High-girth near-Ramanujan graphs with localized eigenvectors

We show that for every prime $d$ and $alphain (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2alpha log_{d}(|V|)(1-o_d(1))$, second adjacency matrix eigenvalue bounded by $(3/sqrt{2})sqrt{d}$, and many eigenvectors fully localized on small sets of size $O(|V|^alpha)$. This strengthens the results of Ganguly-Srivastava, who constructed high girth (but not expanding) graphs with similar properties, and may be viewed as a discrete analogue of the scarring phenomenon observed in the study of quantum ergodicity on manifolds. Key ingredients in the proof are a technique of Kahale for bounding the growth rate of eigenfunctions of graphs, discovered in the context of vertex expansion and a method of ErdH{o}s and Sachs for constructing high girth regular graphs.