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.

Let X be a symmetric random matrix with independent but nonidentically distributed centered Gaussian entries. We show thatwhere Sp denotes the p-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case p = ∞, a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on ℓ 2 . Along the way, we obtain optimal dimension-free bounds on the moments (E X p Sp ) 1/p that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.2000 Mathematics Subject Classification. 60B20; 46B09; 46L53; 15B52.

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.