Random Matrices: The Circular Law

Let x be a complex random variable with mean zero and bounded variance σ 2 . Let N n be a random matrix of order n with entries being i.i.d. copies of x. Let λ 1 , …, λ n be the eigenvalues of [Formula: see text]. Define the empirical spectral distributionμ n of N n by the formula [Formula: see text] The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μ n converges to the uniform distribution μ ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 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.