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

Abstract

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) $⟨\cdot ,\cdot {⟩}_{\mathcal{L}}:{\Lambda }^{\mathbb{R}}\times {\Lambda }^{\mathbb{R}}\to \mathbb{R}$, $(f,g)\mapsto {\int }_{\mathbb{R}}f(s)g(s)\exp (-\mathcal{N}V(s))ds$, $\mathcal{N}\in \mathbb{N}$, 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 $\{1,z^{-1},z,z^{-2},z^2,\dots ,z^{-k},z^k,\dots \}$ with respect to $⟨\cdot ,\cdot {⟩}_{\mathcal{L}}$ yields the even degree and odd degree orthonormal Laurent polynomials $\{{\varphi }_m(z){\}}_{m=0}^{\infty }$: ${\varphi }_{2n}(z)={\xi }_{-n}^{(2n)}{z}^{-n}+\cdots +{\xi }_n^{(2n)}{z}^n$, ${\xi }_n^{(2n)}>0$, and ${\varphi }_{2n+1}(z)={\xi }_{-n-1}^{(2n+1)}{z}^{-n-1}+\cdots +{\xi }_n^{(2n+1)}{z}^n$, ${\xi }_{-n-1}^{(2n+1)}>0$. Asymptotics in the double-scaling limit as $\mathcal{N},n\to \infty$ such that $\mathcal{N}/n=1+o(1)$ of ${\xi }_n^{(2n)}$ and ${\varphi }_{2n}(z)$ (in the entire complex plane) are obtained by formulating the even degree orthonormal Laurent polynomial problem as a matrix Riemann-Hilbert problem on $\mathbb{R}$, and then extracting the large-$n$ behaviour by applying the Deift-Zhou non-linear steepest-descent method in conjunction with the extension of Deift-Venakides-Zhou.