Perturbed Hankel Determinants

Abstract

In this short note, we compute, for large n the determinant of a class of n x n Hankel matrices, which arise from a smooth perturbation of the Jacobi weight. For this purpose, we employ the same idea used in previous papers, where the unknown determinant, D_n[w_{α,β}h] is compared with the known determinant D_n[w_{α,β}]. Here w_{α,β} is the Jacobi weight and w_{α,β}h, where h=h(x),x\in[-1,1] is strictly positive and real analytic, is the smooth perturbation on the Jacobi weight w_{α,β}(x):=(1-x)^α(1+x)^β. Applying a previously known formula on the distribution function of linear statistics, we compute the large n asymptotics of D_n[w_{α,β}h] and supply a missing constant of the expansion.

10 pages

Cited by in corpus (10)

• Painlevé V and time dependent Jacobi polynomials
• Orthogonal Polynomials, Asymptotics and Heun Equations
• Perturbed Hankel determinant, correlation functions and Painlevé equations
• Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrix Models
• Asymptotics of determinants of Hankel matrices via non-linear difference equations
• On the Distribution of MIMO Mutual Information: An In-Depth Painlevé Based Characterization
• Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions
• Exact equivalences and phase discrepancies between random matrix ensembles
• Moduli space of supersymmetric QCD in the Veneziano limit
• Painlevé VI, Painlevé III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble