Large, lengthy graphs look locally like lines

Abstract

We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like $\mathbb{R}$ in the sense that the rescaled ball is close to a line segment in the Gromov-Hausdorff metric.

12 pages. V2: Some minor corrections. V3: Added a substantial amount of additional expository material. Accepted version to appear in the Bulletin of the London Mathematical Society