论文标题
循环pólya在单一矩阵及其光谱统计上合奏
Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
论文作者
论文摘要
球形变换和Pólya集合的框架在以统一的方式得出随机矩阵的总和和产物的结构化分析结果方面具有实用性。在目前的工作中，我们将继续此框架来研究单一矩阵的产品。这些不是通过HAAR度量分布的，而是从特征值和特征向量统计因素分配的分布中得出的。它们包括圆形的雅各比集合，在Toeplitz决定因素理论中与Fisher-Hartwig奇异性有关，以及在单一组上进行的Brownian运动的热内核。我们定义了循环polya频率函数，并显示它们与循环pólya合奏的关系，给相应权重的唯一性语句，并得出固定矩阵维度的特征值统计的确定点过程。在研究当地频谱统计数据时可能会遇到问题的大纲。
The framework of spherical transforms and Pólya ensembles is of utility in deriving structured analytic results for sums and products of random matrices in a unified way. In the present work, we will carry over this framework to study products of unitary matrices. Those are not distributed via the Haar measure, but still are drawn from distributions where the eigenvalue and eigenvector statistics factorise. They include the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, give a uniqueness statement for the corresponding weights, and derive the determinantal point processes of the eigenvalue statistics at fixed matrix dimension. An outline is given of problems one may encounter when investigating the local spectral statistics.
