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We answer a question of Cs\\'aki, Cs\\\"org\\\"o, F\\\"oldes, R\\'ev\\'esz, and Tusn\\'ady by showing that the expected number of vertices visited by a random walk on the comb after n steps is (1/(2\\sqrt{2\\pi})+o(1))\\sqrt n\\log n. This contradicts a claim of Weiss and Havlin."},"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":"1309.6360","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-24T22:33:09Z","cross_cats_sorted":[],"title_canon_sha256":"930b1c6e46cd129cd5529b557b1f1fa49902084cf83df358d47413ccbd83c6ad","abstract_canon_sha256":"50bcc30ff245c143df5ff80987a55aea7223e5c9f6d7b7d0d7ccd1e9e1a8ffe8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:21.466241Z","signature_b64":"Y8kr2CG42Y0Lxg/1r4VzGvg26GQCePwmFReQOmneOEIXA7Z7lB1P9vFMtmVt8NeML+1gV+XMyG3GuwsPle/oCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f10e7c2256213cc2ab21e1a11c7653af44fcf8aed7b2be2082bd3ec408a47f7","last_reissued_at":"2026-05-18T03:12:21.465520Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:21.465520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The range of a random walk on a comb","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"G\\'abor Tardos, J\\'anos Pach","submitted_at":"2013-09-24T22:33:09Z","abstract_excerpt":"The graph obtained from the integer grid Z x Z by the removal of all horizontal edges that do not belong to the x-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex v, in the next step it will visit one of the neighbors of v, each with probability 1/d(v), where d(v) denotes the degree of v. We answer a question of Cs\\'aki, Cs\\\"org\\\"o, F\\\"oldes, R\\'ev\\'esz, and Tusn\\'ady by showing that the expected number of vertices visited by a random walk on the comb after n steps is (1/(2\\sqrt{2\\pi})+o(1))\\sqrt n\\log n. 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