Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Details Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

2006-01-24

arxiv.org/abs/math/0601595v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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) $<\boldsymbol{\cdot},\boldsymbol{\cdot} >_{\mathscr{L}} \colon \Lambda^{\mathbb{R}} \times \Lambda^{\mathbb{R}} \to \mathbb{R}$, $(f,g) \mapsto \int_{\mathbb{R}}f(s)g(s) \exp (-\mathscr{N} V(s)) \md s$, $\mathscr{N} \in \mathbb{N}$, where the external field $V$ satisfies: (i) $V$ is real analytic on $\mathbb{R} \setminus \{0\}$; (ii) $\lim_{| x | \to \infty}(V(x)/\ln (x^{2} + 1)) = +\infty$; and (iii) $\lim_{| x | \to 0}(V(x)/\ln (x^{-2} + 1)) = +\infty$. Orthogonalisation of the (ordered) base $\lbrace 1,z^{-1},z,z^{-2},z^{2},...c,z^{-k},z^{k},...c \rbrace$ with respect to $<\boldsymbol{\cdot},\boldsymbol{\cdot} >_{\mathscr{L}}$ yields the even degree and odd degree orthonormal Laurent polynomials $\lbrace \phi_{m}(z) \rbrace_{m=0}^{\infty}$: $\phi_{2n}(z) = \xi^{(2n)}_{-n}z^{-n} + ...b + \xi^{(2n)}_{n}z^{n}$, $\xi^{(2n)}_{n} > 0$, and $\phi_{2n+1}(z) = \xi^{(2n+1)}_{-n-1} z^{-n-1} + ...b + \xi^{(2n+1)}_{n}z^{n}$, $\xi^{(2n+1)}_{-n-1} > 0$. Asymptotics in the double-scaling limit as $\mathscr{N},n \to \infty$ such that $\mathscr{N}/n = 1 + o(1)$ of $\xi^{(2n+1)}_{-n-1}$ and $\phi_{2n+1}(z)$ (in the entire complex plane) are obtained by formulating the odd degree orthonormal Laurent polynomial problem as a matrix Riemann-Hilbert problem on $\mathbb{R}$, and then extracting the large-N behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2,3].

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