RANDOM MATRICES: THE CIRCULAR LAW

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of . Define the empirical spectral distributionμn of Nn by the formula The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn 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, Gotze–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.

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