Around the circular law

TL;DR

This paper explores the circular law theorem for random matrices, providing proofs in Gaussian and universal cases, extending to heavy-tailed entries, and introducing new analytical tools and transforms.

Contribution

It offers a comprehensive proof of the circular law, extends the law to heavy-tailed entries, and introduces quaternionic transforms and geometric analysis methods.

Findings

Empirical spectral distribution converges to the uniform law on the unit disk.

Heavy-tailed entries lead to a different limiting law related to Poisson weighted trees.

Weak control of smallest singular values is achieved under minimal assumptions.

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.