Proceedings of Symposia in Applied Mathematics
Volume 72, 2014
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