On limits of graphs sphere packed in Euclidean space and applications

The core of this note is the observation that links between circle packings of graphs and potential theory developed in Benjamini and Schramm (2001) [4] and He and Schramm (1995) [11] can be extended to higher dimensions. In particular, it is shown that every limit of finite graphs sphere packed in Rd with a uniformly chosen root is d-parabolic. We then derive a few geometric corollaries. For example, every infinite graph packed in Rd has either strictly positive isoperimetric Cheeger constant or admits arbitrarily large finite sets W with boundary size which satisfies |∂W|⩽|W|d−1d+o(1). Some open problems and conjectures are gathered at the end.