Asymptotics of Recurrence Relation Coefficients, Hankel Determinant Ratios, and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

Abstract

Orthogonalisation of the (ordered) base $\lbrace 1,z^{-1},z,z^{-2},z^{2}, >...c,z^{-k},z^{k},...c \rbrace$ with respect to the real inner product $(f,g) \mapsto \int_{\mathbb{R}}f(s)g(s) \exp (-\mathscr{N} V(s)) \md s$, $\mathscr{N} \in \mathbb{N}$, where $V$ is real analytic on $\mathbb{R} \setminus \{0\}$, $\lim_{| x | \to \infty}(V(x)/ \ln (x^{2} + 1)) = +\infty$, and $\lim_{| x | \to 0} (V(x)/\ln (x^{-2} + 1)) = +\infty$, yields the even degree and odd degree orthonormal Laurent polynomials (OLPs), $\phi_{2n}(z) = \sum_{k=-n}^{n} \xi^{(2n)}_{k}z^{k}$, with $\xi^{(2n)}_{n} > 0$, and $\phi_{2n+1}(z) = \sum_{k=-n-1}^{n} \xi^{(2n+1)}_{k}z^{k}$, with $\xi^{(2n+1)}_{-n-1} > 0$, respectively. Associated with the even degree and odd degree OLPs are two pairs of three- and five-term recurrence relations. Asymptotics in the double-scaling limit as $\mathscr{N},n \to \infty$ such that $\mathscr{N}/n = 1 + o(1)$ of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios, and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on $\mathbb{R}$, and then extracting the large-N behaviours by applying the non-linear steepest-descent method introduced in [1] and further developed in [2,3].