{"paper":{"title":"Percolation of finite clusters and infinite surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander E. Holroyd, Gady Kozma, Geoffrey R. Grimmett","submitted_at":"2013-03-07T12:10:06Z","abstract_excerpt":"Two related issues are explored for bond percolation on the d-dimensional cubic lattice (with d > 2) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point p_fin satisfies p_fin >= p_c, and strict inequality is proved when either d is sufficiently large, or d >= 7 and the model is sufficiently spread out. It is not known whether d >= 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point p_surf sat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.1657","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"}