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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
Author(s) -
Deift P.,
Kriecherbauer T.,
McLaughlin K. TR,
Venakides S.,
Zhou X.
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199911)52:11<1335::aid-cpa1>3.0.co;2-1
Subject(s) - mathematics , orthogonal polynomials , random matrix , universality (dynamical systems) , mehler–heine formula , exponential function , real line , discrete orthogonal polynomials , pure mathematics , classical orthogonal polynomials , method of steepest descent , orthonormal basis , mathematical analysis , gegenbauer polynomials , eigenvalues and eigenvectors , physics , quantum mechanics
We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w n ( 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 dμ V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.