{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:E7T7WQBVFYILXGURMGPFR4E6XD","short_pith_number":"pith:E7T7WQBV","schema_version":"1.0","canonical_sha256":"27e7fb40352e10bb9a91619e58f09eb8c1cfdf443722e62e86036466c83df4f1","source":{"kind":"arxiv","id":"1403.6558","version":2},"attestation_state":"computed","paper":{"title":"Exploring hypergraphs with martingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"B\\'ela Bollob\\'as, Oliver Riordan","submitted_at":"2014-03-26T02:12:20Z","abstract_excerpt":"Recently, we adapted exploration and martingale arguments of Nachmias and Peres, in turn based on ideas of Martin-L\\\"of, Karp and Aldous, to prove asymptotic normality of the number $L_1$ of vertices in the largest component $C$ of the random $r$-uniform hypergraph throughout the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of $L_1$, and joint asymptotic normality of $L_1$ and the number $M_1$ of edges of $C$. These results are used in a separate paper \"Counting connected hypergraphs via the probabilistic m"},"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":"1403.6558","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-03-26T02:12:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"20e04774e562a8cab5bba58b7a6801a9737013151da1aa62ff917ca1e73c2845","abstract_canon_sha256":"6eb1fe95ea5fb4a497e6ec2b25d98ccf498c3f21c6f17be04a0b3f143fb7e576"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:42.338640Z","signature_b64":"pWQzG+j1zg4VjRzTwj1nLNondZEAKu+cLHQ6wYV2Cp3tGqoKz6wvJmCu4Z3yVruN9kxijznPWoWwPf/oEJWMAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27e7fb40352e10bb9a91619e58f09eb8c1cfdf443722e62e86036466c83df4f1","last_reissued_at":"2026-05-18T00:41:42.337834Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:42.337834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exploring hypergraphs with martingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"B\\'ela Bollob\\'as, Oliver Riordan","submitted_at":"2014-03-26T02:12:20Z","abstract_excerpt":"Recently, we adapted exploration and martingale arguments of Nachmias and Peres, in turn based on ideas of Martin-L\\\"of, Karp and Aldous, to prove asymptotic normality of the number $L_1$ of vertices in the largest component $C$ of the random $r$-uniform hypergraph throughout the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of $L_1$, and joint asymptotic normality of $L_1$ and the number $M_1$ of edges of $C$. These results are used in a separate paper \"Counting connected hypergraphs via the probabilistic m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.6558","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":"1403.6558","created_at":"2026-05-18T00:41:42.337978+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.6558v2","created_at":"2026-05-18T00:41:42.337978+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.6558","created_at":"2026-05-18T00:41:42.337978+00:00"},{"alias_kind":"pith_short_12","alias_value":"E7T7WQBVFYIL","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_16","alias_value":"E7T7WQBVFYILXGUR","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_8","alias_value":"E7T7WQBV","created_at":"2026-05-18T12:28:25.294606+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/E7T7WQBVFYILXGURMGPFR4E6XD","json":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD.json","graph_json":"https://pith.science/api/pith-number/E7T7WQBVFYILXGURMGPFR4E6XD/graph.json","events_json":"https://pith.science/api/pith-number/E7T7WQBVFYILXGURMGPFR4E6XD/events.json","paper":"https://pith.science/paper/E7T7WQBV"},"agent_actions":{"view_html":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD","download_json":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD.json","view_paper":"https://pith.science/paper/E7T7WQBV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.6558&json=true","fetch_graph":"https://pith.science/api/pith-number/E7T7WQBVFYILXGURMGPFR4E6XD/graph.json","fetch_events":"https://pith.science/api/pith-number/E7T7WQBVFYILXGURMGPFR4E6XD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/action/storage_attestation","attest_author":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/action/author_attestation","sign_citation":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/action/citation_signature","submit_replication":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/action/replication_record"}},"created_at":"2026-05-18T00:41:42.337978+00:00","updated_at":"2026-05-18T00:41:42.337978+00:00"}