{"paper":{"title":"Random generators of the symmetric group: diameter, mixing time and spectral gap","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.GR","authors_text":"\\'Akos Seress, Andrzej Zuk, Harald A. Helfgott","submitted_at":"2013-11-26T17:18:50Z","abstract_excerpt":"Let $g$, $h$ be a random pair of generators of $G=Sym(n)$ or $G=Alt(n)$. We show that, with probability tending to $1$ as $n\\to \\infty$, (a) the diameter of $G$ with respect to $S = \\{g,h,g^{-1},h^{-1}\\}$ is at most $O(n^2 (\\log n)^c)$, and (b) the mixing time of $G$ with respect to $S$ is at most $O(n^3 (\\log n)^c)$. (Both $c$ and the implied constants are absolute.)\n  These bounds are far lower than the strongest worst-case bounds known (in Helfgott--Seress, 2013); they roughly match the worst known examples. We also give an improved, though still non-constant, bound on the spectral gap.\n  O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6742","kind":"arxiv","version":3},"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"}