{"paper":{"title":"On the Cycle Structure of Mallows Permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Alexey Gladkich, Ron Peled","submitted_at":"2016-01-26T12:24:32Z","abstract_excerpt":"We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $\\pi \\in \\mathbb{S}_n$ is proportional to $q^{\\textrm{inv}(\\pi)}$ where $0<q\\le 1$ and $\\textrm{inv}(\\pi)$ is the number of inversions in $\\pi$.\n  We show that the expected length of the cycle containing a given point is of order $\\min\\{(1-q)^{-2}, n\\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2} \\ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity wit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06991","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"}