The circular law for random regular digraphs

2017/03/16 by Cook, Nicholas A.
#05C80 #15B52 #60B20 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1703.05839

Abstract

Let log^Cn≤ d≤ 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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