{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:KJPKMTQZEF7P4ZWBFG2GF62RFM","short_pith_number":"pith:KJPKMTQZ","schema_version":"1.0","canonical_sha256":"525ea64e19217efe66c129b462fb512b07809c448f306a7fb0c2a52d0f7a7538","source":{"kind":"arxiv","id":"1405.1702","version":2},"attestation_state":"computed","paper":{"title":"A note on the vacant set of random walks on the hypercube and other regular graphs of high degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Frieze, Colin Cooper","submitted_at":"2014-05-07T18:56:47Z","abstract_excerpt":"We consider a random walk on a $d$-regular graph $G$ where $d\\to\\infty$ and $G$ satisfies certain conditions. Our prime example is the $d$-dimensional hypercube, which has $n=2^d$ vertices. We explore the likely component structure of the vacant set, i.e. the set of unvisited vertices. Let $\\Lambda(t)$ be the subgraph induced by the vacant set of the walk at step $t$. We show that if certain conditions are satisfied then the graph $\\Lambda(t)$ undergoes a phase transition at around $t^*=n\\log_ed$. Our results are that if $t\\leq(1-\\epsilon)t^*$ then w.h.p. as the number vertices $n\\to\\infty$, t"},"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":"1405.1702","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-07T18:56:47Z","cross_cats_sorted":[],"title_canon_sha256":"bd8e62addcf138fbeabd9f83ffbec7641f42cc74ce43a4aaf8cea4b97a884c40","abstract_canon_sha256":"2540f0b3ed31128af862e7ea30090aae40e44493b4743559197c0e38378067b6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:53.939430Z","signature_b64":"pNFLeY1M+aJLzfLwN9nSYdhm5uWdAHksmuKcH+z06NcLPK8jkbBkEwg78sD27ZdmuVQZZX2PgIINRLQ+hH5HAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"525ea64e19217efe66c129b462fb512b07809c448f306a7fb0c2a52d0f7a7538","last_reissued_at":"2026-05-18T02:40:53.938863Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:53.938863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the vacant set of random walks on the hypercube and other regular graphs of high degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Frieze, Colin Cooper","submitted_at":"2014-05-07T18:56:47Z","abstract_excerpt":"We consider a random walk on a $d$-regular graph $G$ where $d\\to\\infty$ and $G$ satisfies certain conditions. Our prime example is the $d$-dimensional hypercube, which has $n=2^d$ vertices. We explore the likely component structure of the vacant set, i.e. the set of unvisited vertices. Let $\\Lambda(t)$ be the subgraph induced by the vacant set of the walk at step $t$. We show that if certain conditions are satisfied then the graph $\\Lambda(t)$ undergoes a phase transition at around $t^*=n\\log_ed$. Our results are that if $t\\leq(1-\\epsilon)t^*$ then w.h.p. as the number vertices $n\\to\\infty$, t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1702","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1405.1702","created_at":"2026-05-18T02:40:53.938942+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.1702v2","created_at":"2026-05-18T02:40:53.938942+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1702","created_at":"2026-05-18T02:40:53.938942+00:00"},{"alias_kind":"pith_short_12","alias_value":"KJPKMTQZEF7P","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"KJPKMTQZEF7P4ZWB","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"KJPKMTQZ","created_at":"2026-05-18T12:28:35.611951+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/KJPKMTQZEF7P4ZWBFG2GF62RFM","json":"https://pith.science/pith/KJPKMTQZEF7P4ZWBFG2GF62RFM.json","graph_json":"https://pith.science/api/pith-number/KJPKMTQZEF7P4ZWBFG2GF62RFM/graph.json","events_json":"https://pith.science/api/pith-number/KJPKMTQZEF7P4ZWBFG2GF62RFM/events.json","paper":"https://pith.science/paper/KJPKMTQZ"},"agent_actions":{"view_html":"https://pith.science/pith/KJPKMTQZEF7P4ZWBFG2GF62RFM","download_json":"https://pith.science/pith/KJPKMTQZEF7P4ZWBFG2GF62RFM.json","view_paper":"https://pith.science/paper/KJPKMTQZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.1702&json=true","fetch_graph":"https://pith.science/api/pith-number/KJPKMTQZEF7P4ZWBFG2GF62RFM/graph.json","fetch_events":"https://pith.science/api/pith-number/KJPKMTQZEF7P4ZWBFG2GF62RFM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KJPKMTQZEF7P4ZWBFG2GF62RFM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KJPKMTQZEF7P4ZWBFG2GF62RFM/action/storage_attestation","attest_author":"https://pith.science/pith/KJPKMTQZEF7P4ZWBFG2GF62RFM/action/author_attestation","sign_citation":"https://pith.science/pith/KJPKMTQZEF7P4ZWBFG2GF62RFM/action/citation_signature","submit_replication":"https://pith.science/pith/KJPKMTQZEF7P4ZWBFG2GF62RFM/action/replication_record"}},"created_at":"2026-05-18T02:40:53.938942+00:00","updated_at":"2026-05-18T02:40:53.938942+00:00"}