Uniform Asymptotics for Orthogonal Polynomials

. We consider asymptotics of orthogonal polynomials with respect to a weight e \GammaQ(x) dx on R, where either Q(x) is a polynomial of even order with positive leading coefficient, or Q(x) = NV (x), where V (x) is real analytic on R and grows sufficiently rapidly as jxj ! 1. We formulate the orthogonal polynomial problem as a Riemann-Hilbert problem following the work of Fokas, Its and Kitaev. We employ the steepest descent-type method for Riemann-Hilbert problems introduced by Deift and Zhou, and further developed by Deift, Venakides and Zhou, in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients and the recurrence coefficients of the orthogonal polynomials. These asymptotics are also used to prove various universality conjectures in the theory of random matrices. 1991 Mathematics Subject Classification: 33D45, 60F99, 15A52, 45E05. Keywords and Phrases: orthogonal polynomia..