Tight Bounds for Non-localization of Eigenvectors of High Girth Graphs

We prove tight (up to small constant factors) results on how localized an eigenvector of a high girth regular graph can be (the girth is the length of the shortest cycle). This study was initiated by Brooks and Lindenstrauss [BL13] who relied on the key observation that certain suitably normalized averaging operators on high girth graphs are hyper-contractive (have small `1 → `∞ norm) and can be used to approximate projectors onto the eigenspaces of the graph. Informally, their delocalization result in the contrapositive states that for any ε ∈ (0, 1) and a positive integer k, if a d + 1−regular graph has an eigenvector which supports ε fraction of the `2 2 mass on a subset of k vertices, then the graph must have a cycle of size O(logd(k)/ε 2). In this paper, we improve the upper bound to O(logd(k)/ε) and moreover present a construction, showing that this is sharp. 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.