Around the circular law

Abstract

These expository notes are centered around the circular lawtheorem, which states that the empirical spectral distribution of a n × nrandom matrix with i.i.d. entries of variance 1/n tends to the uniform lawon the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circularlimit for Wigner random Hermitian matrices, and the quarter circular limitfor Marchenko-Pastur random covariance matrices. We present a proof ina Gaussian case, due to Silverstein, based on a formula by Ginibre, and aproof of the universal case by revisiting the approach of Tao and Vu, basedon the Hermitization of Girko, the logarithmic potential, and the control ofthe small singular values. Beyond the finite variance model, we also considerthe case where the entries have heavy tails, by using the objective methodof Aldous and Steele borrowed from randomized combinatorial optimiza-tion. The limiting law is then no longer the circular law and is related to thePoisson weighted infinite tree. We provide a weak control of the smallest sin-gular value under weak assumptions, using asymptotic geometric analysistools. We also develop a quaternionic Cauchy-Stieltjes transform borrowedfrom the Physics literature.