论文标题
广义的Cauchy随机矩阵和连续Hahn多项式的矩
Moments of Generalized Cauchy Random Matrices and continuous-Hahn Polynomials
论文作者
论文摘要
在本文中，我们证明，经过适当的续订，$ \ mathbb {e} _ {n}^{（s）} \ left（tr \ left（| \ mathbf {h} |^{h} |^{2k+2}+2}+| \ | \ mathbf {h} $ hmitian）矩阵$ \ mathbf {h} $根据广义的cauchy（也称为Hua-pickrell）合奏，带有参数$ s> 0 $是可变$ k $中的连续hahn polyenmial。这完成了Cunden，Mezzadri，O'Connell和Simm始于调查的情况，他们为其他三个经典的随机矩阵，高斯，Laguerre和Jacobi获得了类似的结果。我们的举证策略与以前使用的策略有所不同，因为概括的库奇是唯一具有有限数量整数时刻的古典合奏。我们的论点也直接修改也适用于高斯，拉瓜尔和雅各比案件。最终，我们获得了该集合特征值分布的单点密度函数的微分方程，并建立了矩的大$ n $渐近图。
In this paper we prove that, after an appropriate rescaling, the sum of moments $\mathbb{E}_{N}^{(s)} \left( Tr \left( |\mathbf{H}|^{2k+2}+|\mathbf{H}|^{2k}\right) \right)$ of an $N\times N$ Hermitian matrix $\mathbf{H}$ sampled according to the generalized Cauchy (also known as Hua-Pickrell) ensemble with parameter $s>0$ is a continuous-Hahn polynomial in the variable $k$. This completes the picture of the investigation that began by Cunden, Mezzadri, O'Connell and Simm who obtained analogous results for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Jacobi. Our strategy of proof is somewhat different from the one employed previously due to the fact that the generalized Cauchy is the only classical ensemble which has a finite number of integer moments. Our arguments also apply, with straightforward modifications, to the Gaussian, Laguerre and Jacobi cases as well. We finally obtain a differential equation for the one-point density function of the eigenvalue distribution of this ensemble and establish the large $N$ asymptotics of the moments.
