{"paper":{"title":"The polynomial growth of the infinite long-range percolation cluster","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Johannes B\\\"aumler","submitted_at":"2023-11-24T08:53:07Z","abstract_excerpt":"We study independent long-range percolation on $\\mathbb{Z}^d$ where the nearest-neighbor edges are always open and the probability that two vertices $x,y$ with $\\|x-y\\|>1$ are connected by an edge is proportional to $\\frac{\\beta}{\\|x-y\\|^s}$, where $\\beta>0$ and $s> 0$ are parameters. We show that the ball of radius $k$ centered at the origin in the graph metric grows polynomially if and only if $s\\geq 2d$. For the critical case $s=2d$, we show that the volume growth exponent is inversely proportional to the distance growth exponent. Furthermore, we provide sharp upper and lower bounds on the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.14352","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2311.14352/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}