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Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy
Author(s) -
Claeys Tom,
Krasovsky Igor,
Its Alexander
Publication year - 2010
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20284
Subject(s) - mathematics , hierarchy , fredholm determinant , kernel (algebra) , eigenvalues and eigenvectors , order (exchange) , fredholm theory , random matrix , distribution (mathematics) , unitary state , enhanced data rates for gsm evolution , pure mathematics , fredholm integral equation , mathematical analysis , integral equation , political science , law , telecommunications , physics , finance , quantum mechanics , economics , computer science , market economy
We study Fredholm determinants related to a family of kernels that 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. © 2009 Wiley Periodicals, Inc.