The smallest singular value of a shifted d-regular random square matrix

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 C1<d<cn/log2n and let Mn,d be the set of all nxn square matrices with 0/1 entries, such that each row and each column of every matrix in Mn,d has exactly d ones. Let M be a random matrix uniformly distributed on Mn,d. Then the smallest singular value sn(M) of M is greater than n-6 with probability at least 1-C2log2d/ where c, C1, and C2 are absolute positive constants independent of any other parameters. Analogous estimates are obtained for matrices of the form M-zId, where Id is the identity matrix and z is a fixed complex number.