Uniform Asymptotics for Discrete Orthogonal Polynomials with Respect to Varying Exponential Weights on a Regular Infinite Lattice

TL;DR

This paper develops uniform asymptotic formulas for discrete orthogonal polynomials on an infinite lattice with exponential weights, using Riemann-Hilbert problem techniques and steepest descent analysis.

Contribution

It introduces a novel approach to analyze large-$N$ asymptotics of discrete orthogonal polynomials via Riemann-Hilbert problem formulation and steepest descent methods.

Findings

Derived uniform asymptotic formulas for discrete orthogonal polynomials.

Established a Riemann-Hilbert problem framework for the analysis.

Provided insights into the asymptotic behavior on a regular infinite lattice.

Abstract

We consider the large-$N$ asymptotics of a system of discrete orthogonal polynomials on an infinite regular lattice of mesh $\frac{1}{N}$, with weight $e^{-NV(x)}$, where $V(x)$ is a real analytic function with sufficient growth at infinity. The proof is based on formulation of an interpolation problem for discrete orthogonal polynomials, which can be converted to a Riemann-Hilbert problem, and steepest descent analysis of this Riemann-Hilbert problem.