{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:CE34G3VWVRY724YETAV7HLSQTZ","short_pith_number":"pith:CE34G3VW","schema_version":"1.0","canonical_sha256":"1137c36eb6ac71fd7304982bf3ae509e60c12e7fddaeb55b244c4644322d37d2","source":{"kind":"arxiv","id":"2508.19814","version":1},"attestation_state":"computed","paper":{"title":"Collisions of random walks on comb graphs with a planar base","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Carlo Scali, Maximilian Nitzschner, Umberto De Ambroggio","submitted_at":"2025-08-27T11:57:51Z","abstract_excerpt":"In this article we study collisions of two independent random walks on comb graphs $\\mathrm{Comb}(\\tilde{G},f)$ for a large class of recurrent planar graphs $\\tilde{G}$ and profile functions $f$, the latter governing the length of vertical segments (called \"teeth\") attached to vertices of the base graph $\\tilde{G}$. We prove that the number of collisions of two random walks starting from the same site undergoes a phase transition depending on the growth of $f$. As a benchmark example, we show that for $\\mathrm{Comb}(\\mathbb{Z}^2,f_\\gamma)$ with $f_\\gamma(z) = \\log^{\\gamma}(\\|z\\|_\\infty\\vee 1)$"},"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":"2508.19814","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-08-27T11:57:51Z","cross_cats_sorted":[],"title_canon_sha256":"b39d175ab6aba40d3369c43898d97350827aed719bd5f973e04b1441122ae7c6","abstract_canon_sha256":"2a9af12318079c2c67619ed60d34ec3fe57f94767040368d2721444be85fa8f8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:00:18.165788Z","signature_b64":"cmT9QykCExIH2dobcpPWpRnawmRMtq+jEzzKzhxpQqzsXL0zdpQ5AvWtXDTgbCDYnKzaFd+Ke3W2hHUvfdh4DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1137c36eb6ac71fd7304982bf3ae509e60c12e7fddaeb55b244c4644322d37d2","last_reissued_at":"2026-07-05T12:00:18.165350Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:00:18.165350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Collisions of random walks on comb graphs with a planar base","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Carlo Scali, Maximilian Nitzschner, Umberto De Ambroggio","submitted_at":"2025-08-27T11:57:51Z","abstract_excerpt":"In this article we study collisions of two independent random walks on comb graphs $\\mathrm{Comb}(\\tilde{G},f)$ for a large class of recurrent planar graphs $\\tilde{G}$ and profile functions $f$, the latter governing the length of vertical segments (called \"teeth\") attached to vertices of the base graph $\\tilde{G}$. We prove that the number of collisions of two random walks starting from the same site undergoes a phase transition depending on the growth of $f$. As a benchmark example, we show that for $\\mathrm{Comb}(\\mathbb{Z}^2,f_\\gamma)$ with $f_\\gamma(z) = \\log^{\\gamma}(\\|z\\|_\\infty\\vee 1)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.19814","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/2508.19814/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":"2508.19814","created_at":"2026-07-05T12:00:18.165411+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.19814v1","created_at":"2026-07-05T12:00:18.165411+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.19814","created_at":"2026-07-05T12:00:18.165411+00:00"},{"alias_kind":"pith_short_12","alias_value":"CE34G3VWVRY7","created_at":"2026-07-05T12:00:18.165411+00:00"},{"alias_kind":"pith_short_16","alias_value":"CE34G3VWVRY724YE","created_at":"2026-07-05T12:00:18.165411+00:00"},{"alias_kind":"pith_short_8","alias_value":"CE34G3VW","created_at":"2026-07-05T12:00:18.165411+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/CE34G3VWVRY724YETAV7HLSQTZ","json":"https://pith.science/pith/CE34G3VWVRY724YETAV7HLSQTZ.json","graph_json":"https://pith.science/api/pith-number/CE34G3VWVRY724YETAV7HLSQTZ/graph.json","events_json":"https://pith.science/api/pith-number/CE34G3VWVRY724YETAV7HLSQTZ/events.json","paper":"https://pith.science/paper/CE34G3VW"},"agent_actions":{"view_html":"https://pith.science/pith/CE34G3VWVRY724YETAV7HLSQTZ","download_json":"https://pith.science/pith/CE34G3VWVRY724YETAV7HLSQTZ.json","view_paper":"https://pith.science/paper/CE34G3VW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.19814&json=true","fetch_graph":"https://pith.science/api/pith-number/CE34G3VWVRY724YETAV7HLSQTZ/graph.json","fetch_events":"https://pith.science/api/pith-number/CE34G3VWVRY724YETAV7HLSQTZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CE34G3VWVRY724YETAV7HLSQTZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CE34G3VWVRY724YETAV7HLSQTZ/action/storage_attestation","attest_author":"https://pith.science/pith/CE34G3VWVRY724YETAV7HLSQTZ/action/author_attestation","sign_citation":"https://pith.science/pith/CE34G3VWVRY724YETAV7HLSQTZ/action/citation_signature","submit_replication":"https://pith.science/pith/CE34G3VWVRY724YETAV7HLSQTZ/action/replication_record"}},"created_at":"2026-07-05T12:00:18.165411+00:00","updated_at":"2026-07-05T12:00:18.165411+00:00"}