{"paper":{"title":"Cutoff on all Ramanujan graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Eyal Lubetzky, Yuval Peres","submitted_at":"2015-07-16T19:40:55Z","abstract_excerpt":"We show that on every Ramanujan graph $G$, the simple random walk exhibits cutoff: when $G$ has $n$ vertices and degree $d$, the total-variation distance of the walk from the uniform distribution at time $t=\\frac{d}{d-2}\\log_{d-1} n + s\\sqrt{\\log n}$ is asymptotically $\\mathbb{P}(Z > c\\, s)$ where $Z$ is a standard normal variable and $c=c(d)$ is an explicit constant. Furthermore, for all $1 \\leq p \\leq \\infty$, $d$-regular Ramanujan graphs minimize the asymptotic $L^p$-mixing time for SRW among all $d$-regular graphs. Our proof also shows that, for every vertex $x$ in $G$ as above, its distan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04725","kind":"arxiv","version":5},"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"}