{"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"}