Uniform Asymptotics for Orthogonal Polynomials

We consider asymptotics of orthogonal polynomials with respect to a weight e ?Q(x) dx on R, where either Q(x) is a polynomial of even order with positive leading coeecient, or Q(x) = NV (x), where V (x) is real analytic on R and grows suuciently 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 coeecients and the recurrence coeecients of the orthogonal poly-nomials. These asymptotics are also used to prove various universality conjectures in the theory of random matrices. Let w(x)dx = e ?Q(x) dx be a measure on the real line. Denote by n (x; Q) = n (x) = x n + : : : the n-th monic orthogonal polynomial with respect to the measure , and by p n (x; Q) = p n (x) = n n (x), n > 0, the normalized n-th orthogonal polynomial, or simply the n-th orthogonal polynomial, i.e. Furthermore, denote by (a n) n2N , (b n) n2N the coeecients of the associated three term recurrence relation, namely, xp n (x) = b n p n+1 (x) + a n p n (x) + b n?1 p n?1 (x),