随机矩阵理论中波动公式的精确结果综述

IF 1.3 Q2 STATISTICS & PROBABILITY Probability Surveys Pub Date : 2022-04-07 DOI:10.1214/23-ps15
P. Forrester
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引用次数: 9

摘要

点过程的线性统计的协方差和方差可以写成截断两点相关函数上的积分。当点过程由随机矩阵系综的特征值组成时,在平滑之后,这种相关性通常存在大的$N$通用形式,这导致了线性统计量波动的特别简单的限制公式。我们回顾了这些极限公式,它们在最简单的情况下作为截断两点相关性的显式知识的推论而导出。一个大的$N$极限是缩放特征值,使得极限支持是紧凑的,并且线性统计随支持的规模而变化。这是一个全球性的规模。另一种情况是,首先取热力学极限,使特征值之间的间距为一阶单位,然后对测试函数施加一个标度,使其缓慢变化,这就是体标度。在Dyson和Mehta的开创性工作中,后者已经被确定为量子光谱的随机矩阵特征探针。
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A review of exact results for fluctuation formulas in random matrix theory
Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often large $N$ universal forms for this correlation after smoothing, which results in particularly simple limiting formulas for the fluctuation of the linear statistics. We review these limiting formulas, derived in the simplest cases as corollaries of explicit knowledge of the truncated two-point correlation. One of the large $N$ limits is to scale the eigenvalues so that limiting support is compact, and the linear statistics vary on the scale of the support. This is a global scaling. The other, where a thermodynamic limit is first taken so that the spacing between eigenvalues is of order unity, and then a scale imposed on the test functions so they are slowly varying, is the bulk scaling. The latter was already identified as a probe of random matrix characteristics for quantum spectra in the pioneering work of Dyson and Mehta.
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来源期刊
Probability Surveys
Probability Surveys STATISTICS & PROBABILITY-
CiteScore
4.70
自引率
0.00%
发文量
9
期刊最新文献
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