{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:ZV5D6QESOYCZWGSR6YFF5VTZH5","short_pith_number":"pith:ZV5D6QES","schema_version":"1.0","canonical_sha256":"cd7a3f409276059b1a51f60a5ed6793f4077874bb43e533ef4f33495be7584cd","source":{"kind":"arxiv","id":"1202.5172","version":2},"attestation_state":"computed","paper":{"title":"Phase transition and level-set percolation for the Gaussian free field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Alain-Sol Sznitman, Pierre-Fran\\c{c}ois Rodriguez","submitted_at":"2012-02-23T13:17:14Z","abstract_excerpt":"We consider level-set percolation for the Gaussian free field on Z^d, with d bigger or equal to 3, and prove that there is a non-trivial critical level h_* such that for h > h_*, the excursion set above level h does not percolate, and for h < h_*, the excursion set does percolate. It is known from the work of Bricmont-Lebowitz-Maes that h_* is non-negative for all d bigger or equal to 3, and finite, when d=3. We prove here that h_* is finite for all d bigger or equal to 3. In fact, we introduce a second critical parameter h_**, which is bigger or equal to h_*. We show that h_** is finite for a"},"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":"1202.5172","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-23T13:17:14Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"8a32aca7b7395646cb698d42d86342a0357e467fc64ac64b79484ff7171d6b99","abstract_canon_sha256":"dd9259eb50b8c6a1ca61c59b3e5e8d1ed41f19656d5747cde93af7bc99cdcef3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:57.497355Z","signature_b64":"IC206NjXybiOEyxva0ffCgA6Hi3r7/u1HJvK76BLprvw3/BqHgiiKkuWaFBE/tBvoalgOjxuvvlYipPZFVZ2Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd7a3f409276059b1a51f60a5ed6793f4077874bb43e533ef4f33495be7584cd","last_reissued_at":"2026-05-18T03:17:57.496702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:57.496702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Phase transition and level-set percolation for the Gaussian free field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Alain-Sol Sznitman, Pierre-Fran\\c{c}ois Rodriguez","submitted_at":"2012-02-23T13:17:14Z","abstract_excerpt":"We consider level-set percolation for the Gaussian free field on Z^d, with d bigger or equal to 3, and prove that there is a non-trivial critical level h_* such that for h > h_*, the excursion set above level h does not percolate, and for h < h_*, the excursion set does percolate. It is known from the work of Bricmont-Lebowitz-Maes that h_* is non-negative for all d bigger or equal to 3, and finite, when d=3. We prove here that h_* is finite for all d bigger or equal to 3. In fact, we introduce a second critical parameter h_**, which is bigger or equal to h_*. We show that h_** is finite for a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5172","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":"1202.5172","created_at":"2026-05-18T03:17:57.496786+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.5172v2","created_at":"2026-05-18T03:17:57.496786+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.5172","created_at":"2026-05-18T03:17:57.496786+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZV5D6QESOYCZ","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZV5D6QESOYCZWGSR","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZV5D6QES","created_at":"2026-05-18T12:27:30.460161+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/ZV5D6QESOYCZWGSR6YFF5VTZH5","json":"https://pith.science/pith/ZV5D6QESOYCZWGSR6YFF5VTZH5.json","graph_json":"https://pith.science/api/pith-number/ZV5D6QESOYCZWGSR6YFF5VTZH5/graph.json","events_json":"https://pith.science/api/pith-number/ZV5D6QESOYCZWGSR6YFF5VTZH5/events.json","paper":"https://pith.science/paper/ZV5D6QES"},"agent_actions":{"view_html":"https://pith.science/pith/ZV5D6QESOYCZWGSR6YFF5VTZH5","download_json":"https://pith.science/pith/ZV5D6QESOYCZWGSR6YFF5VTZH5.json","view_paper":"https://pith.science/paper/ZV5D6QES","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.5172&json=true","fetch_graph":"https://pith.science/api/pith-number/ZV5D6QESOYCZWGSR6YFF5VTZH5/graph.json","fetch_events":"https://pith.science/api/pith-number/ZV5D6QESOYCZWGSR6YFF5VTZH5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZV5D6QESOYCZWGSR6YFF5VTZH5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZV5D6QESOYCZWGSR6YFF5VTZH5/action/storage_attestation","attest_author":"https://pith.science/pith/ZV5D6QESOYCZWGSR6YFF5VTZH5/action/author_attestation","sign_citation":"https://pith.science/pith/ZV5D6QESOYCZWGSR6YFF5VTZH5/action/citation_signature","submit_replication":"https://pith.science/pith/ZV5D6QESOYCZWGSR6YFF5VTZH5/action/replication_record"}},"created_at":"2026-05-18T03:17:57.496786+00:00","updated_at":"2026-05-18T03:17:57.496786+00:00"}