High-girth near-Ramanujan graphs with lossy vertex expansion

Kahale proved that linear sized sets in $d$-regular Ramanujan graphs have vertex expansion $\sim\frac{d}{2}$ and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than $\frac{d}{2}$. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether high-girth Ramanujan graphs have improved vertex expansion. Our results are two-fold: 1. For every $d = p+1$ for prime $p$ and infinitely many $n$, we exhibit an $n$-vertex $d$-regular graph with girth $\Omega(\log{d-1} n)$ and vertex expansion of sublinear sized sets bounded by $\frac{d+1}{2}$ whose nontrivial eigenvalues are bounded in magnitude by $2\sqrt{d-1}+O\left(\frac{1}{\log n}\right)$. 2. In any Ramanujan graph with girth $C\log n$, all sets of size bounded by $n^{0.99C/4}$ have vertex expansion $(1-od(1))d$. The tools in analyzing our construction include the nonbacktracking operator of an infinite graph, the Ihara--Bass formula, a trace moment method inspired by Bordenave's proof of Friedman's theorem, and a method of Kahale to study dispersion of eigenvalues of perturbed graphs.