{"paper":{"title":"On the joint behaviour of speed and entropy of random walks on groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.GR","authors_text":"Gideon Amir","submitted_at":"2015-09-01T12:26:28Z","abstract_excerpt":"For every $3/4\\le \\delta, \\beta< 1$ satisfying $\\delta\\leq \\beta < \\frac{1+\\delta}{2}$ we construct a finitely generated group $\\Gamma$ and a (symmetric, finitely supported) random walk $X_n$ on $\\Gamma$ so that its expected distance from its starting point satisfies $E|X_n|\\asymp n^{\\beta}$ and its entropy satisfies $H(X_n)\\asymp n^\\delta$. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions $f,h$ up to a constant factor as long as the functions satisfy the relation $n^{\\frac{3}{4}}\\leq h(n)\\leq f(n)\\leq \\sqrt{{nh(n)}/{\\log (n+1)}}\\leq n^\\gamm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00256","kind":"arxiv","version":1},"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"}