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Zbl 0789.35152
Tracy, Craig A.; Widom, Harold
Level-spacing distributions and the Airy kernel. (English)
[J] Commun. Math. Phys. 159, No.1, 151-174 (1994). ISSN 0010-3616; ISSN 1432-0916/e

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 2000:: *35Q58 Other completely integrable PDE
15A52 Random matrices
37J35 Completely integrable systems, etc.
37K10 Completely integrable systems etc.

Keywords: random matrices; random matrix models; limit; completely integrable system; Fredholm determinant

Cited in: Zbl 1235.42023 Zbl 1182.62180 Zbl 1140.47036 Zbl 1135.47046 Zbl 1136.82021 Zbl 1108.62014 Zbl 1085.60014 Zbl 1070.15013 Zbl 1060.60025 Zbl 1045.15012 Zbl 1039.60037 Zbl 1031.60084 Zbl 1042.82019 Zbl 1062.82502

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