Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

P. Deift, T. Kriecherbauer, Kenneth D T Mclaughlin, S. Venakides, X. Zhou

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Abstract

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e-nV(x)dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμv for V as analyzed in [8].

Original languageEnglish (US)
Pages (from-to)1335-1425
Number of pages91
JournalCommunications on Pure and Applied Mathematics
Volume52
Issue number11
StatePublished - Nov 1999

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Exponential Weights
Uniform Asymptotics
Random Matrix Theory
Orthogonal Polynomials
Universality
Polynomials
Riemann-Hilbert Problem
Steepest descent method
Orthonormal Polynomials
Equilibrium Measure
Steepest Descent Method
Matrix Models
Coefficient
Random Matrices
Recurrence
Infinity
Sufficient
Line

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. / Deift, P.; Kriecherbauer, T.; Mclaughlin, Kenneth D T; Venakides, S.; Zhou, X.

In: Communications on Pure and Applied Mathematics, Vol. 52, No. 11, 11.1999, p. 1335-1425.

Research output: Contribution to journalArticle

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