High-girth near-Ramanujan graphs with localized eigenvectors

We show that for every prime d and α ∈ (0, 1/6), there is an infinite sequence of (d + 1)-regular graphs G = (V, E) with high girth Ω(α logd(∣V∣), second adjacency matrix eigenvalue bounded by $$(3/\sqrt 2)\sqrt $$ , many eigenvectors fully localized on small sets size O(mα). This strengthens the results [GS18], who constructed (but not expanding) similar properties, may be viewed as a discrete analogue “scarring” phenomenon observed in study quantum ergodicity manifolds. Key ingredients proof are technique Kahale [Kah92] bounding growth rate eigenfunctions graphs, discovered context vertex expansion method Erdős Sachs constructing regular graphs.