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

K. T.R. McLaughlin, Arthur H. Vartanian, X. Zhou

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9 Scopus citations


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
StatePublished - 2006
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics


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