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 (2009) who relied on the observation that certain suitably normalized averaging operators on high 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 which supports $\varepsilon$ fraction of the $\ell_2^2$ mass on a subset of $k$ vertices, then the graph must have a cycle of size $\tilde{O}(\log_{d}(k)/\varepsilon^2)$, suppressing logarithmic terms in $1/\varepsilon$. In this paper, we improve the upper bound to $\tilde{O}(\log_{d}(k)/\varepsilon)$ and present a construction showing a lower bound of $\Omega(\log_d(k)/\varepsilon)$. 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.

翻译：我们证明,对于高 girth 普通图形的偏移值,我们可以改进了范围, 并展示了一些例子, 表明这些界限接近于锐利。 本研究由布鲁克斯和林登斯特拉斯(2009年)发起, 其依据的观察是, 高 girth 图形上某些适当正常化平均操作器的分级值是超合的, 可以用来将投影器近似于这些图形的偏移空间。 非正式地, 它们的离地化导致相反的状态, 任何$\varepsilon\ in (0, 1美元) 和正整值美元, 如果 $( d+1) 和正整值的正整值值值值值是接近的 。 在本文中, 我们改进了 $\ 2\\\ 平面 质量 部分的比值部分, 然后在构建中显示 $\ rqrqlus 的上限值 。