Circular law for sparse random regular digraphs

@article{Litvak2018CircularLF,
  title={Circular law for sparse random regular digraphs},
  author={Alexander E. Litvak and Anna Lytova and Konstantin E. Tikhomirov and Nicole Tomczak-Jaegermann and Pierre Youssef},
  journal={arXiv: Probability},
  year={2018}
}
Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical… Expand
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