Delocalized eigenvectors of transitive graphs and beyond

TL;DR

This paper demonstrates delocalization of eigenvectors in vertex-transitive graphs and general symmetric matrices using elementary spectral estimates, extending known results and applying to graphs with few short loops.

Contribution

It introduces elementary spectral projector estimates for eigenvector delocalization, recovering known results and extending to approximate eigenvectors in large graphs with few short loops.

Findings

Eigenvectors of vertex-transitive graphs are delocalized.

Most approximate eigenvectors in certain spectral windows are delocalized in L^q norms.

Delocalization results apply to large graphs with few short loops, including random lifts.

Abstract

We prove delocalization of eigenvectors of vertex-transitive graphs via elementary estimates of the spectral projector. We recover in this way known results which were formerly proved using representation theory. Similar techniques show that for general symmetric matrices, most approximate eigenvectors spectrally localized in a given window containing sufficiently many eigenvalues are delocalized in $L^q$ norms. Building upon this observation, we prove a delocalization result for approximate eigenvectors of large graphs containing few short loops, under an assumption on the resolvent which is verified in some standard cases, for instance random lifts of a fixed base graph.