{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:CRHUTPACWFXHPMUKIN6OKBHMKC","short_pith_number":"pith:CRHUTPAC","schema_version":"1.0","canonical_sha256":"144f49bc02b16e77b28a437ce504ec50ad6242caa8cdf83c8c602bc9a33895cb","source":{"kind":"arxiv","id":"1310.4457","version":2},"attestation_state":"computed","paper":{"title":"Maximum percolation time in two-dimensional bootstrap percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fabricio Benevides, Micha{\\l} Przykucki","submitted_at":"2013-10-16T17:34:30Z","abstract_excerpt":"We consider a classic model known as bootstrap percolation on the $n \\times n$ square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least $2$ already infected neighbours. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap percolation process can take to eventually infect the entire vertex set of the grid is $13n^2/18+O(n)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1310.4457","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-16T17:34:30Z","cross_cats_sorted":[],"title_canon_sha256":"2cfac493ce2b00aa5394be32886fd17d95e2479dc534a94072aa43e0ae630046","abstract_canon_sha256":"47c00df8802799a7bb7531253d38100a8eeae80744a4a277979caa788f47a4ca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:37.373166Z","signature_b64":"7aUfYwnkP7GP3/9cfGWcCBDMWSBClzGuiGeUAnOHf31iFqn8GUkGQR8ImQDu93W1XayJ8nki2cAnCAqYoY5vCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"144f49bc02b16e77b28a437ce504ec50ad6242caa8cdf83c8c602bc9a33895cb","last_reissued_at":"2026-05-18T02:38:37.372759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:37.372759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximum percolation time in two-dimensional bootstrap percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fabricio Benevides, Micha{\\l} Przykucki","submitted_at":"2013-10-16T17:34:30Z","abstract_excerpt":"We consider a classic model known as bootstrap percolation on the $n \\times n$ square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least $2$ already infected neighbours. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap percolation process can take to eventually infect the entire vertex set of the grid is $13n^2/18+O(n)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4457","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1310.4457","created_at":"2026-05-18T02:38:37.372816+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.4457v2","created_at":"2026-05-18T02:38:37.372816+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.4457","created_at":"2026-05-18T02:38:37.372816+00:00"},{"alias_kind":"pith_short_12","alias_value":"CRHUTPACWFXH","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_16","alias_value":"CRHUTPACWFXHPMUK","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_8","alias_value":"CRHUTPAC","created_at":"2026-05-18T12:27:40.988391+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC","json":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC.json","graph_json":"https://pith.science/api/pith-number/CRHUTPACWFXHPMUKIN6OKBHMKC/graph.json","events_json":"https://pith.science/api/pith-number/CRHUTPACWFXHPMUKIN6OKBHMKC/events.json","paper":"https://pith.science/paper/CRHUTPAC"},"agent_actions":{"view_html":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC","download_json":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC.json","view_paper":"https://pith.science/paper/CRHUTPAC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.4457&json=true","fetch_graph":"https://pith.science/api/pith-number/CRHUTPACWFXHPMUKIN6OKBHMKC/graph.json","fetch_events":"https://pith.science/api/pith-number/CRHUTPACWFXHPMUKIN6OKBHMKC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC/action/storage_attestation","attest_author":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC/action/author_attestation","sign_citation":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC/action/citation_signature","submit_replication":"https://pith.science/pith/CRHUTPACWFXHPMUKIN6OKBHMKC/action/replication_record"}},"created_at":"2026-05-18T02:38:37.372816+00:00","updated_at":"2026-05-18T02:38:37.372816+00:00"}