{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:HZIIKPXTZ443MLZLKU66AXGU3H","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":"fc4a769f24dcb90b336b1f3ec5dba49eaea6fc78d422b6315adabdec97c161c9","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-04-19T15:45:50Z","title_canon_sha256":"349707f501cbfe07fb4776098285059b65d625247954ecb9fea176a65ae082b3"},"schema_version":"1.0","source":{"id":"0704.2560","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0704.2560","created_at":"2026-07-04T17:30:03Z"},{"alias_kind":"arxiv_version","alias_value":"0704.2560v3","created_at":"2026-07-04T17:30:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.2560","created_at":"2026-07-04T17:30:03Z"},{"alias_kind":"pith_short_12","alias_value":"HZIIKPXTZ443","created_at":"2026-07-04T17:30:03Z"},{"alias_kind":"pith_short_16","alias_value":"HZIIKPXTZ443MLZL","created_at":"2026-07-04T17:30:03Z"},{"alias_kind":"pith_short_8","alias_value":"HZIIKPXT","created_at":"2026-07-04T17:30:03Z"}],"graph_snapshots":[{"event_id":"sha256:16215e3499989818836d2e9e74e664a13f234cc829c5a9f3258808a871efd8e6","target":"graph","created_at":"2026-07-04T17:30:03Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0704.2560/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder with base a d-1-dimensional discrete torus of side-length N, or the set of points visited by simple random walk on the d-dimensional discrete torus of side-length N by times of order uN^d. We study the percolative properties of the ","authors_text":"Alain-Sol Sznitman","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-04-19T15:45:50Z","title":"Vacant Set of Random Interlacements and Percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.2560","kind":"arxiv","version":3},"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":"sha256:188b1121cb503804b42733c5c4a4bd7b598d689ab7deff1192234e85989dca31","target":"record","created_at":"2026-07-04T17:30:03Z","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":"fc4a769f24dcb90b336b1f3ec5dba49eaea6fc78d422b6315adabdec97c161c9","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-04-19T15:45:50Z","title_canon_sha256":"349707f501cbfe07fb4776098285059b65d625247954ecb9fea176a65ae082b3"},"schema_version":"1.0","source":{"id":"0704.2560","kind":"arxiv","version":3}},"canonical_sha256":"3e50853ef3cf39b62f2b553de05cd4d9f011a67875629f926aeb6c81243d6a1c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e50853ef3cf39b62f2b553de05cd4d9f011a67875629f926aeb6c81243d6a1c","first_computed_at":"2026-07-04T17:30:03.781281Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T17:30:03.781281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r/VOxAe4c7FoFBNQzk8XGPyEOlGQWCAKtCnEShL5lu9ET+c7YI/6amwH/Efd8F/h9dbcshHWFxzD9GHr0mrfDA==","signature_status":"signed_v1","signed_at":"2026-07-04T17:30:03.781751Z","signed_message":"canonical_sha256_bytes"},"source_id":"0704.2560","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:188b1121cb503804b42733c5c4a4bd7b598d689ab7deff1192234e85989dca31","sha256:16215e3499989818836d2e9e74e664a13f234cc829c5a9f3258808a871efd8e6"],"state_sha256":"281ba502455dc46673b5d6863cbc1d71cbeecafe4eab50745e951d4e5288466c"}