Structure of eigenvectors of random regular digraphs

Alexander Litvak, Anna Lytova, Konstantin Tikhomirov, Nicole Tomczak-Jaegermann, Pierre Youssef

Published 2018-01-17Version 1

Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and sloping with many levels. As a corollary of this theorem, we show that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: all subsets of coordinates with the same value have cardinality less than $Cn\ln^2 d/\ln n$. For a large constant $d$ this provides a principally new structural information on eigenvectors, implying that the number of their level sets grows to infinity with $n$, and -- combined with additional arguments -- that the rank of the adjacency matrix $M$ is at least $n-1$, with probability going to 1 as $n\to\infty$. Moreover, our main result for $d$ slowly growing to infinity with $n$ contributes a crucial step towards establishing the circular law for the empirical spectral distribution of sparse random $d$-regular directed graphs. As a key technical device we introduce a decomposition of ${\mathcal C}^n$ into vectors of different degree of "structuredness", which is an alternative to the decomposition based on the least common denominator in the regime when the underlying random matrix is very sparse.