Circular law, Extreme Singular values and Potential theory

非厄米随机带状矩阵在带宽 $N^{2/3+c}$ 以内的圆律

The circular law for non-Hermitian random band matrices up to bandwidth $N^{2/3+c}$

本文证明了非厄米随机带状矩阵在带宽 $N^{2/3+c}$ 及更宽条件下的圆律，覆盖双随机方差分布与周期标量条带，无需下方差界或密度假设，通过粗奇异值计数等方法实现。

Around the circular law

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal ca…

Dimension Results for the Spectral Measure of the Circular Beta Ensembles

We study the dimension properties of the spectral measure of the Circular $β$-Ensembles. For $β\geq 2$ it it was previously shown by Simon that the spectral measure is almost surely singular continuous with respect to Lebesgue measure on $\partial \mathbb{D}$ and the dimension of its support is $1 - 2/β$. We reprove this result with a combination of probabilistic techniques and the so-called Jitomirskaya-Last inequalities. Our method is simpler in nature and mostly self-contained, with an emphasis on the probabilistic aspects rather than the analytic. We also extend the method to prove a large…

The circular law for random regular digraphs

Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral distribution of $A_n$, suitably rescaled, is governed by the Circular Law. A key step is to obtain quantitative lower tail bounds for the smallest singular value of additive perturbations of $A_n$.

Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups

We analyse the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by M. Meckes for the Abelian case. We show that for regular sequences of groups the limiting distribution of eigenvalues (resp. singular values) is a mixture of eigenvalue (resp. singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular for the sequence of symmetric groups, the limiting distri…

随机带矩阵的圆形定律：改进的带宽

The circular law for random band matrices with arbitrary doubly stochastic variance profiles

本文改进了随机带矩阵圆形定律的带宽阈值，适用于任意双随机方差剖面，并推导了特征向量弱去局域化估计。

The singular values and vectors of low rank perturbations of large rectangular random matrices

In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix. As in the prequel, where we considered the eigenvalue aspect of the problem, the non-random limiting value is shown to depend explicitly on the limiting singular value distribution of the unperturbed matrix via an integral transforms that linearizes rectangular additive convolution in free …

Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists $τ\in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1-τ}$ and the nonzero entries are iid random variables with mean zero and slightly more than four finite moments, then the limiting empirical eigenvalue distribution of $X$, when properly normalized, converges in probability to the uniform distribution on the unit disk in the complex plane. The key technical result is a least singular value bound for shifted random band block ma…