Structure of eigenvectors of random regular digraphs
Abstract
Let $d$ and $n$ be integers satisfying $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in \mathbb{C}$. Denote by $M$ the adjacency matrix of a random $d$regular directed graph on $n$ vertices. In this paper, we study the structure of the kernel of submatrices of $Mz\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two nonintersecting types of vectors, which we call very steep and gradual with many levels. As a corollary, we show, in particular, that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: its level sets 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 $n$. As a key technical ingredient of our proofs we introduce a decomposition of $\mathbb{C}^n$ into vectors of different degrees 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.05575
 Bibcode:
 2018arXiv180105575L
 Keywords:

 Mathematics  Probability;
 60B20;
 15B52;
 46B06;
 05C80
 EPrint:
 Accepted in Transactions of the American Mathematical Society