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arXiv · 0907.2609

On limits of Graphs Sphere Packed in Euclidean Space and Applications

Itai Benjamini·Nicolas Curien
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Abstract

The core of this note is the observation that links between circle packings of graphs and potential theory developed in \cite{BeSc01} and \cite{HS} 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 $ |\partial W| \leq |W|^{\frac{d-1}{d}+o(1)}$. Some open problems and conjectures are gathered at the end.

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Itai Benjamini, Nicolas Curien. 2010-10-13. On limits of Graphs Sphere Packed in Euclidean Space and Applications. https://arxiv.org/abs/0907.2609

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