Asymptotics of orthogonal polynomials and the Painlevé transcendents

Abstract

In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlevé transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the last part of this article, we list several open problems.

23 pages, 2 figures