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Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights. (English) Zbl 1112.33010

Let \(\Lambda^{\mathbb R}\) denote the linear space over \(\mathbb R\) spanned by \(z^k\), \(k\in\mathbb Z\). Define the real inner product (with varying exponential weights) \(\langle\cdot,\cdot\rangle_{{\mathcal L}}:\Lambda^{\mathbb R}\times \Lambda^{\mathbb R}\to \mathbb R\), \((f,g)\mapsto \int_{\mathbb R}f(s) g(s)\exp(-{\mathcal N}{\mathcal V}(s))\,ds\), \({\mathcal N}\in\mathbb N\), where the external field \({\mathcal V}\) satisfies the following: (i) \({\mathcal V}\) is real analytic on \(\mathbb R\setminus\{0\}\); (ii) \(\lim_{|x|\to\infty}({\mathcal V}(x)/\ln(x^2+ 1))= +\infty\); and (iii) \(\lim_{|x|\to 0}({\mathcal V}(x)/\ln(x^{-2}+ 1))=+\infty\). Orthogonalisation of the (ordered) base \(\{1, z^{-1},z, z^{-2},z^2,\dots, z^{-k}, z^k,\dots\}\) with respect to \(\langle\cdot, \cdot\rangle_{{\mathcal L}}\) yields the even degree and odd degree orthonormal Laurent polynomials\(\{\phi_m(z)\}^\infty_{m=0}: \phi_{2n}(z)= \xi^{(2n)}_{-n} z^{-n}+\cdots+ \xi^{(2n)}_n z^n,\xi^{(2n)}_n> 0\), and \(\phi_{2n+1}(z)= \xi^{(2n+1)}_{-n-1} z^{-n-1}+\cdots+ \xi^{(2n+1)}_n z^n\), \(\xi^{(2n+1)}_{n-1}> 0\). Define the even degree and odd degree monic orthogonal Laurent polynomials: \(\pi_{2n}(z):= (\xi^{(2n)}_n)^{-1} \phi_{2n}(z)\) and \(\pi_{2n+1}(z):= (\xi^{(2n+1)}_{-n-1})^{-1} \phi_{2n+1}(z)\). Asymptotics in the double-scaling limit as \({\mathcal N}\), \(n\to\infty\) such that \({\mathcal N}/n= 1 + o(1)\) of \(\pi_{2n}(z)_i\) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence \(\{c_k= \int_{\mathbb R} s^k\exp(-{\mathcal N}{\mathcal V}(s))\,ds\}_{k\in\mathbb Z}\) are obtained by formulating the even degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on \(\mathbb R\), 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.
This research paper contains 5 sections: 1. Introduction and Background; 2. Hyperelliptic Riemann surfaces, the Riemann-Hilbert problems and summary of results; 3. The equilibrium measure, the variational problem, and the transformed RHP; 4. The model RHP and parametrices; 5. Asymptotic (as \(n\to\infty\)) solution of the RHP for \(y(z)\); Appendix. Small-\(z\) asymptotics for \(Jy(z)\). And a list of 108 references. The paper is very technical, use matrix analysis, complex variable, integration in one and several variables.

MSC:

33C47 Other special orthogonal polynomials and functions
30E05 Moment problems and interpolation problems in the complex plane
44A60 Moment problems
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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