Around the circular law

Abstract

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 case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.Comment: Added: one reference and few comment

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

“…For β = 2 and 4, (3) turns out to be the eigenvalues joint distribution for normal complex and normal self-dual matrix ensembles respectively [8,32,33,42,58]. Remarkably, at inverse temperature β = 2, (3) is a determinantal point process.…”

“…This distribution on the disk is known as circular law in random matrix theory [6,8,36,37]. The electrostatic energy and the entropy of the gas are then easy to compute,…”

Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices

“…Many authors have proved versions of the circular law under various assumptions on the atom variable ξ; see for instance [2,8,16,26,33,34] and references therein. We present the most general version due to Tao and Vu [34].…”

Random antagonistic matrices

“…In the limit n → ∞, with proper assumptions on the moments of the probability law, the density of eigenvalues of the matrixM converges weakly to the uniform distribution on the disk {z ∈ C, |z| ≤ σ √ n}. This is known as the circular law; a survey is [9]. Provided that −µ + σ √ n ≤ 0, the eigenvalues of M are predicted with large probability to have negative real part.…”