Math @ Duke

Publications [#244275] of Xin Zhou
Papers Published
 McLaughlin, KTR; Vartanian, AH; Zhou, X, Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights,
International Mathematics Research Papers, vol. 2006
(2006), ISSN 16873017 [doi]
(last updated on 2018/10/20)
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(x2 + 1)) = +∞. Orthogonalisation of the (ordered) base {1, z1,z,z2,z2,..., zk, zk,...} with respect to 〈̇,̇〉∫ yields the even degree and odd degree orthonormal Laurent polynomials {Φm (z)}m=0∞: Φ2n (z) = ξn(2n) zn + ... + ξn(2n) zn, ξn(2n) > 0, and Φ2n+1 (z) = ξn1(2n+1) zn1+ ⋯ + ξn(2n+1) zn, ξn1(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) := (ξn 1(2n+1))1 Φ2n+1 (z). Asymptotics in the doublescaling 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 realvalued, biinfinite, 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 RiemannHilbert problem on ℝ, and then extracting the largen behaviour by applying the nonlinear steepestdescent method introduced by P. Deift and X. Zhou and further developed by P. Deift, S. Venakides, and X. Zhou.


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