论文标题
Log-Concave合奏的极端特征值
Extreme eigenvalues of Log-concave Ensemble
论文作者
论文摘要
在本文中，我们考虑了随机矩阵的日志conconcave集合，这是一类协方差型矩阵$ xx^*$带有摩托型log-concave $ x $ -columns。一个主要的例子是各向同性凸体均匀度量的协方差估计器。在文献中已经获得了极端特征值的非反应估计和一阶渐近限。在本文中，随着对数孔的最新进展，\ cite \ cite {chen，kl22}，我们迈出了一步，以几乎最佳的精度定位特征值，即，该集合的光谱刚度被得出。基于频谱刚度和额外的``无条件假设''，我们进一步得出了$ xx^*$的极端特征值的Tracy-Widom定律，以及对于极端特征值的高斯法律，以防强烈的尖峰。
In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices $XX^*$ with isotropic log-concave $X$-columns. A main example is the covariance estimator of the uniform measure on isotropic convex body. Non-asymptotic estimates and first order asymptotic limits for the extreme eigenvalues have been obtained in the literature. In this paper, with the recent advancements on log-concave measures \cite{chen, KL22}, we take a step further to locate the eigenvalues with a nearly optimal precision, namely, the spectral rigidity of this ensemble is derived. Based on the spectral rigidity and an additional ``unconditional" assumption, we further derive the Tracy-Widom law for the extreme eigenvalues of $XX^*$, and the Gaussian law for the extreme eigenvalues in case strong spikes are present.
