  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].
