The circular law

The circular law theorem states that the empirical spectral distribution of a n×n 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. In these expository notes, 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. We also discuss some related models. These notes constitute an abridged and updated version of the probability survey [BC12], prepared at the occasion of the American Mathematical Society short course on Random Matrices, organized by Van H. Vu for the 2013 AMSMAA Joint Mathematics Meeting held in January 9–13 in San Diego, USA. Section 1 introduces the notion of eigenvalues and singular values and discusses their relationships. Section 2 states the circular law theorem. Section 3 is devoted to the Gaussian model known as the Complex Ginibre Ensemble, for which the law of the spectrum is known and leads to the circular law. Section 4 provides the proof of the circular law theorem in the universal case, using the approach of Tao and Vu based on the Hermitization of Girko and the logarithmic potential. Section 5 gathers finally some few comments on related problems and models. All random variables are defined on a unique common probability space (Ω,A,P). A typical element of Ω is denoted ω. We write a.s., a.a., and a.e. for almost surely, Lebesgue almost all, and Lebesgue almost everywhere respectively. 1. Two kinds of spectra The eigenvalues of A ∈ Mn(C) are the roots in C of its characteristic polynomial PA(z) := det(A − zI). We label them λ1(A), . . . , λn(A) so that |λ1(A)| ≥ · · · ≥ |λn(A)| with growing phases. The spectral radius is |λ1(A)|. The eigenvalues form the algebraic spectrum of A. The singular values of A are defined by