# High-girth near-Ramanujan graphs with localized eigenvectors

## TL;DR

This paper constructs infinite families of high-girth, near-Ramanujan regular graphs with localized eigenvectors, revealing a discrete analogue of quantum scarring phenomena and advancing understanding of graph eigenvector localization.

## Contribution

It introduces a method to build high-girth, near-Ramanujan graphs with localized eigenvectors, extending prior work and connecting to quantum ergodicity phenomena.

## Key findings

- Existence of infinite high-girth, near-Ramanujan graphs with localized eigenvectors
- Eigenvectors can be fully localized on small vertex sets
- Girth scales logarithmically with graph size

## Abstract

We show that for every prime $d$ and $\alpha\in (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2\alpha \log_{d}(|V|)(1-o_d(1))$, 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 Erd\H{o}s and Sachs for constructing high girth regular graphs.