{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:J3SAW3LTCP6KCTZFBBVQQGOAYC","short_pith_number":"pith:J3SAW3LT","schema_version":"1.0","canonical_sha256":"4ee40b6d7313fca14f25086b0819c0c09dd3b915bc21f24c5d1fddd2d31a95b4","source":{"kind":"arxiv","id":"1007.0902","version":1},"attestation_state":"computed","paper":{"title":"On the fragmentation of a torus by random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Augusto Teixeira, David Windisch","submitted_at":"2010-07-06T14:20:52Z","abstract_excerpt":"We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the random walk in its first [uN^d] steps. We prove the existence of two distinct phases of the vacant set in the following sense: if u > 0 is chosen large enough, all components of the vacant set contain no more than a power of log(N) vertices with high probability as N tends to infinity. On the other hand, for small u > 0, there exists a macroscopic"},"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":"1007.0902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-06T14:20:52Z","cross_cats_sorted":[],"title_canon_sha256":"335b53de4e19253bb114b4e134f78cb40fafe53b4050154d131eb6e37edf15fd","abstract_canon_sha256":"c4c962fc963977b1458ae46a2530517b4062217511b829d340dab1647b9f299c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:59.275715Z","signature_b64":"zyUjyXgSe6CIN/a501dF5BrxCoC0+RUhKeg28Eq5eVlsNXnN203V7Rs4mUuUW2JfxKJGkIzWzFckMpdNLzGxCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ee40b6d7313fca14f25086b0819c0c09dd3b915bc21f24c5d1fddd2d31a95b4","last_reissued_at":"2026-05-18T03:16:59.275025Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:59.275025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the fragmentation of a torus by random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Augusto Teixeira, David Windisch","submitted_at":"2010-07-06T14:20:52Z","abstract_excerpt":"We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the random walk in its first [uN^d] steps. We prove the existence of two distinct phases of the vacant set in the following sense: if u > 0 is chosen large enough, all components of the vacant set contain no more than a power of log(N) vertices with high probability as N tends to infinity. On the other hand, for small u > 0, there exists a macroscopic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0902","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":""},"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":"1007.0902","created_at":"2026-05-18T03:16:59.275130+00:00"},{"alias_kind":"arxiv_version","alias_value":"1007.0902v1","created_at":"2026-05-18T03:16:59.275130+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.0902","created_at":"2026-05-18T03:16:59.275130+00:00"},{"alias_kind":"pith_short_12","alias_value":"J3SAW3LTCP6K","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_16","alias_value":"J3SAW3LTCP6KCTZF","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_8","alias_value":"J3SAW3LT","created_at":"2026-05-18T12:26:09.077623+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/J3SAW3LTCP6KCTZFBBVQQGOAYC","json":"https://pith.science/pith/J3SAW3LTCP6KCTZFBBVQQGOAYC.json","graph_json":"https://pith.science/api/pith-number/J3SAW3LTCP6KCTZFBBVQQGOAYC/graph.json","events_json":"https://pith.science/api/pith-number/J3SAW3LTCP6KCTZFBBVQQGOAYC/events.json","paper":"https://pith.science/paper/J3SAW3LT"},"agent_actions":{"view_html":"https://pith.science/pith/J3SAW3LTCP6KCTZFBBVQQGOAYC","download_json":"https://pith.science/pith/J3SAW3LTCP6KCTZFBBVQQGOAYC.json","view_paper":"https://pith.science/paper/J3SAW3LT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1007.0902&json=true","fetch_graph":"https://pith.science/api/pith-number/J3SAW3LTCP6KCTZFBBVQQGOAYC/graph.json","fetch_events":"https://pith.science/api/pith-number/J3SAW3LTCP6KCTZFBBVQQGOAYC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J3SAW3LTCP6KCTZFBBVQQGOAYC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J3SAW3LTCP6KCTZFBBVQQGOAYC/action/storage_attestation","attest_author":"https://pith.science/pith/J3SAW3LTCP6KCTZFBBVQQGOAYC/action/author_attestation","sign_citation":"https://pith.science/pith/J3SAW3LTCP6KCTZFBBVQQGOAYC/action/citation_signature","submit_replication":"https://pith.science/pith/J3SAW3LTCP6KCTZFBBVQQGOAYC/action/replication_record"}},"created_at":"2026-05-18T03:16:59.275130+00:00","updated_at":"2026-05-18T03:16:59.275130+00:00"}