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International Mathematics Research Papers (2006) Vol. 2006 : article ID 62815, 216 pages, doi:10.1155/IMRP/2006/62815
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

K. T.-R. McLaughlin, A. H. Vartanian and X. Zhou

Department of Mathematics, The University of Arizona 617 North Santa Rita Avenue, P.O. Box 210089, Tucson, AZ 85721--0089, USA E-mail address: mcl{at}math.arizona.edu
Department of Mathematics, University of Central Florida P.O. Box 161364, Orlando, FL 32816--1364, USA E-mail address: arthurv{at}math.ucf.edu
Department of Mathematics, Duke University P.O. Box 90320, Durham, NC 27708--0320, USA E-mail address: zhou{at}math.duke.edu

Let {Lambda}R denote the linear space over R spanned by zk, k isin Z. Define the real inner product (with varying exponential weights) <·,·>L:{Lambda}R x {Lambda}R -> R, (f,g) ↦ {int}R f(s)g(s) exp(–NV(s))ds, N isin N, where the external field V satisfies the following: (i) V is real analytic on R \ {0}; (ii) lim|x|->{infty}(V(x)/ln(x2 + 1)) = +{infty}; and (iii) lim|x|->0(V(x)/ln(x–2 + 1)) = +{infty}. Orthogonalisation of the (ordered) base {1,z–1,z,z–2,z2,...,z–k,zk,...} with respect to <·,·>L yields the even degree and odd degree orthonormal Laurent polynomials $${\left\{{\Phi }_{m}\left(z\right)\right\}}_{m=0}^{\infty }:{\Phi }_{2n}\left(z\right)={\xi }_{-n}^{\left(2n\right)}{z}^{-n}+\dots +{\xi }_{n}^{\left(2n\right)}{z}^{n},{\xi }_{n}^{\left(2n\right)}>0$$, and $${\Phi }_{2n+1}\left(z\right)={\xi }_{-n-1}^{\left(2n+1\right)}{z}^{-n-1}+\dots +{\xi }_{n}^{\left(2n+1\right)}{z}^{n}$$, $${\xi }_{-n-1}^{\left(2n+1\right)}>0$$. Define the even degree and odd degree monic orthogonal Laurent polynomials: $${\mathbf{\pi }}_{2n}\left(z\right):={\left({\xi }_{n}^{\left(2n\right)}\right)}^{-1}{\Phi }_{2n}\left(z\right)$$ and $${\mathbf{\pi }}_{2n+1}\left(z\right):={\left({\xi }_{-n-1}^{\left(2n+1\right)}\right)}^{-1}{\Phi }_{2n+1}\left(z\right)$$. Asymptotics in the double-scaling limit as N, n -> {infty} such that N/n = 1 + o(1) of {pi}2n(z) (in the entire complex plane), $${\xi }_{n}^{\left(2n\right)}$$, {Phi}2n(z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence {ck = {int}Rsk exp(–NV(s))ds}kisinZ are obtained by formulating the even degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on R, and then extracting the large-n behaviour by applying the nonlinear steepest-descent method introduced by P. Deift and X. Zhou and further developed by P. Deift, S. Venakides, and X. Zhou.

Current address: Department of Mathematics, College of Charleston, Charleston, SC 29424, USA



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