{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:SFXKWMX7EZEVVK44VYCTRLHBWO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"14d28160d9482159dbce5227996b7380f4af1e471d0d83a63a70b73fa62ba8fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-10T22:06:02Z","title_canon_sha256":"bdc2964d2849622ee47acf2e330ec58e6fb6a570981cbba8d070ad5ad5e04448"},"schema_version":"1.0","source":{"id":"1101.1978","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.1978","created_at":"2026-05-18T04:31:50Z"},{"alias_kind":"arxiv_version","alias_value":"1101.1978v1","created_at":"2026-05-18T04:31:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.1978","created_at":"2026-05-18T04:31:50Z"},{"alias_kind":"pith_short_12","alias_value":"SFXKWMX7EZEV","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"SFXKWMX7EZEVVK44","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"SFXKWMX7","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:c885ceac9662f01dbfd197fa8a217da0493b35d1ffeb8bfd19e832377075c90b","target":"graph","created_at":"2026-05-18T04:31:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider the simple random walk on a random d-regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. It was shown in [arXiv:1012.5117] that this so-called vacant set exhibits a phase transition at u = u*: there is a giant component if u < u* and only small components when u > u*. In this paper we show the existence of a critical window of size n^(-1/3) around u*. In this window the size of the largest cluster is of order n^(2/3).","authors_text":"Augusto Teixeira, Jiri Cerny","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-10T22:06:02Z","title":"Critical window for the vacant set left by random walk on random regular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1978","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:69e74413603f954a7ba0a88534169ef34acfe3ddefa7f44e035244ff45ec23fe","target":"record","created_at":"2026-05-18T04:31:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"14d28160d9482159dbce5227996b7380f4af1e471d0d83a63a70b73fa62ba8fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-10T22:06:02Z","title_canon_sha256":"bdc2964d2849622ee47acf2e330ec58e6fb6a570981cbba8d070ad5ad5e04448"},"schema_version":"1.0","source":{"id":"1101.1978","kind":"arxiv","version":1}},"canonical_sha256":"916eab32ff26495aab9cae0538ace1b3b12316f7a41a4bc1f3221bc88c2aec4d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"916eab32ff26495aab9cae0538ace1b3b12316f7a41a4bc1f3221bc88c2aec4d","first_computed_at":"2026-05-18T04:31:50.929545Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:50.929545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VzYHdgg0bpzBbgRmmb0LOvdiwFB/SyExOoe9x+XvtDTVwhYiTbGBYU+NqhrGgByzVM8AylQsJ4bULy5QMXLzCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:50.929960Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.1978","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69e74413603f954a7ba0a88534169ef34acfe3ddefa7f44e035244ff45ec23fe","sha256:c885ceac9662f01dbfd197fa8a217da0493b35d1ffeb8bfd19e832377075c90b"],"state_sha256":"3e44e83bdab3745043ca42512dc2cdf0a6ce13fd816f81ddff870830cf17aa69"}