On Non-localization of Eigenvectors of High Girth Graphs

Abstract

Abstract We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss [6] who relied on the observation that certain suitably normalized averaging operators o nhigh girth graphs are hyper-contractive and can be used to approximate projectors onto the eigenspaces of such graphs. Informally, their delocalization result in the contrapositive states that for any $\varepsilon \in (0,1)$ and positive integer $k,$ if a $(d+1)-$regular graph has an eigenvector that supports $\varepsilon $ fraction of the $\ell _2^2$ mass on a subset of $k$ vertices, then the graph must have a cycle of size $\log _{d}(k)/\varepsilon ^2)$, up to multiplicative universal constants and additive logarithmic terms in $1/\varepsilon $. In this paper, we improve the upper bound to $\log _{d}(k)/\varepsilon $ up to similar logarithmic correction terms; and present a construction showing a lower bound of $\log _d(k)/\varepsilon $ up to multiplicative constants. Our construction is probabilistic and involves gluing together a pair of trees while maintaining high girth as well as control on the eigenvectors and could be of independent interest.