Adjacency matrices of random digraphs: singularity and anti-concentration

@article{Litvak2015AdjacencyMO,
  title={Adjacency matrices of random digraphs: singularity and anti-concentration},
  author={Alexander E. Litvak and Anna Lytova and Konstantin E. Tikhomirov and Nicole Tomczak-Jaegermann and Pierre Youssef},
  journal={arXiv: Probability},
  year={2015}
}
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, 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… 
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