Anti-concentration property for random digraphs and invertibility of their adjacency matrices

Let Dn,d be the set of all directed d-regular 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-Cln³d/d for C≤d≤cn/ln²n, where c, C are positive absolute constants. To this end, we establish a few properties of directed d-regular 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|≤cn/d. Let δ_i be the indicator of the event that the vertex i is connected to J and δ=(δ₁, δ₂, . ., δ_n)∈[0, 1]ⁿ. Then δ is not concentrated around any vertex of the cube. This property holds even if a part of the graph is fixed.