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

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.

arXiv comment18 pages, 3 figures, addition of important reference, changes in introduction