On the almost eigenvectors of random regular graphs

TL;DR

This paper demonstrates that almost eigenvectors of large random regular graphs have entry distributions close to Gaussian, with joint distributions also Gaussian if eigenvalues are close, using graph limits and information theory.

Contribution

It establishes Gaussianity of almost eigenvectors and their joint distributions in random regular graphs, extending understanding of eigenvector behavior.

Findings

Entry distributions of almost eigenvectors are close to Gaussian.

Joint distributions of nearby eigenvectors are also Gaussian.

Results apply to factor of i.i.d. processes on infinite regular trees.

Abstract

Let $d\geq 3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector $v$ of $G$ (with entry sum 0 and normalized to have length $\sqrt{n}$) is close to some Gaussian distribution $N(0,\sigma)$ in the weak topology where $0\leq\sigma\leq 1$. Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph. Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenvalues are close. Our proof uses graph limits and information theory. Our results have consequences for factor of i.i.d.\ processes on the infinite regular tree.