For each n, let Mn be an n × n random matrix with independent ±1 entries. We show that P{Mn is singular} = (1/2 + on(1)) n , which settles an old problem. Some generalizations are considered.

Let A = (a ij ) be an n × n random matrix with i.i.d. entries such that Ea 11 = 0 and Ea 11 2 = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B n 2 of cardinality at most exp(δn) such that with probability very close to one we haveIn fact, a stronger statement holds true. As an application, we show that for some L ′ > 0 and u ∈ [0, 1) depending only on the distribution law of a 11 , the smallest singular value s n of the matrix A satisfies P{s n (A) ≤ εn −1/2 } ≤ L ′ ε + u n for all ε > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

Let D n,d be the set of all d-regular directed graphs on n vertices. Let G be a graph chosen uniformly at random from D n,d and M be its adjacency matrix. We show that M is invertible with probability at least 1where c, C are positive absolute constants. To this end, we establish a few properties of d-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let J be a subset of vertices of G with |J| ≈ n/d. Let δ i be the indicator of the event that the vertex i is connected to J and define δ = (δ 1 , δ 2 , ..., δ n ) ∈ {0, 1} n . Then for every v ∈ {0, 1} n the probability that δ = v is exponentially small. This property holds even if a part of the graph is "frozen." AMS 2010 Classification: 60C05, 60B20, 05C80, 15B52, 46B06.

We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let C 1 < d < cn/ log 2 n and let M n,d be the set of all n × n square matrices with 0/1 entries, such that each row and each column of every matrix in M n,d has exactly d ones. Let M be a random matrix uniformly distributed on M n,d . Then the smallest singular value s n (M ) of M is greater than n −6 with probability at least 1 − C 2 log 2 d/ √ d, where c, C 1 , and C 2 are absolute positive constants independent of any other parameters. Analogous estimates are obtained for matrices of the form M − z Id, where Id is the identity matrix and z is a fixed complex number.

Let p > 2, B ≥ 1, N ≥ n and let X be a centered n-dimensional random vector with the identity covariance matrix such that sup a∈S n−1 E| X, a | p ≤ B. Further, let X 1 , X 2 , . . . , X N be independent copies of X, and Σ N :T be the sample covariance matrix. We prove that