Communications in Mathematical Physics

, Volume 159, Issue 1, pp 151–174 | Cite as

Level-spacing distributions and the Airy kernel

  • Craig A. Tracy
  • Harold Widom


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.


Neural Network Distribution Function Statistical Physic Complex System Sine 
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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Craig A. Tracy
    • 1
  • Harold Widom
    • 2
  1. 1.Department of Mathematics and Institute of Theoretical DynamicsUniversity of CaliforniaDavisUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSanta CruzUSA

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