Random Matrices: The circular Law

Abstract

Let $\a$ be a complex random variable with mean zero and bounded variance ${\sigma }^2$. Let $N_n$ be a random matrix of order $n$ with entries being i.i.d. copies of $\a$. Let ${\lambda }_1,...,{\lambda }_n$ be the eigenvalues of $\frac{1}{\sigma \sqrt{n}}{N}_n$. Define the empirical spectral distribution ${\mu }_n$ of $N_n$ by the formula $$ μ_n(s,t) := \frac{1}{n} # \{k \leq n| \Re(λ_k) \leq s; \Im(λ_k) \leq t \}.$$ The Circular law conjecture asserts that ${\mu }_n$ converges to the uniform distribution ${\mu }_{\infty }$ over the unit disk as $n$ tends to infinity. We prove this conjecture under the slightly stronger assumption that the $(2+\eta )þ$-moment of $\a$ is bounded, for any $\eta >0$. Our method builds and improves upon earlier work of Girko, Bai, Götze-Tikhomirov, and Pan-Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

46 pages, no figures, submitted. More minor corrections

References in corpus (2)

• Smoothed analysis of algorithms
• Circular law, Extreme Singular values and Potential theory

Cited by in corpus (1)

• Correlation Functions of Asymmetric Real Matrices