{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:MHDB3BMJYDTXEQROYR7ARRZOXU","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":"61f11e43cc25ee564994a057f618296c40cc9f555b267323c6a3b2f0f2c98990","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-03-01T13:01:23Z","title_canon_sha256":"77ad627657f6e5d3d0383866412f7ada3fa66eb5821614c75a1dfa79fd663689"},"schema_version":"1.0","source":{"id":"1003.0334","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.0334","created_at":"2026-05-18T04:14:27Z"},{"alias_kind":"arxiv_version","alias_value":"1003.0334v1","created_at":"2026-05-18T04:14:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.0334","created_at":"2026-05-18T04:14:27Z"},{"alias_kind":"pith_short_12","alias_value":"MHDB3BMJYDTX","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MHDB3BMJYDTXEQRO","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MHDB3BMJ","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:ea62c39b2df147d0831284d1205dc087c11f5a0753ab1cad76977b8a6c59d499","target":"graph","created_at":"2026-05-18T04:14:27Z","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 investigate the percolative properties of the vacant set left by random interlacements on Z^d, when d is large. A non-negative parameter u controls the density of random interlacements on Z^d. It is known from arXiv:0704.2560, and arXiv:0808.3344, that there is a non-degenerate critical value u_*, such that the vacant set at level u percolates when u < u_*, and does not percolate when u > u_*. Little is known about u_*, however for large d, random interlacements on Z^d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, for which the corresponding critical parame","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":"2010-03-01T13:01:23Z","title":"A lower bound on the critical parameter of interlacement percolation in high dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.0334","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:ee077c2af199bacb26139a21ce25639a45de94278aa581ef7cfc5aedbbf6afba","target":"record","created_at":"2026-05-18T04:14:27Z","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":"61f11e43cc25ee564994a057f618296c40cc9f555b267323c6a3b2f0f2c98990","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-03-01T13:01:23Z","title_canon_sha256":"77ad627657f6e5d3d0383866412f7ada3fa66eb5821614c75a1dfa79fd663689"},"schema_version":"1.0","source":{"id":"1003.0334","kind":"arxiv","version":1}},"canonical_sha256":"61c61d8589c0e772422ec47e08c72ebd0cdf30b7fc814d6cf10665d37cd585e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61c61d8589c0e772422ec47e08c72ebd0cdf30b7fc814d6cf10665d37cd585e3","first_computed_at":"2026-05-18T04:14:27.631775Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:27.631775Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NQrORm2Wq9O0P5Vyif/HoIzPd7UvmuEeTubY0rH/oGKSmYD7eY7zj3IvHM3hbV4qOaGydWqlcdNxYRNZoxg2Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:27.632280Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.0334","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee077c2af199bacb26139a21ce25639a45de94278aa581ef7cfc5aedbbf6afba","sha256:ea62c39b2df147d0831284d1205dc087c11f5a0753ab1cad76977b8a6c59d499"],"state_sha256":"6e4d7a3d1fe7e63d78bdf7c3a12f6531a8be35cc17fa621be3e027ab2ec8993e"}