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Bulletin of the American Mathematical Society

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From the Littlewood-Offord problem to the Circular Law: Universality of the spectral distribution of random matrices

Authors: Terence Tao and Van Vu
Journal: Bull. Amer. Math. Soc. 46 (2009), 377-396
MSC (2000): Primary 15A52, 60G50
Published electronically: February 24, 2009
MathSciNet review: 2507275
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Abstract: The famous circular law asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of the normalized matrix $\frac {1}{\sqrt {n}} M_n$ converges both in probability and almost surely to the uniform distribution on the unit disk $\{ z \in \mathbf {C}: |z| \leq 1 \}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the circular law is now known to be true for arbitrary distributions with mean zero and unit variance. In this survey we describe some of the key ingredients used in the establishment of the circular law at this level of generality, in particular recent advances in understanding the Littlewood-Offord problem and its inverse.

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Additional Information

Terence Tao
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095-1555
MR Author ID: 361755
ORCID: 0000-0002-0140-7641

Van Vu
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Received by editor(s): October 16, 2008
Received by editor(s) in revised form: January 1, 2009
Published electronically: February 24, 2009
Additional Notes: The first author is supported by NSF grant CCF-0649473 and a grant from the MacArthur Foundation
The second author is supported by an NSF Career Grant 0635606
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.