On limits of Graphs Sphere Packed in Euclidean Space and Applications

2009/07/15 by Itai Benjamini, Nicolas Curien, Benjamini, Itai +1
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR) #math.MG #math.PR

paper · pdf · doi:10.48550/arxiv.0907.2609

arxiv created 2010/10/13 · arxiv updated 2010/10/14

Abstract

The core of this note is the observation that links between circle packings of graphs and potential theory developed in \citeBeSc01 and \citeHS can be extended to higher dimensions. In particular, it is shown that every limit of finite graphs sphere packed in \R^d with a uniformly-chosen root is d-parabolic. We then derive few geometric corollaries. E.g. every infinite graph packed in \R^d has either strictly positive isoperimetric Cheeger constant or admits arbitrarily large finite sets W with boundary size which satisfies |∂ W| ≤ |W|^{(d-1)/(d)+o(1)}. Some open problems and conjectures are gathered at the end.

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