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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn075, 66 pages, doi:10.1093/imrn/rnn075 published on July 21, 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.

The Formula Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights

K. T.-R. McLaughlin1 and P. D. Miller2

1 Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
2 Department of Mathematics, University of Michigan, East Hall, 530 Church Street, Ann Arbor, MI 48109-1109, USA

Correspondence: Correspondence to be sent to: mcl{at}math.arizona.edu

We obtain Plancherel–Rotach-type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form Formula on the real line, assuming that V has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application, we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. We develop a new technique of asymptotic analysis for matrix Riemann–Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.


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