{"paper":{"title":"Bootstrap percolation on the Hamming torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Christopher Hoffman, David Sivakoff, James Pfeiffer, Janko Gravner","submitted_at":"2012-02-24T00:47:02Z","abstract_excerpt":"The Hamming torus of dimension $d$ is the graph with vertices $\\{1,\\dots,n\\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\\theta$ starts with a random set of open vertices, to which every vertex belongs independently with probability $p$, and at each time step the open set grows by adjoining every vertex with at least $\\theta$ open neighbors. We assume that $n$ is large and that $p$ scales as $n^{-\\alpha}$ for some $\\alpha>1$, and study the probability that an $i$-dimensional subgraph ever becomes open. For large $\\theta$, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5351","kind":"arxiv","version":5},"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"}