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].
Categories: math.CA math-ph math.MP