{"paper":{"title":"On groups, slow heat kernel decay yields Liouville property and sharp entropy bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.PR","authors_text":"Tianyi Zheng, Yuval Peres","submitted_at":"2016-09-16T18:57:53Z","abstract_excerpt":"Let $\\mu$ be a symmetric probability measure of finite entropy on a group $G$. We show that if $-\\log \\mu^{(2n)}(id)=o(n^{1/2})$, then the pair $(G,\\mu)$ has the Liouville property (all bounded $\\mu$-harmonic functions on $G$ are constant). Furthermore, if $-\\log \\mu^{(2n)}(id)=O(n^{\\beta})$ where $\\beta\\in(0,1/2)$, then the entropy of the $n$-fold convolution power $\\mu^{(n)}$ satisfies $H(\\mu^{(n)})=O\\left(n^{\\frac{\\beta}{1-\\beta}}\\right)$. This improves earlier results of Gournay and of Saloff-Coste and the second author. We extend the bounds to transitive graphs and illustrate their sharpn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05174","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"}