{"paper":{"title":"Clustering in the three and four color cyclic particle systems in one dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech"],"primary_cat":"math.PR","authors_text":"Eric Foxall, Hanbaek Lyu","submitted_at":"2017-11-13T18:18:39Z","abstract_excerpt":"We study the $\\kappa$-color cyclic particle system on the one-dimensional integer lattice $\\mathbb{Z}$, first introduced by Bramson and Griffeath in \\cite{bramson1989flux}. In that paper they show that almost surely, every site changes its color infinitely often if $\\kappa\\in \\{3,4\\}$ and only finitely many times if $\\kappa\\ge 5$. In addition, they conjecture that for $\\kappa\\in \\{3,4\\}$ the system clusters, that is, for any pair of sites $x,y$, with probability tending to 1 as $t\\to\\infty$, $x$ and $y$ have the same color at time $t$. Here we prove that conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04741","kind":"arxiv","version":3},"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"}