Abstract. We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n −1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum � k akXk lies near some number v. For arbitrary coefficients ak of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p. 1.

Given an n × n complex matrix A, let

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. 1 copies of x. Let λ1,..., λn be the eigenvalues of

The famous circular law asserts that if Mn is an n×n matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of the normalized matrix 1 √ Mn converges both in probability and n almost surely to the uniform distribution on the unit disk {z ∈ C: |z | ≤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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We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have moment of order E |Xjk | 2 ϕ(x), with some positive function ϕ(x) which is growing of order (log(1 + |x|)) 7, or that they are sparsely nonzero. The results are based on and extend previous work of Bai, Rudelson and the authors. 1

We study the empirical measure LAn of the eigenvalues of non-normal square matrices of the form An = UnDnVn with Un,Vn independent Haar distributed on the unitary group and Dn real diagonal. We show that when the empirical measure of the eigenvalues of Dn converges, and Dn satisfies some technical conditions, LAn converges towards a rotationally invariant measure on the complex plane whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem [5]. We also consider the case where Un,Vn are independent Haar distributed on the orthogonal group. 1 The problem Horn [15] asked the question of describing the eigenvalues of a square matrix with prescribed singular values. If A is a n × n matrix with singular values s1 ≥... ≥

Let (Xi,j)1�i,j< ∞ be an infinite array of i.i.d. complex random variables, with mean m = 0, variance σ 2 = 1, and say with finite fourth moment. The famous circular law theorem states that the empirical spectral distribution 1 n (δ λ1(X)+ · · ·+δ λn(X)) of X = (n −1/2 Xi,j)1�i,j�n converges almost surely, as n → ∞, to the uniform law over the unit disc {z ∈ C; |z | � 1}. For now, most efforts where focused on the improvement of moments hypotheses for the centered case m = 0. Regarding the non-central case m ̸ = 0, Silverstein has already observed that almost surely, the eigenvalue of X of largest module goes to + ∞ as n → ∞, while the rest of the spectrum remains bounded. We show in this note that the circular law theorem remains valid when m ̸ = 0, by using logarithmic potentials and bounds on extremal singular values.

Given an n × n complex matrix A, let