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

Kenneth D T Mclaughlin, Arthur H. Vartanian, X. Zhou

Research output: Contribution to journalArticle

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Abstract

Let Λ 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) ∫ 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 = ∫ 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.

Original languageEnglish (US)
Article number62815
JournalInternational Mathematics Research Papers
Volume2006
DOIs
StatePublished - 2006
Externally publishedYes

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Orthogonal Laurent Polynomials
Exponential Weights
Monic
Laurent Polynomials
Argand diagram
Odd
Entire
Moment Sequence
Hankel Determinant
Orthonormal Polynomials
Steepest Descent Method
Orthogonalization
Riemann-Hilbert Problem
Scaling Limit
Linear Space
Scalar, inner or dot product
External Field
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ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: International Mathematics Research Papers, Vol. 2006, 62815, 2006.

Research output: Contribution to journalArticle

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abstract = "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.",
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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.

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