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PDEs for the Gaussian ensemble with external source and the Pearcey distribution
Author(s) -
Adler Mark,
van Moerbeke Pierre
Publication year - 2007
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.20175
Subject(s) - mathematics , hermitian matrix , eigenvalues and eigenvectors , random matrix , gaussian , quartic function , mathematical analysis , diagonal , diagonal matrix , matrix (chemical analysis) , nonlinear system , brownian motion , statistical physics , physics , quantum mechanics , pure mathematics , geometry , statistics , materials science , composite material
The present paper studies a Gaussian Hermitian random matrix ensemble with external source, given by a fixed diagonal matrix with two eigenvalues ± a . As a first result, the probability that the eigenvalues of the ensemble belong to an interval E satisfies a fourth‐order PDE with quartic nonlinearity; the variables are the eigenvalue a and the boundary of E . This equation enables one to find a PDE for the Pearcey distribution. The latter describes the statistics of the eigenvalues near the closure of a gap, i.e., when the support of the equilibrium measure for large‐size random matrices has a gap that can be made to close. The Gaussian Hermitian random matrix ensemble with external source, described above, has this feature. The Pearcey distribution is shown to satisfy a fourth‐order PDE with cubic nonlinearity. This also gives the PDE for the transition probability of the Pearcey process, a limiting process associated with nonintersecting Brownian motions on ℝ. © 2006 Wiley Periodicals, Inc.