{"paper":{"title":"Subcritical $\\mathcal{U}$-bootstrap percolation models have non-trivial phase transitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"B\\'ela Bollob\\'as, Micha{\\l} Przykucki, Paul Balister, Paul Smith","submitted_at":"2013-11-22T20:55:06Z","abstract_excerpt":"We prove that there exist natural generalizations of the classical bootstrap percolation model on $\\mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property.\n  Van Enter (in the case $d=r=2$) and Schonmann (for all $d \\geq r \\geq 2$) proved that $r$-neighbour bootstrap percolation models have trivial critical probabilities on $\\mathbb{Z}^d$ for every choice of the parameters $d \\geq r \\geq 2$: that is, an initial set of density $p$ almost surely percolates $\\mathbb{Z}^d$ for every $p>0$. These results eff"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.5883","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":""},"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"}