{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:FPO4DBED3RXJGLOHC76NZK7TL4","short_pith_number":"pith:FPO4DBED","canonical_record":{"source":{"id":"0902.3842","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-23T03:34:33Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d516c74719e3d8dbc04de791c66f985a2f383f35e203b94e218e9c7fd28a89ce","abstract_canon_sha256":"906143bdc538bef9b3c097e3fe41e29b57e0984136c674cf3d0f863ee97202a0"},"schema_version":"1.0"},"canonical_sha256":"2bddc18483dc6e932dc717fcdcabf35f26e0a1ff48b23a627aea941dbaf32ebf","source":{"kind":"arxiv","id":"0902.3842","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0902.3842","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"arxiv_version","alias_value":"0902.3842v3","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.3842","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"pith_short_12","alias_value":"FPO4DBED3RXJ","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FPO4DBED3RXJGLOH","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FPO4DBED","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:FPO4DBED3RXJGLOHC76NZK7TL4","target":"record","payload":{"canonical_record":{"source":{"id":"0902.3842","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-23T03:34:33Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d516c74719e3d8dbc04de791c66f985a2f383f35e203b94e218e9c7fd28a89ce","abstract_canon_sha256":"906143bdc538bef9b3c097e3fe41e29b57e0984136c674cf3d0f863ee97202a0"},"schema_version":"1.0"},"canonical_sha256":"2bddc18483dc6e932dc717fcdcabf35f26e0a1ff48b23a627aea941dbaf32ebf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:59.641375Z","signature_b64":"fZMclulp43id/gKM3NyIewGQrXLaxuSKOI8ziTXMZ8YYgEd0ZbnVx/dSZoUOpOUpyii1/V7DVbklqmhsxQkKAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2bddc18483dc6e932dc717fcdcabf35f26e0a1ff48b23a627aea941dbaf32ebf","last_reissued_at":"2026-05-18T04:39:59.640739Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:59.640739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0902.3842","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:39:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"frWOn7hQs1Ax7HP53YpvPjDOJID8kHnYBPOP/KUwKFFY3eEE7qw2ruVRjT+QMPr4CIMqrgletT6wXX+uEelYDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T20:34:18.963275Z"},"content_sha256":"afdba225d7988453a12ad20b87c65600906ba0e0a9cd181331ba696a378e54ad","schema_version":"1.0","event_id":"sha256:afdba225d7988453a12ad20b87c65600906ba0e0a9cd181331ba696a378e54ad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:FPO4DBED3RXJGLOHC76NZK7TL4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Thick points of 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":"Jason Miller, Xiaoyu Hu, Yuval Peres","submitted_at":"2009-02-23T03:34:33Z","abstract_excerpt":"Let $U\\subseteq\\mathbf{C}$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\\int_U\\nabla f(x)\\cdot \\nabla g(x)\\,dx$. The set $T(a;U)$ of $a$-thick points of $F$ consists of those $z\\in U$ such that the average of $F$ on a disk of radius $r$ centered at $z$ has growth $\\sqrt{a/\\pi}\\log\\frac{1}{r}$ as $r\\to 0$. We show that for each $0\\leq a\\leq2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$, that $\\nu_{2-a}(T(a;U))=\\infty$ when $0<a\\leq2$ and $\\nu_2(T(0;U))=\\nu_2(U)$ almost "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.3842","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:39:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TWOvBV7N4lk8ljnmXKAHOhQ29z1mQUAOefJzb+036XlodR6f8p8TBiZ7VPWKBjwP4Vv2et0x6oWB4wgVr1wCCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T20:34:18.963735Z"},"content_sha256":"35d2ee231b5d41b6cf5bcf7a5b19e18966e4a24821126f23e96e14f72d075b32","schema_version":"1.0","event_id":"sha256:35d2ee231b5d41b6cf5bcf7a5b19e18966e4a24821126f23e96e14f72d075b32"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FPO4DBED3RXJGLOHC76NZK7TL4/bundle.json","state_url":"https://pith.science/pith/FPO4DBED3RXJGLOHC76NZK7TL4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FPO4DBED3RXJGLOHC76NZK7TL4/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":"2026-08-16T20:34:18Z","links":{"resolver":"https://pith.science/pith/FPO4DBED3RXJGLOHC76NZK7TL4","bundle":"https://pith.science/pith/FPO4DBED3RXJGLOHC76NZK7TL4/bundle.json","state":"https://pith.science/pith/FPO4DBED3RXJGLOHC76NZK7TL4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FPO4DBED3RXJGLOHC76NZK7TL4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:FPO4DBED3RXJGLOHC76NZK7TL4","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":"906143bdc538bef9b3c097e3fe41e29b57e0984136c674cf3d0f863ee97202a0","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-23T03:34:33Z","title_canon_sha256":"d516c74719e3d8dbc04de791c66f985a2f383f35e203b94e218e9c7fd28a89ce"},"schema_version":"1.0","source":{"id":"0902.3842","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0902.3842","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"arxiv_version","alias_value":"0902.3842v3","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.3842","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"pith_short_12","alias_value":"FPO4DBED3RXJ","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FPO4DBED3RXJGLOH","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FPO4DBED","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:35d2ee231b5d41b6cf5bcf7a5b19e18966e4a24821126f23e96e14f72d075b32","target":"graph","created_at":"2026-05-18T04:39:59Z","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":"Let $U\\subseteq\\mathbf{C}$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\\int_U\\nabla f(x)\\cdot \\nabla g(x)\\,dx$. The set $T(a;U)$ of $a$-thick points of $F$ consists of those $z\\in U$ such that the average of $F$ on a disk of radius $r$ centered at $z$ has growth $\\sqrt{a/\\pi}\\log\\frac{1}{r}$ as $r\\to 0$. We show that for each $0\\leq a\\leq2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$, that $\\nu_{2-a}(T(a;U))=\\infty$ when $0<a\\leq2$ and $\\nu_2(T(0;U))=\\nu_2(U)$ almost ","authors_text":"Jason Miller, Xiaoyu Hu, Yuval Peres","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-23T03:34:33Z","title":"Thick points of the Gaussian free field"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.3842","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:afdba225d7988453a12ad20b87c65600906ba0e0a9cd181331ba696a378e54ad","target":"record","created_at":"2026-05-18T04:39:59Z","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":"906143bdc538bef9b3c097e3fe41e29b57e0984136c674cf3d0f863ee97202a0","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-23T03:34:33Z","title_canon_sha256":"d516c74719e3d8dbc04de791c66f985a2f383f35e203b94e218e9c7fd28a89ce"},"schema_version":"1.0","source":{"id":"0902.3842","kind":"arxiv","version":3}},"canonical_sha256":"2bddc18483dc6e932dc717fcdcabf35f26e0a1ff48b23a627aea941dbaf32ebf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bddc18483dc6e932dc717fcdcabf35f26e0a1ff48b23a627aea941dbaf32ebf","first_computed_at":"2026-05-18T04:39:59.640739Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:59.640739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fZMclulp43id/gKM3NyIewGQrXLaxuSKOI8ziTXMZ8YYgEd0ZbnVx/dSZoUOpOUpyii1/V7DVbklqmhsxQkKAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:59.641375Z","signed_message":"canonical_sha256_bytes"},"source_id":"0902.3842","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:afdba225d7988453a12ad20b87c65600906ba0e0a9cd181331ba696a378e54ad","sha256:35d2ee231b5d41b6cf5bcf7a5b19e18966e4a24821126f23e96e14f72d075b32"],"state_sha256":"504af669d7ab697994ce051e1ed394f0b556ae3f2bc1d6d32a5bc8b3c11c29b4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K4DadI7aU6OVypgHjsxLk4pQPU5hgdyuLHKOTTQrVaW9T0ZmG7TQ8A3VDF/vDWJgBoNGj7C+EKbKeuQZZbPfCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T20:34:18.967415Z","bundle_sha256":"2f34981a32af684daf4a1925cd1103d4162f2ec8587948660b4f96532cad9138"}}