### Abstract

Let Λ ^{ℝ} denote the linear space over ℝ spanned by z^{k} k ℤ. Define the (real) inner product Ċ,Ċ L : Λ ^{ℝ}× Λ ^{ℝ} ℝ, (f,g) ∫_{ℝ} f(s)g(s) exp(- N V(s)) ds, N ℕ, where V satisfies: (i) V is real analytic on ℝ 0; (ii) lim x (V(x)/ln(x^{2}} + 1)) = + and (iii) limx 0(V(x)/ln (x^{2}} + 1)) = +. Orthogonalisation of the (ordered) base 1,z^{-1},z,z^{-2}z^{2}},z ^{-k}},z^{k} with respect to , {{ L}} yields the even degree and odd degree orthonormal Laurent polynomials φ{m}(z)_{m=0}: φ_{2n}(z) = ξ^{(2n)}z^{-n} + + ξ^{(2n)}_{n}z^{n} ξ^{(2n)}_{n} > 0, and φ^{{2n+1}}(z) = ξ^{(2n+1)} _{-n-1}z^{-n-1} + + ξ^{(2n+1)n}z ^{n} ξ^{(2n+1)}_{-n-1} > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π_{2n}(z) := (ξ^{(2n)}_{n}^{-1}} φ_{2n}(z) and π^{{2n+1}}(z) := (ξ^{(2n+1)}_{-n-1}^{-1} φ_{2n+1}(z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π_{2n+1}(z) (in the entire complex plane), ξ^{(2n+1)}_{-n-1}, and φ_{2n+1}(z)(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on 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].

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### Keywords

- Asymptotics
- Equilibrium measures
- Hankel determinants
- Laurent polynomials
- Laurent-Jacobi matrices
- Riemann-Hilbert problems

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis

### Cite this

*Constructive Approximation*,

*27*(2), 149-202. https://doi.org/10.1007/s00365-007-0675-z