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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn037, 51 pages, doi:10.1093/imrn/rnn037 published on April 25, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Janossy Densities for Unitary Ensembles at the Spectral Edge

Brian Rider1

Xin Zhou2

1 Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309, USA
2 Department of Mathematics, Duke University, Durham, NC 27708, USA

Correspondence: Correspondence to be sent to: Brian.Rider{at}Colorado.edu

For a broad class of unitary ensembles of random matrices, we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in question. The approach is based on a representation of the Janossy densities in terms of a system of orthogonal polynomials, plus the steepest descent method of Deift and Zhou for the asymptotic analysis of the associated Riemann.Hilbert problem.


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