论文标题
排名$ 1 $ $扰动在随机矩阵理论中 - 精确结果的评论
Rank $1$ perturbations in random matrix theory -- a review of exact results
论文作者
论文摘要
许多随机矩阵合奏允许精确确定其特征值和特征向量统计信息，将此属性维持在等级$ 1 $扰动下。在这篇评论中考虑的是Hermitian Gaussian合奏的添加等级$ 1 $扰动，多种等级$ 1 $ $ 1 $的Wishart合奏扰动，排名$ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $的Hermitian和单位矩阵扰动，从而增加了对特征值的二维支持。整个过程的重点是精确公式，这通常是各种可集成结构的结果。最简单的是确定点过程，而其他与某些随机的三角形矩阵有关的偏微分方程则与部分相关。在排名$ 1 $扰动的情况下，还要注意特征向量重叠。
A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the Hermitian Gaussian ensembles, the multiplicative rank $1$ perturbation of the Wishart ensembles, and rank $1$ perturbations of Hermitian and unitary matrices giving rise to a two-dimensional support for the eigenvalues. The focus throughout is on exact formulas, which are typically the result of various integrable structures. The simplest is that of a determinantal point process, with others relating to partial differential equations implied by a formulation in terms of certain random tridiagonal matrices. Attention is also given to eigenvector overlaps in the setting of a rank $1$ perturbation.
