Communications in Mathematical Physics

, Volume 159, Issue 1, pp 151–174

Level-spacing distributions and the Airy kernel


  • Craig A. Tracy
    • Department of Mathematics and Institute of Theoretical DynamicsUniversity of California
  • Harold Widom
    • Department of MathematicsUniversity of California

DOI: 10.1007/BF02100489

Cite this article as:
Tracy, C.A. & Widom, H. Commun.Math. Phys. (1994) 159: 151. doi:10.1007/BF02100489


Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry 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 P.D.E.'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 generaln, of the probability that an interval contains preciselyn eigenvalues.

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© Springer-Verlag 1994