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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 $N\to \infty$ leads to the Fredholm determinant of the sine kernel $sin\pi \left(x-y\right)/\pi \left(x-y\right)$. Similarly a scaling limit at the “edge of the spectrum” leads to the Airy kernel $\left[\text{Ai}\left(x\right)\text{Ai}\left(y\right)-{\text{Ai}}^{\text{'}}\left(x\right)\text{Ai}\left(y\right)\right]/\left(x-y\right)$. 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) 15A52 Random matrices (MSC2000) 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
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