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Level-Spacing Distributions and the Airy Kernel | | Craig A. Tracy
; Harold Widom
; | | Date: |
14 Oct 1992 | | Journal: | Phys.Lett. B305 (1993) 115-118 | | Subject: | High Energy Physics - Theory; Exactly Solvable and Integrable Systems; Mathematical Physics | hep-th cond-mat math-ph math.MP nlin.SI solv-int | | Abstract: | Scaling level-spacing distribution functions in the ``bulk of the spectrum’’ in random matrix models of $N imes N$ hermitian matrices and then going to the limit $N oinfty$, leads to the Fredholm determinant of the sine kernel $sinpi(x-y)/pi (x-y)$. Similarly a double scaling limit at the ``edge of the spectrum’’ leads to the Airy kernel $[{
m Ai}(x) {
m Ai}’(y) -{
m Ai}’(x) {
m Ai}(y)]/(x-y)$. We announce analogies 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, M{ôri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{é 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. | | Source: | arXiv, hep-th/9210074 | | Other source: | [GID 1046698] hep-th/9211141 | | Services: | Forum | Review | PDF | Favorites | |
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