### 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) |
---|---|

Article number | 62815 |

Journal | International Mathematics Research Papers |

Volume | 2006 |

DOIs | |

State | Published - 2006 |

Externally published | Yes |

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### Cite this

*International Mathematics Research Papers*,

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

**Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights.** / Mclaughlin, Kenneth D T; Vartanian, Arthur H.; Zhou, X.

Research output: Contribution to journal › Article

*International Mathematics Research Papers*, vol. 2006, 62815. https://doi.org/10.1155/IMRP/2006/62815

}

TY - JOUR

T1 - Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

AU - Mclaughlin, Kenneth D T

AU - Vartanian, Arthur H.

AU - Zhou, X.

PY - 2006

Y1 - 2006

N2 - Let Λℝ denote the linear space over ℝ spanned by zk, k ∈ ℤ. Define the real inner product (with varying exponential weights) 〈̇,̇〉ℒ : ΛRdbl; x ΛRdbl;. → 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(x2 + 1)) = + ∞; and (iii) lim x →0 (V(x)/ln(x-2 + 1)) = +∞. Orthogonalisation of the (ordered) base {1, z-1,z,z-2,z2,..., z-k, zk,...} with respect to 〈̇,̇〉∫ yields the even degree and odd degree orthonormal Laurent polynomials {Φm (z)}m=0∞: Φ2n (z) = ξ-n(2n) z-n + ... + ξn(2n) zn, ξn(2n) > 0, and Φ2n+1 (z) = ξ-n-1(2n+1) z-n-1+ ⋯ + ξn(2n+1) zn, ξ-n-1(2n+1) > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n (z) := (ξn(2n))-1 Φ2n (z) and π2n+1 (z) := (ξ-n- 1(2n+1))-1 Φ2n+1 (z). Asymptotics in the double-scaling limit as N, n → ∞ such that N/n = 1 + o(1) of π2n (z) (in the entire complex plane), ξn(2n), Φ2n (z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence {ck = ∫ℝ sk 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.

AB - Let Λℝ denote the linear space over ℝ spanned by zk, k ∈ ℤ. Define the real inner product (with varying exponential weights) 〈̇,̇〉ℒ : ΛRdbl; x ΛRdbl;. → 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(x2 + 1)) = + ∞; and (iii) lim x →0 (V(x)/ln(x-2 + 1)) = +∞. Orthogonalisation of the (ordered) base {1, z-1,z,z-2,z2,..., z-k, zk,...} with respect to 〈̇,̇〉∫ yields the even degree and odd degree orthonormal Laurent polynomials {Φm (z)}m=0∞: Φ2n (z) = ξ-n(2n) z-n + ... + ξn(2n) zn, ξn(2n) > 0, and Φ2n+1 (z) = ξ-n-1(2n+1) z-n-1+ ⋯ + ξn(2n+1) zn, ξ-n-1(2n+1) > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n (z) := (ξn(2n))-1 Φ2n (z) and π2n+1 (z) := (ξ-n- 1(2n+1))-1 Φ2n+1 (z). Asymptotics in the double-scaling limit as N, n → ∞ such that N/n = 1 + o(1) of π2n (z) (in the entire complex plane), ξn(2n), Φ2n (z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence {ck = ∫ℝ sk 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.

UR - http://www.scopus.com/inward/record.url?scp=33749680768&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749680768&partnerID=8YFLogxK

U2 - 10.1155/IMRP/2006/62815

DO - 10.1155/IMRP/2006/62815

M3 - Article

VL - 2006

JO - International Mathematics Research Papers

JF - International Mathematics Research Papers

SN - 1687-3017

M1 - 62815

ER -