{"paper":{"title":"Precise cover times for branching random walks on Hamming graphs: (iterated) logarithmic corrections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Zhenyuan Zhang","submitted_at":"2026-07-26T18:27:51Z","abstract_excerpt":"We prove tight asymptotics of the cover time $\\tau_{\\mathrm{cov}}(d)$ of a continuous-time branching random walk on the Hamming graph $\\{0,1,\\dots,b-1\\}^d$, as $d\\to\\infty$. We focus on the slow-branching regime, where particles move at rate one and branch at rate $\\lambda\\in(0,1)$. For $b>2$, we show that $\\tau_{\\mathrm{cov}}(d)=x_\\star d+\\lambda^{-1}\\log d+O_{\\mathbb P}(1)$. For $b=2$, we show that $\\tau_{\\mathrm{cov}}(d)=x_\\star d+\\chi^{-1}\\log\\log d+O_{\\mathbb P}(1)$. Here, $x_\\star$ and $\\chi$ are explicit positive constants depending only on $b$ and $\\lambda$. Our results sharpen previou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23791","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.23791/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}