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

P. Deift, T. Kriecherbauer, K. T.R. McLaughlin, S. Venakides, X. Zhou

Research output: Contribution to journalArticle

436 Scopus citations

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
DOIs
StatePublished - Nov 1999

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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