Large, lengthy graphs look locally like lines
Metric Geometry
2020-12-02 v3 Combinatorics
Probability
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 in the sense that the rescaled ball is close to a line segment in the Gromov-Hausdorff metric.
Cite
@article{arxiv.1905.00316,
title = {Large, lengthy graphs look locally like lines},
author = {Itai Benjamini and Tom Hutchcroft},
journal= {arXiv preprint arXiv:1905.00316},
year = {2020}
}
Comments
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