Structure of eigenvectors of random regular digraphs

TL;DR

This paper investigates the structure of eigenvectors of random regular directed graphs, revealing they are either very steep or gradual, and establishes a weak delocalization property for these eigenvectors.

Contribution

It introduces a novel structural analysis of eigenvectors of sparse random regular digraphs, identifying two main types and their properties, which was previously unexplored.

Findings

Eigenvectors are either very steep or gradual with many levels.

Except for the trivial eigenvector, eigenvectors exhibit weak delocalization.

Number of level sets of eigenvectors grows with graph size.

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 $M-z\,{\rm Id}$, 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 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.