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

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 --> infinity. 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 r ecently in [31] and also in [4]. An additional application concerns the asy mptotics of the recurrence coefficients and leading coefficients for the or thonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert proble m following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [1 1, 13]. A critical role in our method is played by the equilibrium measure d(mu V) for V as analyzed in [8]. (C) 1999 John Wiley & Sons, Inc.