酉矩阵上的循环Pólya系综及其谱统计
Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
AI总结：
本研究将球变换与Pólya系综框架拓展至酉矩阵乘积研究，定义了循环Pólya频率函数并明确其与循环Pólya系综的关联，给出对应权重唯一性结论，推导了固定维度下特征值的行列式点过程，还概述了局部谱统计研究的待解问题。
摘要：
球变换与Pólya系综框架可用于以统一方式推导随机矩阵和与积的结构化解析结果。在本工作中，我们将该框架拓展用于研究酉矩阵的乘积。这些矩阵不服从Haar测度分布，但仍取自特征值与特征向量统计量可因式分解的分布族，其中包括与Toeplitz行列式理论中Fisher-Hartwig奇点相关的circular Jacobi ensemble（圆雅可比系综），以及酉群上布朗运动的heat kernel（热核）。我们定义了cyclic Pólya frequency functions（循环Pólya频率函数），阐明了其与cyclic Pólya ensembles（循环Pólya系综）的关联，给出了对应权重的唯一性表述，并推导了固定矩阵维度下特征值统计量的determinantal point processes（行列式点过程）。本文还概述了研究局部谱统计时可能遇到的问题。
英文摘要：
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.
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