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PDEs for the Gaussian ensemble with external source and the Pearcey distribution

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.