{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:CRDQWWG5FHLHMUFUTLRRUEI626","short_pith_number":"pith:CRDQWWG5","canonical_record":{"source":{"id":"1710.02096","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-05T16:18:39Z","cross_cats_sorted":[],"title_canon_sha256":"9faf5fc4df12562568afb9d30b2df63e3b4960c6ee8e49e9eb2afc1f79748183","abstract_canon_sha256":"a24f8509a951a61226ced170829c63175650ef43d22f09e702fa42dd7cb0bf75"},"schema_version":"1.0"},"canonical_sha256":"14470b58dd29d67650b49ae31a111ed7b5f75d709611a45813673d381b07622a","source":{"kind":"arxiv","id":"1710.02096","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.02096","created_at":"2026-05-17T23:57:03Z"},{"alias_kind":"arxiv_version","alias_value":"1710.02096v3","created_at":"2026-05-17T23:57:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.02096","created_at":"2026-05-17T23:57:03Z"},{"alias_kind":"pith_short_12","alias_value":"CRDQWWG5FHLH","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_16","alias_value":"CRDQWWG5FHLHMUFU","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_8","alias_value":"CRDQWWG5","created_at":"2026-05-18T12:31:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:CRDQWWG5FHLHMUFUTLRRUEI626","target":"record","payload":{"canonical_record":{"source":{"id":"1710.02096","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-05T16:18:39Z","cross_cats_sorted":[],"title_canon_sha256":"9faf5fc4df12562568afb9d30b2df63e3b4960c6ee8e49e9eb2afc1f79748183","abstract_canon_sha256":"a24f8509a951a61226ced170829c63175650ef43d22f09e702fa42dd7cb0bf75"},"schema_version":"1.0"},"canonical_sha256":"14470b58dd29d67650b49ae31a111ed7b5f75d709611a45813673d381b07622a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:03.262356Z","signature_b64":"yZvvmy9SOpYo2jZApij9K6YpXfPIzxO49XU3Wbugbg9OY2s07sxR3PC79cmqKjV9gI9eI2tJRDxGmF2JJM2gDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14470b58dd29d67650b49ae31a111ed7b5f75d709611a45813673d381b07622a","last_reissued_at":"2026-05-17T23:57:03.261842Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:03.261842Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1710.02096","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":"2026-05-17T23:57:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tFMcRLJQujP0WollC767visu9eCgvpYACWwpXM7RGpmm4ygQRrYT4S0CmWsNRB2lVya7VuvX/pwUUmL1gyEKDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T20:08:46.362821Z"},"content_sha256":"f70358f22402bcb80bc3a27df83ab63e80deb7b0d8e3f026095f028db03ec43f","schema_version":"1.0","event_id":"sha256:f70358f22402bcb80bc3a27df83ab63e80deb7b0d8e3f026095f028db03ec43f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:CRDQWWG5FHLHMUFUTLRRUEI626","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Tail expansion of Gaussian multiplicative chaos and the Liouville reflection coefficient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"R\\'emi Rhodes, Vincent Vargas","submitted_at":"2017-10-05T16:18:39Z","abstract_excerpt":"In this short note, we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d GFF on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main interest of our work consists of two points. First, our derivation is based on a simple method which we believe is universal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.02096","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":"2026-05-17T23:57:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QOBsNRP5TSts0fRI15bbnV0V5HwJ9V3lKDL6kyJp+/cg55YRgfWVl2t34o6qC4PM4Ge50RLpvrsWSG72KYr4Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T20:08:46.363507Z"},"content_sha256":"c1483ba2bd10cc363236f4d05c440edc4d066b41f2c0099ce2cfe543b331c7cc","schema_version":"1.0","event_id":"sha256:c1483ba2bd10cc363236f4d05c440edc4d066b41f2c0099ce2cfe543b331c7cc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CRDQWWG5FHLHMUFUTLRRUEI626/bundle.json","state_url":"https://pith.science/pith/CRDQWWG5FHLHMUFUTLRRUEI626/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CRDQWWG5FHLHMUFUTLRRUEI626/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":"2026-06-08T20:08:46Z","links":{"resolver":"https://pith.science/pith/CRDQWWG5FHLHMUFUTLRRUEI626","bundle":"https://pith.science/pith/CRDQWWG5FHLHMUFUTLRRUEI626/bundle.json","state":"https://pith.science/pith/CRDQWWG5FHLHMUFUTLRRUEI626/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CRDQWWG5FHLHMUFUTLRRUEI626/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:CRDQWWG5FHLHMUFUTLRRUEI626","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":"a24f8509a951a61226ced170829c63175650ef43d22f09e702fa42dd7cb0bf75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-05T16:18:39Z","title_canon_sha256":"9faf5fc4df12562568afb9d30b2df63e3b4960c6ee8e49e9eb2afc1f79748183"},"schema_version":"1.0","source":{"id":"1710.02096","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.02096","created_at":"2026-05-17T23:57:03Z"},{"alias_kind":"arxiv_version","alias_value":"1710.02096v3","created_at":"2026-05-17T23:57:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.02096","created_at":"2026-05-17T23:57:03Z"},{"alias_kind":"pith_short_12","alias_value":"CRDQWWG5FHLH","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_16","alias_value":"CRDQWWG5FHLHMUFU","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_8","alias_value":"CRDQWWG5","created_at":"2026-05-18T12:31:10Z"}],"graph_snapshots":[{"event_id":"sha256:c1483ba2bd10cc363236f4d05c440edc4d066b41f2c0099ce2cfe543b331c7cc","target":"graph","created_at":"2026-05-17T23:57:03Z","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":"In this short note, we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d GFF on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main interest of our work consists of two points. First, our derivation is based on a simple method which we believe is universal ","authors_text":"R\\'emi Rhodes, Vincent Vargas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-05T16:18:39Z","title":"The Tail expansion of Gaussian multiplicative chaos and the Liouville reflection coefficient"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.02096","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:f70358f22402bcb80bc3a27df83ab63e80deb7b0d8e3f026095f028db03ec43f","target":"record","created_at":"2026-05-17T23:57:03Z","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":"a24f8509a951a61226ced170829c63175650ef43d22f09e702fa42dd7cb0bf75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-05T16:18:39Z","title_canon_sha256":"9faf5fc4df12562568afb9d30b2df63e3b4960c6ee8e49e9eb2afc1f79748183"},"schema_version":"1.0","source":{"id":"1710.02096","kind":"arxiv","version":3}},"canonical_sha256":"14470b58dd29d67650b49ae31a111ed7b5f75d709611a45813673d381b07622a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14470b58dd29d67650b49ae31a111ed7b5f75d709611a45813673d381b07622a","first_computed_at":"2026-05-17T23:57:03.261842Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:03.261842Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yZvvmy9SOpYo2jZApij9K6YpXfPIzxO49XU3Wbugbg9OY2s07sxR3PC79cmqKjV9gI9eI2tJRDxGmF2JJM2gDQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:03.262356Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.02096","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f70358f22402bcb80bc3a27df83ab63e80deb7b0d8e3f026095f028db03ec43f","sha256:c1483ba2bd10cc363236f4d05c440edc4d066b41f2c0099ce2cfe543b331c7cc"],"state_sha256":"70a6522fcf8a283803a78e51665592389b6166f8f795b4f90457dbeb6d01e4f4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i5gPyOp+EtZLJMImP1jzMF1KwJ/HzEzOE9Ir6zocA6WIG9OhDmFMCPi1sIacf255cGMDDkxr8zjywMne0m3TAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T20:08:46.367185Z","bundle_sha256":"826da6ef39a68962cb85e46ca2f14f63c5810c5954eec1bfa0babf90de5731de"}}