Large, lengthy graphs look locally like lines
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 R in the sense that the rescaled ball is close to a line segment in the Gromov–Hausdorff metric.
Additional Information© 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence. Received 5 July 2019; revised 23 September 2020; published online 25 November 2020. We thank Jonathan Hermon and Matthew Tointon for helpful comments on a draft. We also thank the anonymous referee for their helpful suggestions.
Accepted Version - 1905.00316.pdf