# The Circular Law for random regular digraphs

@article{Cook2019TheCL,
title={The Circular Law for random regular digraphs},
author={Nicholas A. Cook},
journal={Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
year={2019}
}
• Nicholas A. Cook
• Published 16 March 2017
• Mathematics
• Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral distribution of $A_n$, suitably rescaled, is governed by the Circular Law. A key step is to obtain quantitative lower tail bounds for the smallest singular value of additive perturbations of $A_n$.

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