# On the singularity of adjacency matrices for random regular digraphs

@article{Cook2014OnTS, title={On the singularity of adjacency matrices for random regular digraphs}, author={Nicholas A. Cook}, journal={Probability Theory and Related Fields}, year={2014}, volume={167}, pages={143-200} }

We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming $$\min (d,n-d)\ge C\log ^2n$$min(d,n-d)≥Clog2n for a sufficiently large constant $$C>0$$C>0. The proof makes use of a coupling of random regular digraphs formed by “shuffling” the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges, proved in Cook (Random Struct Algorithms. arXiv:1410…

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