Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach

有限秩正交扰动下大旋转不变非厄米特矩阵的本征值和本征向量

The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices

研究有限秩正交扰动对大旋转不变非厄米特随机矩阵的影响，扩展BBP框架，分析非厄米特随机矩阵加有限秩正交扰动模型中异常本征值的出现与波动，描述对应的本征向量行为，提供统一框架涵盖厄米特和非厄米特情形。

Large deviations for the largest eigenvalues and eigenvectors of spiked random matrices

We consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $θ$, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. We obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $θ>1$, in large deviations characterized by a small value of $u$, i.e. $u<1-1/θ$, the second-largest eigenvalue pops out from the Wigner semi-circ…

带尖峰的相关随机矩阵特征向量之间的重叠：从矩阵PCA到随机高斯景观

Overlaps between eigenvectors of spiked, correlated random matrices: from matrix PCA to random Gaussian landscapes

研究成对相关GOE矩阵在秩一扰动下离群值特征向量的典型重叠，探讨其对带尖峰矩阵信号恢复及高维随机景观问题的意义。

Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles

Statistical properties of eigenvectors in non-Hermitian random matrix ensembles are discussed, with an emphasis on correlations between left and right eigenvectors. Two approaches are described. One is an exact calculation for Ginibre's ensemble, in which each matrix element is an independent, identically distributed Gaussian complex random variable. The other is a simpler calculation using $N^{-1}$ as an expansion parameter, where $N$ is the rank of the random matrix: this is applied to Girko's ensemble. Consequences of eigenvector correlations which may be of physical importance in applicati…

The Perturbative Resolvent Method: spectral densities of random matrix ensembles via perturbation theory

We present a simple, perturbative approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit we call the Perturbative Resolvent Method (PRM). The PRM is based on constructing a linear system of equations and calculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-circle Law for symmetric matrices, the Marchenko-Pastur Law for Wishart matrices, the spectral density for a product Wish…

Localization transition in random Levy matrices : multifractality of eigenvectors in the localized phase and at criticality

For random Lévy matrices of size $N \times N$, where matrix elements are drawn with some heavy-tailed distribution $P(H_{ij}) \propto N^{-1} |H_{ij} |^{-1-μ}$ with $0<μ<2$ (infinite variance), there exists an extensive number of finite eigenvalues $E=O(1)$, while the maximal eigenvalue grows as $E_{max} \sim N^{\frac{1}μ}$. Here we study the localization properties of the corresponding eigenvectors via some strong disorder perturbative expansion that remains consistent within the localized phase and that yields their Inverse Participation Ratios (I.P.R.) $Y_q$ as a function of the continuous p…