{"paper":{"title":"How fast does a random walk cover a torus?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Peter Grassberger","submitted_at":"2017-04-17T17:22:27Z","abstract_excerpt":"We present high statistics simulation data for the average time $\\langle T_{\\rm cover}(L)\\rangle$ that a random walk needs to cover completely a 2-dimensional torus of size $L\\times L$. They confirm the mathematical prediction that $\\langle T_{\\rm cover}(L)\\rangle \\sim (L \\ln L)^2$ for large $L$, but the prefactor {\\it seems} to deviate significantly from the supposedly exact result $4/\\pi$ derived by A. Dembo {\\it et al.}, Ann. Math. {\\bf 160}, 433 (2004), if the most straightforward extrapolation is used. On the other hand, we find that this scaling does hold for the time $ T_{\\rm N(t)=1}(L)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.05039","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"}