Higher order analogues of the Tracy-Widom distribution and the Painleve II hierarchy

Abstract

We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painlevé I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlevé II hierarchy. In addition we compute large gap asymptotics for the Fredholm determinants.

arXiv comment45 pages

References in corpus (6)

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• Level Spacing of Random Matrices in an External Source
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