{"paper":{"title":"The critical activation density in graph bootstrap percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For every graph H the critical H-percolation threshold p_c(n,H) is located in terms of the limiting density ρ(H) of the graphs that activate an edge most efficiently.","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Brett Kolesnik, Maksim Zhukovskii, Pawe{\\l} Pra{\\l}at, Rajko Nenadov, Tam\\'as Makai, Xavier P\\'erez-Gim\\'enez","submitted_at":"2026-05-14T16:56:35Z","abstract_excerpt":"In graph bootstrap percolation, edges of an Erd\\H{o}s-R\\'enyi random graph ${\\mathcal G}_{n,p}$ are initially active. Activation spreads to other edges of the complete graph $K_n$ by an iterative process governed by a fixed graph $H$, whereby an edge becomes active whenever it is the only inactive edge in a copy of $H$. If all edges of $K_n$ are eventually activated, we say the process $H$-percolates. The case $H=K_3$ corresponds to the classical sharp threshold for connectivity in ${\\mathcal G}_{n,p}$. When $H=K_4$, there are close connections with $2$-neighbor bootstrap percolation from stat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for every graph H, we locate the critical H-percolation threshold p_c(n,H)","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the critical limiting density ρ(H) of graphs that most efficiently activate a given edge is well-defined and finite for every H, and that the threshold is determined by this quantity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For any fixed graph H the critical percolation threshold p_c(n,H) equals the value determined by the minimal activation density ρ(H) of graphs that efficiently activate an edge.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For every graph H the critical H-percolation threshold p_c(n,H) is located in terms of the limiting density ρ(H) of the graphs that activate an edge most efficiently.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"de4d5078fc9a83ec4ba0afa448f4e8bef3c934ded18058d468407166489f8e52"},"source":{"id":"2605.15066","kind":"arxiv","version":1},"verdict":{"id":"ac963031-7fbb-4e57-a814-21e4cf41cf3d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:57:30.338857Z","strongest_claim":"for every graph H, we locate the critical H-percolation threshold p_c(n,H)","one_line_summary":"For any fixed graph H the critical percolation threshold p_c(n,H) equals the value determined by the minimal activation density ρ(H) of graphs that efficiently activate an edge.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the critical limiting density ρ(H) of graphs that most efficiently activate a given edge is well-defined and finite for every H, and that the threshold is determined by this quantity.","pith_extraction_headline":"For every graph H the critical H-percolation threshold p_c(n,H) is located in terms of the limiting density ρ(H) of the graphs that activate an edge most efficiently."},"references":{"count":30,"sample":[{"doi":"","year":1988,"title":"M. Aizenman and J. L. Lebowitz,Metastability effects in bootstrap percolation, J. Phys. A21(1988), no. 19, 3801–3813","work_id":"32a08250-75c2-4560-b60c-880d47df233d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1985,"title":"N. Alon,An extremal problem for sets with applications to graph theory, J. Combin. Theory Ser. A40(1985), no. 1, 82–89","work_id":"0e5f2e1d-37cf-4e60-9169-d2eaaaa69502","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"N. Alon and J. H. Spencer,The probabilistic method, fourth ed., Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2016","work_id":"4499de7c-30a3-4c53-87d6-15694724ad69","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"O. Angel and B. Kolesnik,Sharp thresholds for contagious sets in random graphs, Ann. Appl. Probab.28(2018), no. 2, 1052–1098","work_id":"41cb610d-ae93-4915-8d84-8e2d75c86b6b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":",Large deviations for subcritical bootstrap percolation on the Erd˝ os–Rényi graph, J. Stat. Phys.185(2021), no. 2, Paper No. 8, 16","work_id":"dc0e7b4d-8ff3-4b1d-8ac6-666e7cfc27e3","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"ded220e1a6d40e552d9364e3a039c3f426434a349ce47a064514e90b340e29ee","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"}