{"paper":{"title":"Monotone cellular automata in a random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Andrew Uzzell, B\\'ela Bollob\\'as, Paul Smith","submitted_at":"2012-04-18T05:42:10Z","abstract_excerpt":"In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in $\\mathbb{Z}^d$ with random initial configurations. Formally, we are given a set $\\mathcal{U}=\\{X_1,\\dots,X_m\\}$ of finite subsets of $\\mathbb{Z}^d\\setminus\\{\\mathbf{0}\\}$, and an initial set $A_0\\subset\\mathbb{Z}^d$ of `infected' sites, which we take to be random according to the product measure with density $p$. At time $t\\in\\mathbb{N}$, the set of infected sites $A_t$ is the union of $A_{t-1}$ and the set of all $x\\in\\mathbb{Z}^d$ such that $x+X\\in A_{t-1}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3980","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"}