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  and also in . An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also ). 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  and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμv for V as analyzed in .
|Original language||English (US)|
|Number of pages||91|
|Journal||Communications on Pure and Applied Mathematics|
|State||Published - Nov 1999|
ASJC Scopus subject areas
- Applied Mathematics