Random matrices: Universality of ESDs and the circular law

By Terence Tao, Van H. Vu, Manjunath Krishnapur. 2008

494 citations recorded by the source corpus.

Given an n x n complex matrix A, let mu(A)(x, y) := 1/n vertical bar{1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y}vertical bar be the empirical spectral distribution (ESD) of its eigenvalues lambda(i) is an element of C, i = l, ... , n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD mu(1/root n An) of a random matrix A(n) = (a(ij))(1 <= i, j <= n), where the random variables a(ij) - E(a(ij)) are i.i.d. copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely, that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1/root n A(n) - zI for complex z. As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that mu(1/root n An) converges to the uniform measure on the unit disc when the a(ij) have zero mean.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d1fb239b65eab0ee1d1be949a0c2afe992af3da6

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Citations in the corpus

Delocalization of eigenvectors of random matrices with independent entries (2013, unknown): “Proof of Theorem 4.1.”
The ecology of the microbiome: Networks, competition, and stability (2015, background): “(52), showed that this conclusion is also applicable to non-Normal distributions, save for one eigenvalue that is approximately equal to the average row sum of M⇤, and further noted that setting each of the diagonal entries a ii to s will move the center of this ellipse to ( s, 0), thus the system w”
The stability–complexity relationship at age 40: a random matrix perspective (2015, background): “1 (Theorem 1.10 of Tao et al. 2010).”
On the Asymptotic Spectrum of Products of Independent Random Matrices. (2010, methodology): “For m = 1 we recover the well-known circular law for random matrices [9], [15].”
Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, background): “According to [53, 50], see also [17], almost surely the empirical spectral measure of μMN converges weakly to a probability measure β on C which is characterized by its logarithmic potential hβ(z) = 1 2 ∫ log(t)dμz(t).”
Stability criteria for complex ecosystems (2011, background): “For large systems, the eigenvalues are contained in a circl”
On the singularity of adjacency matrices for random regular digraphs (2014, background): “In [34] Tao and Vu proved the circular law for Ξ, which states that almost surely, as n→∞, μ 1 √ n Ξ converges weakly to the uniform measure on the unit disc in C.”
Universality and the circular law for sparse random matrices. (2010, methodology): “In this paper, we will follow the program developed in [27] and prove a universality principle for sparse random matrices.”
Non-asymptotic theory of random matrices: extreme singular values (2010, background): “For an N × n random matrix A whose dimensions are not too close to each other Feldheim and Sodin [25] proved the Tracy–Widom law for the smallest singular value together with a non-asymptotic version of the bound s min ( A ) ∼ √ N − √ n :”