The Gaussian Wave for Graphs of Finite Cone Type

TL;DR

This paper proves that on certain infinite trees and related random graphs, the only typical eigenvector process with covariance from the Green's function is the Gaussian wave, extending prior results to more general structures.

Contribution

It generalizes the Gaussian wave characterization from regular trees to infinite trees of finite cone type under mild expansion conditions.

Findings

Distribution of eigenvector neighborhoods approximates Gaussian wave.

Results apply to random bipartite biregular graphs.

Local eigenvector distributions converge to Gaussian wave in various models.

Abstract

We show that for any infinite tree of finite cone type satisfying a mild expansion condition, the only typical process on its vertices with covariance induced by the Green's function is the Gaussian wave. This generalizes a result of Backhausz and Szegedy, who proved this for the infinite regular tree of degree $d\geq 3$. We do this by giving a reduction to a statement concerning the distribution of the inner product of our process with columns of the Green's function, which in turn are straightforward to calculate. As a consequence, for random bipartite biregular graphs, the distribution of local neighborhoods of eigenvectors must approximate the Gaussian wave. Moreover, for generic configuration models including random lifts, the local distribution of a uniformly chosen eigenvector from any arbitrarily small spectral window likewise converges to the Gaussian wave.