### Abstract

Let Λ^{ℝ} denote the linear space over ℝ spanned by z^{k}, k ∈ ℤ. Define the real inner product (with varying exponential weights) 〈̇,̇〉ℒ : Λ^{Rdbl;} x Λ^{Rdbl;}. → ℝ, (f, g) ∫_{ℝ} f(s)g(s) exp(-NV(s))ds, N ∈ ℕ, where the external field V satisfies the following: (i) V is real analytic on ℝ\{0}; (ii) lim_{ x →∞} (V(x)/ ln(x^{2} + 1)) = + ∞; and (iii) lim_{ x →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 〈̇,̇〉_{∫} yields the even degree and odd degree orthonormal Laurent polynomials {Φ_{m} (z)}_{m=0}^{∞}: Φ_{2n} (z) = ξ_{-n}^{(2n)} z^{-n} + ... + ξ_{n}^{(2n)} z^{n}, ξ_{n}^{(2n)} > 0, and Φ_{2n+1} (z) = ξ_{-n-1}^{(2n+1)} z^{-n-1}+ ⋯ + ξ_{n}^{(2n+1)} z^{n}, ξ_{-n-1}^{(2n+1)} > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π_{2n} (z) := (ξ_{n}^{(2n)})^{-1} Φ_{2n} (z) and π_{2n+1} (z) := (ξ_{-n- 1}^{(2n+1)})^{-1} Φ_{2n+1} (z). Asymptotics in the double-scaling limit as N, n → ∞ such that N/n = 1 + o(1) of π_{2n} (z) (in the entire complex plane), ξ_{n}^{(2n)}, Φ2_{n} (z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence {c_{k} = ∫_{ℝ} s^{k} exp (-NV(s))ds}_{k∈ℤ} are obtained by formulating the even degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on ℝ, and then extracting the large-n behaviour by applying the nonlinear steepest-descent method introduced by P. Deift and X. Zhou and further developed by P. Deift, S. Venakides, and X. Zhou.

Original language | English (US) |
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Article number | 62815 |

Journal | International Mathematics Research Papers |

Volume | 2006 |

DOIs | |

State | Published - Oct 18 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Mathematics Research Papers*,

*2006*, [62815]. https://doi.org/10.1155/IMRP/2006/62815