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) $\langle \boldsymbol{ \cdot},\boldsymbol{ \cdot} \rangle_{ \mathscr{ L}} \colon \Lambda^{ \mathbb{ R}} \times \Lambda^{ \mathbb{ R}} \! \to \! \mathbb{ R}$, $(f,g) \! \mapsto \! \int_{ \mathbb{ R}}f(s)g(s) \exp (-\mathscr{ N} \, V(s)) \, ds$, $\mathscr{ N} \! \in \! \mathbb{ N}$, where the external field $V$ satisfies: (i) $V$ is real analytic on $\mathbb{ R} \setminus \{ 0\}$; (ii) $\lim_{ \vert x \vert \to \infty}(V(x)/\ln (x^{ 2} \! + \! 1)) \! = \! +\infty$; and (iii) $\lim_{ \vert x \vert \to 0}(V(x)/\ln (x^{ -2} \! + \! 1)) \! = \! +\infty$. Orthogonalisation of the (ordered) base $\lbrace 1,z^{ -1},z,z^{ -2},z^{ 2},\dotsc,z^{ -k},z^{ k},\dotsc \rbrace$ with respect to $\langle \boldsymbol{ \cdot},\boldsymbol{ \cdot} \rangle_{ \mathscr{ L}}$ yields the even degree and odd degree orthonormal Laurent polynomials $\lbrace \phi_{ m}(z) \rbrace_{ m=0}^{ \infty}$: $\phi_{ 2n}(z) \! = \! \xi^{ (2n)}_{ -n} z^{ -n} \! + \! \dotsb \! + \! \xi^{ (2n)}_{ n}z^{ n}$, $\xi^{ (2n)}_{ n} \! > \! 0$, and $\phi_{ 2n+1}(z) \! = \! \xi^{ (2n+1)}_{ -n-1}z^{ -n-1} \! + \! \dotsb \! + \! \xi^{ (2n+1)}_{ n}z^{ n}$, $\xi^{ (2n+1)}_{ -n-1} \! > \! 0$. Asymptotics in the double-scaling limit as $\mathscr{ N},n \! \to \! \infty$ such that $\mathscr{ N}/n \! = \! 1 \! + \! o(1)$ of $\xi^{ (2n)}_{ n}$ and $\phi_{ 2n} (z)$ (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.
Categories: math.CA math-ph math.MP