Random matrices: Universality of ESDs and the circular law
Terence Tao, Van Vu, Manjunath Krishnapur
备注：
45 pages, 8 figures, submitted, Acta Math. The main article is by Tao and Vu, the appendix is by Krishnapur, and the figures are by Phillip Wood. A simplified proof of the replacement principle added; some other corrections
Given an $n \times n$ complex matrix $A$, let
$$μ_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re λ_i \le x, \Im λ_i \le y\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues
$λ_i \in \BBC, i=1, ... n$.
We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD $μ_{\frac{1}{\sqrt{n}} A_n}$ of a random matrix $A_n = (a_{ij})_{1 \leq i,j \leq n}$ where the random variables $a_{ij} - \E(a_{ij})$ are iid copies of a fixed random variable $x$ with unit variance. We prove a \emph{universality principle} for such ensembles, namely that the limit distribution in question is {\it independent} of the actual choice of $x$. In particular, in order to compute this distribution, one can assume that $x$ is real of complex gaussian. As a related result, we show how laws for this ESD follow from laws for the \emph{singular} value distribution of $\frac{1}{\sqrt{n}} A_n - zI$ for complex $z$. As a corollary we establish the Circular Law conjecture (in both strong and weak forms), that asserts that $μ_{\frac{1}{\sqrt{n}} A_n}$ converges to the uniform measure on the unit disk when the $a_{ij}$ have zero mean.
In this short note, we extend the celebrated results of Tao and Vu, and Krishnapur on the universality of empirical spectral distributions to a wide class of inhomogeneous complex random matrices, by showing that a technical and hard-to-verify Fourier domination assumption may be replaced simply by a natural uniform anti-concentration assumption. Along the way, we show that inhomogeneous complex random matrices, whose expected squared Hilbert-Schmidt norm is quadratic in the dimension, and whose entries (after symmetrization) are uniformly anti-concentrated at $0$ and infinity, typically have …
The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\frac{1}{\sqrt{n}} M_n$ converges almost surely to the uniform distribution on the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the full circular law was recently established in \cite{TVcir2}. In this survey we describe some of the key ingredients used in the establis…
The universality phenomenon asserts that the distribution of the eigenvalues of random matrix with i.i.d. zero mean, unit variance entries does not depend on the underlying structure of the random entries. For example, a plot of the eigenvalues of a random sign matrix, where each entry is +1 or -1 with equal probability, looks the same as an analogous plot of the eigenvalues of a random matrix where each entry is complex Gaussian with zero mean and unit variance. In the current paper, we prove a universality result for sparse random n by n matrices where each entry is nonzero with probability …
The circular law asserts that if $X_n$ is a $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges almost surely to the uniform distribution on the unit disk as $n$ tends to infinity. Answering a question of Tao, we prove the circular law for a general class of random block matrices with dependent entries. The proof relies on an inverse-type result for the concentration of linear operators and multilinear forms.
An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular …
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One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form $\bY= \bA^{-1} \bB$, where $\bA$ and $\bB$ are independent $N\times N$ matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensemble…
