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Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights
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 
denote the linear space over spanned by zk, k 
. Define the real inner product (with varying exponential weights) 
·,·
: x 
, (f,g) 
f(s)g(s) exp(–NV(s))ds, N 
, where the external field V satisfies the following: (i) V is real analytic on \ {0}; (ii) lim|x|
(V(x)/ln(x2 + 1)) = +; and (iii) lim|x|0(V(x)/ln(x–2 + 1)) = +. Orthogonalisation of the (ordered) base {1,z–1,z,z–2,z2,...,z–k,zk,...} with respect to ·,· 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 such that N/n = 1 + o(1) of 
2n(z) (in the entire complex plane), $${\xi }_{n}^{\left(2n\right)}$$, 
2n(z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence {ck = sk exp(–NV(s))ds}k are obtained by formulating the even degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , 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