论文标题
随机矩阵理论中波动公式的确切结果的综述
A review of exact results for fluctuation formulas in random matrix theory
论文作者
论文摘要
点过程的线性统计数据的协方差和方差可以写为截断的两点相关函数的积分。当点过程由随机矩阵集合的特征值组成时，平滑后，这种相关性通常会有很大的$ n $通用形式，这导致了特别简单的限制公式，以使线性统计的波动。我们回顾了这些限制公式，这些公式在最简单的情况下是对截断的两点相关性明确知识的推论。 $ n $ n $的限制之一是扩展特征值，以使限制支持紧凑，线性统计数据在支持的规模上有所不同。这是一个全球扩展。另一个是首先采用热力学极限，以便特征值之间的间距是阶的，然后对测试函数施加的刻度使其逐渐变化，是批量的缩放。后者已经被确定为戴森和Mehta开创性工作中量子光谱的随机矩阵特征的探针。
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.
