{"paper":{"title":"Sharpness of the percolation transition in the two-dimensional contact process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"J. van den Berg","submitted_at":"2009-07-16T13:53:46Z","abstract_excerpt":"For ordinary (independent) percolation on a large class of lattices it is well known that below the critical percolation parameter $p_c$ the cluster size distribution has exponential decay and that power-law behavior of this distribution can only occur at $p_c$. This behavior is often called ``sharpness of the percolation transition.'' For theoretical reasons, as well as motivated by applied research, there is an increasing interest in percolation models with (weak) dependencies. For instance, biologists and agricultural researchers have used (stationary distributions of) certain two-dimension"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2843","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"}