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}
}
  • Nicholas A. Cook
  • Published 2 November 2014
  • Mathematics, Computer Science
  • Probability Theory and Related Fields
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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