The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights

TL;DR

This paper analyzes the asymptotic behavior of recurrence coefficients for orthogonal polynomials with exponential weights, revealing their expansion patterns using Riemann-Hilbert problem techniques.

Contribution

It provides the first detailed asymptotic expansions of recurrence coefficients for orthogonal polynomials with varying exponential weights in the one-cut regular case.

Findings

Recurrence coefficients have asymptotic expansions in powers of 1/n and 1/n^2.

The method used is the Deift-Zhou steepest descent for Riemann-Hilbert problems.

Results apply to one-cut regular potentials V.

Abstract

We consider orthogonal polynomials $\{p_{n,N}(x)\}_{n=0}^{\infty}$ on the real line with respect to a weight $w(x)=e^{-NV(x)}$ and in particular the asymptotic behaviour of the coefficients $a_{n,N}$ and $b_{n,N}$ in the three term recurrence $x \pi_{n,N}(x) = \pi_{n+1,N}(x) + b_{n,N} \pi_{n,N}(x) + a_{n,N} \pi_{n-1,N}(x)$. For one-cut regular $V$ we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrence coefficients $a_{n,n}$ and $b_{n,n}$ have asymptotic expansions as $n \to \infty$ in powers of $1/n^2$ and powers of $1/n$, respectively.