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Large, lengthy graphs look locally like lines

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arxiv 1905.00316 v3 pith:4O5A5CEK submitted 2019-05-01 math.MG math.COmath.PR

classification math.MGmath.COmath.PR
keywords graphsgraphlargelikemetricapplyballbounded
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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.

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