Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results

TL;DR

This paper develops uniform asymptotic formulas for a broad class of discrete orthogonal polynomials, including Krawtchouk and Hahn, and applies these results to establish universality in related random ensemble models.

Contribution

It introduces a general approach for asymptotics of discrete orthogonal polynomials with non-uniform nodes and extends Riemann-Hilbert methods to handle pole accumulation.

Findings

Asymptotic formulas with error bounds for discrete orthogonal polynomials.

Universality results for correlation functions in discrete polynomial ensembles.

Asymptotic analysis of statistics in random tiling models.

Abstract

We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical discrete orthogonal polynomials, but is far more general. In particular, we consider nodes that are not necessarily equally spaced. The asymptotic results are given with error bound for all points in the complex plane except for a finite union of discs of arbitrarily small but fixed radii. These exceptional discs are the neighborhoods of the so-called band edges of the associated equilibrium measure. As applications, we prove universality results for correlation functions of a general class of discrete orthogonal polynomial ensembles, and in particular we deduce asymptotic formulae with error bound for certain statistics relevant in the random tiling of a hexagon with rhombus-shaped tiles. The discrete orthogonal polynomials are characterized in terms of a a Riemann-Hilbert problem formulated for a meromorphic matrix with certain pole conditions. By extending the methods of [17, 22], we suggest a general and unifying approach to handle Riemann-Hilbert problems in the situation when poles of the unknown matrix are accumulating on some set in the asymptotic limit of interest.