On the smallest singular value of the adjacency matrix of a d-regular random directed graph

We consider the set M_{n,d} of adjacency matrices of d-regular random directed graphs.  This set consists of  0/1-valued n by n matrices such that each row and each column of a matrix has exactly d ones. Probability is given by the normalized counting measure on M_{n,d}. We establish a lower bound for the smallest singular value s_{n} (M)=\min_z||Mz||_2/||z||_2 of M in M_{n,d}. Also we discuss the obtained results in connection with the convergence of the empirical spectral distributions as n,d tend to infinity towards the circular law. This is a joint work with A. Litvak, K. Tikhomirov, N. Tomczak-Jaegermann, and P. Youssef.