{"paper":{"title":"How fast does the range of simple random walk grow?","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Itai Benjamini, Justin Salez","submitted_at":"2026-02-11T17:19:58Z","abstract_excerpt":"Consider a discrete-time simple random walk $(X_t)_{t\\ge 0}$ on an infinite, connected, locally finite simple graph $G$, and let \\[ R_t := |\\{X_0,\\ldots,X_t\\}| \\] denote its range. The main result of this revised note is that positive vertex isoperimetry already forces linear expected range, with no bounded-degree assumption: if \\[ \\iota_V(G) := \\inf_{0<|S|<\\infty} \\frac{|\\partial_V S|}{|S|} >0, \\] then $\\E_x R_t \\ge c(G)(t+1)$ for every starting vertex $x$ and every $t\\ge 0$. The proof is direct: vertex expansion implies an unweighted Dirichlet inequality, which in turn gives a uniform positi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.11051","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.11051/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"}