Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory

Abstract

We consider polynomials orthogonal on $[0,\infty )$ with respect to Laguerre-type weights $w(x)=x^{\alpha }{e}^{-Q(x)}$, where $\alpha >-1$ and where $Q$ denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach type asymptotics in the entire complex plane for the orthonormal polynomials with respect to $w$, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a $2\times 2$ matrix valued Riemann-Hilbert problem, due to Fokas, Its and Kitaev, together with an application of the Deift-Zhou steepest descent method to analyze the Riemann-Hilbert problem asymptotically.

45 pages, 6 figures, error term in formula (2.22) corrected, new references added