##### MSC:
 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.)
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