Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights
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Let $Λ^{\mathbb{R}}$ denote the linear space over $\mathbb{R}$ spanned by $z^{k}$, $k \in \mathbb{Z}$. Define the real inner product (with varying exponential weights) $<\boldsymbol{\cdot},\boldsymbol{\cdot} >_{\mathscr{L}} \colon Λ^{\mathbb{R}} \times Λ^{\mathbb{R}}
\to \mathbb{R}$, $(f,g) \mapsto \int_{\mathbb{R}}f(s)g(s) \exp (-\mathscr{N} V(s)) \md s$, $\mathscr{N} \in \mathbb{N}$, 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 $\lbrace ϕ_{m}(z) \rbrace_{m=0}^{\infty}$: $ϕ_{2n}(z) = ξ^{(2n)}_{-n}z^{-n} + ...b + ξ^{(2n)}_{n}z^{n}$, $ξ^{(2n)}_{n} > 0$, and $ϕ_{2n+1}(z) = ξ^{(2n+1)}_{-n-1} z^{-n-1} + ...b + ξ^{(2n+1)}_{n}z^{n}$, $ξ^{(2n+1)}_{-n-1} > 0$. Asymptotics in the double-scaling limit as $\mathscr{N},n \to \infty$ such that $\mathscr{N}/n = 1 + o(1)$ of $ξ^{(2n+1)}_{-n-1}$ and $ϕ_{2n+1}(z)$ (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].