{"paper":{"title":"On limits of Graphs Sphere Packed in Euclidean Space and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"Itai Benjamini, Nicolas Curien","submitted_at":"2009-07-15T15:30:57Z","abstract_excerpt":"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 ope"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2609","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}