Proceedings of Symposia in Applied Mathematics

Volume 72, 2014

http://dx.doi.org/10.1090/psapm/072/00617

Lecture notes on the circular law

Charles Bordenave and Djalil Chafa¨ı

Abstract. The circular law theorem states that the empirical spectral distri-

bution 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 Mehta and 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 loga-

rithmic potential, and the control of the small singular values. We also discuss

some related models and open problems.

These notes constitute an abridged and updated version of the probability

survey [BC], prepared at the occasion of the American Mathematical Society short

course on Random Matrices, organized by Van H. Vu for the 2013 AMS-MAA Joint

Mathematics Meeting held in January 9–13 in San Diego, CA, 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). An element of Ω is denoted ω. We write a.s., a.a., and a.e. for al-

most surely, Lebesgue almost all, and Lebesgue almost everywhere respectively.

1. Two kinds of spectra

The eigenvalues of a matrix A ∈ Mn(C) are the roots in C of its character-

istic 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

2010 Mathematics Subject Classification. Primary 15B52 (60B20; 60F15).

Key words and phrases. Spectrum, singular values, eigenvalues, random matrices, random

graphs, circular law, ginibre ensemble, non Hermitian matrices, non normal matrices.

c 2014 Bordenave and Chafa¨ı

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