{"paper":{"title":"Upper tail bounds for cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abigail Raz","submitted_at":"2019-03-18T14:49:22Z","abstract_excerpt":"This paper examines bounds on upper tails for cycle counts in $G_{n,p}$. For a fixed graph $H$ define $\\xi_H= \\xi_H^{n,p}$ to be the number of copies of $H$ in $G_{n,p}$. It is a much studied and surprisingly difficult problem to understand the upper tail of the distribution of $\\xi_H$, for example, to estimate \\begin{equation*}\n  \\mathbb{P}(\\xi_H > 2 \\mathbb{E}\\xi_H). \\end{equation*} The best known result for general $H$ and $p$ is due to Janson, Oleszkiewicz, and Ruci\\'nski, who, in 2004, proved \\begin{align}\\label{a:JOR} \\exp[-O_{H, \\eta}(M_H(n,p) \\ln(1/p))]&<\\mathbb{P}(\\xi_H > (1+\\eta)\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07488","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"}