Adjacency matrices of random digraphs: Singularity and anti-concentration

Abstract

Let Dn,d be the set of all d-regular directed graphs on n vertices. Let G be a graph chosen uniformly at random from Dn,d and M be its adjacency matrix. We show that M is invertible with probability at least 1−Cln3⁡d/d for C≤d≤cn/ln2⁡n, where 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 δi be the indicator of the event that the vertex i is connected to J and define δ=(δ1,δ2,...,δn)∈{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.”