{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:J6YJ6SQI34C44Q6GPDOQAKAI52","short_pith_number":"pith:J6YJ6SQI","schema_version":"1.0","canonical_sha256":"4fb09f4a08df05ce43c678dd002808ee95005fa823f4717252101c262e816845","source":{"kind":"arxiv","id":"2602.11051","version":3},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2602.11051","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-02-11T17:19:58Z","cross_cats_sorted":[],"title_canon_sha256":"7061dff8acef98690b21eb2b897006e4d81fdfa5d90bf55b075cc900363412c6","abstract_canon_sha256":"bca1f8b63ff7dbfd4538438aa3a159ce2f1af0beda298a29a546aa7723b91c58"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:51.295557Z","signature_b64":"Bg3xvdvXAm2mQeOG3JipDfnGx8MhKNnd218GHmOExTNi6Lo+KprR9XhUSiZO42vnq6oUtcSysw56yr4HweD2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4fb09f4a08df05ce43c678dd002808ee95005fa823f4717252101c262e816845","last_reissued_at":"2026-06-02T02:04:51.295069Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:51.295069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2602.11051","created_at":"2026-06-02T02:04:51.295126+00:00"},{"alias_kind":"arxiv_version","alias_value":"2602.11051v3","created_at":"2026-06-02T02:04:51.295126+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.11051","created_at":"2026-06-02T02:04:51.295126+00:00"},{"alias_kind":"pith_short_12","alias_value":"J6YJ6SQI34C4","created_at":"2026-06-02T02:04:51.295126+00:00"},{"alias_kind":"pith_short_16","alias_value":"J6YJ6SQI34C44Q6G","created_at":"2026-06-02T02:04:51.295126+00:00"},{"alias_kind":"pith_short_8","alias_value":"J6YJ6SQI","created_at":"2026-06-02T02:04:51.295126+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52","json":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52.json","graph_json":"https://pith.science/api/pith-number/J6YJ6SQI34C44Q6GPDOQAKAI52/graph.json","events_json":"https://pith.science/api/pith-number/J6YJ6SQI34C44Q6GPDOQAKAI52/events.json","paper":"https://pith.science/paper/J6YJ6SQI"},"agent_actions":{"view_html":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52","download_json":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52.json","view_paper":"https://pith.science/paper/J6YJ6SQI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2602.11051&json=true","fetch_graph":"https://pith.science/api/pith-number/J6YJ6SQI34C44Q6GPDOQAKAI52/graph.json","fetch_events":"https://pith.science/api/pith-number/J6YJ6SQI34C44Q6GPDOQAKAI52/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52/action/storage_attestation","attest_author":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52/action/author_attestation","sign_citation":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52/action/citation_signature","submit_replication":"https://pith.science/pith/J6YJ6SQI34C44Q6GPDOQAKAI52/action/replication_record"}},"created_at":"2026-06-02T02:04:51.295126+00:00","updated_at":"2026-06-02T02:04:51.295126+00:00"}