Math @ Duke

Publications [#244283] of Xin Zhou
Papers Published
 with Deift, P; Kriecherbauer, T; McLaughlin, KTR; Venakides, S; Zhou, X, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory,
Communications on Pure and Applied Mathematics, vol. 52 no. 11
(1999),
pp. 13351425 [MR1702716]
(last updated on 2018/07/18)
Abstract: We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = enV(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 PlancherelRotachtype 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 RiemannHilbert problem following [19, 20]. The RiemannHilbert problem is analyzed in turn using the steepestdescent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμv for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.


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