{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:7AXUVZV6HKX4NVB6ED2TKF2S24","short_pith_number":"pith:7AXUVZV6","canonical_record":{"source":{"id":"2409.16273","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-24T17:45:22Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"ddc2c8c28a09b2ae0d50a9e5e2985e2ee57d269e096fd7de9933bd8fcae5b2df","abstract_canon_sha256":"5929aec6832af002381d7e9eb36643480a52d7dd3cc821bbe359d62b6d45bf78"},"schema_version":"1.0"},"canonical_sha256":"f82f4ae6be3aafc6d43e20f5351752d717e2e7c6dee9cbab8a9acdd236731635","source":{"kind":"arxiv","id":"2409.16273","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.16273","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"arxiv_version","alias_value":"2409.16273v2","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.16273","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"pith_short_12","alias_value":"7AXUVZV6HKX4","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"pith_short_16","alias_value":"7AXUVZV6HKX4NVB6","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"pith_short_8","alias_value":"7AXUVZV6","created_at":"2026-05-28T02:04:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:7AXUVZV6HKX4NVB6ED2TKF2S24","target":"record","payload":{"canonical_record":{"source":{"id":"2409.16273","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-24T17:45:22Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"ddc2c8c28a09b2ae0d50a9e5e2985e2ee57d269e096fd7de9933bd8fcae5b2df","abstract_canon_sha256":"5929aec6832af002381d7e9eb36643480a52d7dd3cc821bbe359d62b6d45bf78"},"schema_version":"1.0"},"canonical_sha256":"f82f4ae6be3aafc6d43e20f5351752d717e2e7c6dee9cbab8a9acdd236731635","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T02:04:39.159723Z","signature_b64":"xYAISx1FglBDLp+CHK0cZ155sK5i6A14G1d2zVQWGiHtWprsSs10rlA0vgCwW9ABfl2HJ0V925PG2a7FUflXCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f82f4ae6be3aafc6d43e20f5351752d717e2e7c6dee9cbab8a9acdd236731635","last_reissued_at":"2026-05-28T02:04:39.159078Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T02:04:39.159078Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2409.16273","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-28T02:04:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x4CMmU/urQe0Vk1S4PX043u6tHX+B5DRdFBsor5PmuoeKk8pmiXBfs5OmJ9Yzii+k/vd2btZklLzH9GLQUeTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T20:09:58.948943Z"},"content_sha256":"c96bb23f81d81693752567ef4d12536b6764f9530a1e13dca3cdd6ecfeba3aa0","schema_version":"1.0","event_id":"sha256:c96bb23f81d81693752567ef4d12536b6764f9530a1e13dca3cdd6ecfeba3aa0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:7AXUVZV6HKX4NVB6ED2TKF2S24","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Percolation of discrete GFF in dimension two II. Connectivity properties of two-sided level sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Pierre Nolin, Wei Qian, Yifan Gao","submitted_at":"2024-09-24T17:45:22Z","abstract_excerpt":"We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in 2D. For a DGFF $\\varphi$ defined in a box $B_N$ with side length $N$, for $C$ large enough, there exist low crossings in the set of vertices $z$ where $|\\varphi(z)|\\le C \\sqrt{\\log \\log N}$, with probability tending to $1$ as $N \\to \\infty$, while the average and the maximum of $\\varphi$ are of order $\\sqrt{\\log N}$ and $\\log N$, respectively. As a consequence, we also obtain connectivity properties of the set of thick points of a random walk.\n  We rely on an isomorphism between the DGFF and the random "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.16273","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.16273/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-28T02:04:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MYImTxSlTjedSaOvqtoBj4mbG8AXDe9Fzj3h7QqVmc7GXNVs+/YNtJPLfXO9GsjwXZkfmj8tySIANm1cOzhqCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T20:09:58.949711Z"},"content_sha256":"402f8e9a2007eba86da5c46e50cbe4356347260a2e8dcbae2b1b1f4ef03bb82c","schema_version":"1.0","event_id":"sha256:402f8e9a2007eba86da5c46e50cbe4356347260a2e8dcbae2b1b1f4ef03bb82c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7AXUVZV6HKX4NVB6ED2TKF2S24/bundle.json","state_url":"https://pith.science/pith/7AXUVZV6HKX4NVB6ED2TKF2S24/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7AXUVZV6HKX4NVB6ED2TKF2S24/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T20:09:58Z","links":{"resolver":"https://pith.science/pith/7AXUVZV6HKX4NVB6ED2TKF2S24","bundle":"https://pith.science/pith/7AXUVZV6HKX4NVB6ED2TKF2S24/bundle.json","state":"https://pith.science/pith/7AXUVZV6HKX4NVB6ED2TKF2S24/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7AXUVZV6HKX4NVB6ED2TKF2S24/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:7AXUVZV6HKX4NVB6ED2TKF2S24","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":"5929aec6832af002381d7e9eb36643480a52d7dd3cc821bbe359d62b6d45bf78","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-24T17:45:22Z","title_canon_sha256":"ddc2c8c28a09b2ae0d50a9e5e2985e2ee57d269e096fd7de9933bd8fcae5b2df"},"schema_version":"1.0","source":{"id":"2409.16273","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.16273","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"arxiv_version","alias_value":"2409.16273v2","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.16273","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"pith_short_12","alias_value":"7AXUVZV6HKX4","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"pith_short_16","alias_value":"7AXUVZV6HKX4NVB6","created_at":"2026-05-28T02:04:39Z"},{"alias_kind":"pith_short_8","alias_value":"7AXUVZV6","created_at":"2026-05-28T02:04:39Z"}],"graph_snapshots":[{"event_id":"sha256:402f8e9a2007eba86da5c46e50cbe4356347260a2e8dcbae2b1b1f4ef03bb82c","target":"graph","created_at":"2026-05-28T02:04:39Z","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/2409.16273/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in 2D. For a DGFF $\\varphi$ defined in a box $B_N$ with side length $N$, for $C$ large enough, there exist low crossings in the set of vertices $z$ where $|\\varphi(z)|\\le C \\sqrt{\\log \\log N}$, with probability tending to $1$ as $N \\to \\infty$, while the average and the maximum of $\\varphi$ are of order $\\sqrt{\\log N}$ and $\\log N$, respectively. As a consequence, we also obtain connectivity properties of the set of thick points of a random walk.\n  We rely on an isomorphism between the DGFF and the random ","authors_text":"Pierre Nolin, Wei Qian, Yifan Gao","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-24T17:45:22Z","title":"Percolation of discrete GFF in dimension two II. Connectivity properties of two-sided level sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.16273","kind":"arxiv","version":2},"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:c96bb23f81d81693752567ef4d12536b6764f9530a1e13dca3cdd6ecfeba3aa0","target":"record","created_at":"2026-05-28T02:04:39Z","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":"5929aec6832af002381d7e9eb36643480a52d7dd3cc821bbe359d62b6d45bf78","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-24T17:45:22Z","title_canon_sha256":"ddc2c8c28a09b2ae0d50a9e5e2985e2ee57d269e096fd7de9933bd8fcae5b2df"},"schema_version":"1.0","source":{"id":"2409.16273","kind":"arxiv","version":2}},"canonical_sha256":"f82f4ae6be3aafc6d43e20f5351752d717e2e7c6dee9cbab8a9acdd236731635","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f82f4ae6be3aafc6d43e20f5351752d717e2e7c6dee9cbab8a9acdd236731635","first_computed_at":"2026-05-28T02:04:39.159078Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T02:04:39.159078Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xYAISx1FglBDLp+CHK0cZ155sK5i6A14G1d2zVQWGiHtWprsSs10rlA0vgCwW9ABfl2HJ0V925PG2a7FUflXCA==","signature_status":"signed_v1","signed_at":"2026-05-28T02:04:39.159723Z","signed_message":"canonical_sha256_bytes"},"source_id":"2409.16273","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c96bb23f81d81693752567ef4d12536b6764f9530a1e13dca3cdd6ecfeba3aa0","sha256:402f8e9a2007eba86da5c46e50cbe4356347260a2e8dcbae2b1b1f4ef03bb82c"],"state_sha256":"caceddad732833c0f411bb68e0f255eb997542d1d48c0861852d41deb75bf2a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fhHxe29FgUJoHx0QGrBEsodGYpUt+EnOOHm82RV8TqAxDZYf2/DEJYpQ6ntzlnQWANf8ltMLnBIfEV3SMxKyAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T20:09:58.953463Z","bundle_sha256":"dca075eeeeedc40fa0ad11acd8b60ff7bda0d4f2beaedb4619247c058ce8321e"}}