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

Publication ,  Journal Article
McLaughlin, KTR; Vartanian, AH; Zhou, X
Published in: International Mathematics Research Papers
October 18, 2006

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.

## Published In

International Mathematics Research Papers

1687-3009

1687-3017

October 18, 2006

## Volume

2006

• General Mathematics
• 0101 Pure Mathematics

### Citation

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McLaughlin, K. T. R., Vartanian, A. H., & Zhou, X. (2006). Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights. International Mathematics Research Papers, 2006. https://doi.org/10.1155/IMRP/2006/62815
McLaughlin, K. T. R., A. H. Vartanian, and X. Zhou. “Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights.” International Mathematics Research Papers 2006 (October 18, 2006). https://doi.org/10.1155/IMRP/2006/62815.
McLaughlin KTR, Vartanian AH, Zhou X. Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights. International Mathematics Research Papers. 2006 Oct 18;2006.
McLaughlin, K. T. R., et al. “Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights.” International Mathematics Research Papers, vol. 2006, Oct. 2006. Scopus, doi:10.1155/IMRP/2006/62815.
McLaughlin KTR, Vartanian AH, Zhou X. Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights. International Mathematics Research Papers. 2006 Oct 18;2006.

## Published In

International Mathematics Research Papers

1687-3009

1687-3017

October 18, 2006

2006