Asymptotics for Products of Characteristic Polynomials in Classical beta-Ensembles

作者：Desrosiers, Patrick^*; Liu, Dang-Zheng

来源：Constructive Approximation, 2014, 39(2): 273-322.

DOI：10.1007/s00365-013-9206-2

• Random matrices
• Beta-ensembles
• Jack polynomials
• Multivariate hypergeometric functions
• Steepest descent method

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral), and Jacobi beta-ensembles of NxN random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as N -> a. In the bulk of the spectrum of each beta-ensemble, the same scaling limit is found to be , whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre beta-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when beta is even, scaling limits of the k-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson's lemma and the steepest descent method for integrals of Selberg type.

• 出版日期2014-4
• 单位中国科学技术大学