{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:4UOTPLDCMIFWBBZ7BXA2MQUXRF","short_pith_number":"pith:4UOTPLDC","canonical_record":{"source":{"id":"0803.3750","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-03-26T16:11:30Z","cross_cats_sorted":["math-ph","math.CA","math.MP"],"title_canon_sha256":"e8baeb0cef8cf1a586ae93cd2808814cb9e09a5c864e314d85825aa0a8c6369d","abstract_canon_sha256":"bf084ff547978b3bbbdd5eab98f6bba834a81d7bdb6297228f7c7043def4c45c"},"schema_version":"1.0"},"canonical_sha256":"e51d37ac62620b60873f0dc1a64297896a9127bf9d3efe9566f5e8ac06b94b4c","source":{"kind":"arxiv","id":"0803.3750","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0803.3750","created_at":"2026-05-18T03:34:23Z"},{"alias_kind":"arxiv_version","alias_value":"0803.3750v4","created_at":"2026-05-18T03:34:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0803.3750","created_at":"2026-05-18T03:34:23Z"},{"alias_kind":"pith_short_12","alias_value":"4UOTPLDCMIFW","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"4UOTPLDCMIFWBBZ7","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"4UOTPLDC","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:4UOTPLDCMIFWBBZ7BXA2MQUXRF","target":"record","payload":{"canonical_record":{"source":{"id":"0803.3750","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-03-26T16:11:30Z","cross_cats_sorted":["math-ph","math.CA","math.MP"],"title_canon_sha256":"e8baeb0cef8cf1a586ae93cd2808814cb9e09a5c864e314d85825aa0a8c6369d","abstract_canon_sha256":"bf084ff547978b3bbbdd5eab98f6bba834a81d7bdb6297228f7c7043def4c45c"},"schema_version":"1.0"},"canonical_sha256":"e51d37ac62620b60873f0dc1a64297896a9127bf9d3efe9566f5e8ac06b94b4c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:23.808292Z","signature_b64":"wS9l6tO/kYqLz8Pr0o9BNGztG2TYwfUrO+GaCY9962UkrkVyFBwk4JtW8MBLJQ75DCxVvnziiqSklVLtuP6FCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e51d37ac62620b60873f0dc1a64297896a9127bf9d3efe9566f5e8ac06b94b4c","last_reissued_at":"2026-05-18T03:34:23.807891Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:23.807891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0803.3750","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:34:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7126DQdxlrZAcimofIRg+Oz7pHxsn18XmJI1+KmpW/HrzSJSCZeL7xdpVH1KR+rngpDIdMvPxAcEf3aWXf+GCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:21:23.679143Z"},"content_sha256":"4a3255499bef5d093973c87d93608cfb8259f4335d5689092c54906ba037becd","schema_version":"1.0","event_id":"sha256:4a3255499bef5d093973c87d93608cfb8259f4335d5689092c54906ba037becd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:4UOTPLDCMIFWBBZ7BXA2MQUXRF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Fourier spectrum of critical percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.MP"],"primary_cat":"math.PR","authors_text":"Christophe Garban, G\\'abor Pete, Oded Schramm","submitted_at":"2008-03-26T16:11:30Z","abstract_excerpt":"Consider the indicator function $f$ of a two-dimensional percolation crossing event. In this paper, the Fourier transform of $f$ is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension 31/36 a.s., and the corresponding dimension in the half-plane is 5/9. It is also proved that critical bond percolation on the square grid has exceptional times a.s. Also, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.3750","kind":"arxiv","version":4},"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":"2026-05-18T03:34:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tehuf/PA1GleQYcv4C2SE2d4zWqqyC7cM13ifWRlIdHlxUW5/jtjcxRyFsNhZzJAEmMLdF+HaPBzsQ4/PevXBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:21:23.679527Z"},"content_sha256":"df0f414f85b027973d306b952fdda9d282a1f5605b811e1a292ae44e661661ef","schema_version":"1.0","event_id":"sha256:df0f414f85b027973d306b952fdda9d282a1f5605b811e1a292ae44e661661ef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4UOTPLDCMIFWBBZ7BXA2MQUXRF/bundle.json","state_url":"https://pith.science/pith/4UOTPLDCMIFWBBZ7BXA2MQUXRF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4UOTPLDCMIFWBBZ7BXA2MQUXRF/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":"2026-05-27T00:21:23Z","links":{"resolver":"https://pith.science/pith/4UOTPLDCMIFWBBZ7BXA2MQUXRF","bundle":"https://pith.science/pith/4UOTPLDCMIFWBBZ7BXA2MQUXRF/bundle.json","state":"https://pith.science/pith/4UOTPLDCMIFWBBZ7BXA2MQUXRF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4UOTPLDCMIFWBBZ7BXA2MQUXRF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:4UOTPLDCMIFWBBZ7BXA2MQUXRF","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":"bf084ff547978b3bbbdd5eab98f6bba834a81d7bdb6297228f7c7043def4c45c","cross_cats_sorted":["math-ph","math.CA","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-03-26T16:11:30Z","title_canon_sha256":"e8baeb0cef8cf1a586ae93cd2808814cb9e09a5c864e314d85825aa0a8c6369d"},"schema_version":"1.0","source":{"id":"0803.3750","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0803.3750","created_at":"2026-05-18T03:34:23Z"},{"alias_kind":"arxiv_version","alias_value":"0803.3750v4","created_at":"2026-05-18T03:34:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0803.3750","created_at":"2026-05-18T03:34:23Z"},{"alias_kind":"pith_short_12","alias_value":"4UOTPLDCMIFW","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"4UOTPLDCMIFWBBZ7","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"4UOTPLDC","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:df0f414f85b027973d306b952fdda9d282a1f5605b811e1a292ae44e661661ef","target":"graph","created_at":"2026-05-18T03:34:23Z","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":"Consider the indicator function $f$ of a two-dimensional percolation crossing event. In this paper, the Fourier transform of $f$ is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension 31/36 a.s., and the corresponding dimension in the half-plane is 5/9. It is also proved that critical bond percolation on the square grid has exceptional times a.s. Also, the","authors_text":"Christophe Garban, G\\'abor Pete, Oded Schramm","cross_cats":["math-ph","math.CA","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-03-26T16:11:30Z","title":"The Fourier spectrum of critical percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.3750","kind":"arxiv","version":4},"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":"sha256:4a3255499bef5d093973c87d93608cfb8259f4335d5689092c54906ba037becd","target":"record","created_at":"2026-05-18T03:34:23Z","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":"bf084ff547978b3bbbdd5eab98f6bba834a81d7bdb6297228f7c7043def4c45c","cross_cats_sorted":["math-ph","math.CA","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-03-26T16:11:30Z","title_canon_sha256":"e8baeb0cef8cf1a586ae93cd2808814cb9e09a5c864e314d85825aa0a8c6369d"},"schema_version":"1.0","source":{"id":"0803.3750","kind":"arxiv","version":4}},"canonical_sha256":"e51d37ac62620b60873f0dc1a64297896a9127bf9d3efe9566f5e8ac06b94b4c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e51d37ac62620b60873f0dc1a64297896a9127bf9d3efe9566f5e8ac06b94b4c","first_computed_at":"2026-05-18T03:34:23.807891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:23.807891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wS9l6tO/kYqLz8Pr0o9BNGztG2TYwfUrO+GaCY9962UkrkVyFBwk4JtW8MBLJQ75DCxVvnziiqSklVLtuP6FCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:23.808292Z","signed_message":"canonical_sha256_bytes"},"source_id":"0803.3750","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a3255499bef5d093973c87d93608cfb8259f4335d5689092c54906ba037becd","sha256:df0f414f85b027973d306b952fdda9d282a1f5605b811e1a292ae44e661661ef"],"state_sha256":"e82f1c94ed240cdca7c6cabc2ad454087a131bb8fce9e9086c31ab0669140629"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/uUvKaYIyrgqOBO2z37CRI83GPi7vUExm7M17vrPUTfolA/YCLrLa5hhfp+yVhiLIzARqE53me6m/sqjU7nSCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T00:21:23.681735Z","bundle_sha256":"f56eb86f06a61d9f249fa31f3f01f2b26036dad8b693c505b0b0c7b91c44ab2f"}}