{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:AJ5NU2GCBRL4CKJKU7HHQ3BFLW","short_pith_number":"pith:AJ5NU2GC","canonical_record":{"source":{"id":"1905.12027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-05-28T18:46:54Z","cross_cats_sorted":[],"title_canon_sha256":"25162deaeb26e403985ad82f0016a9a925fdfa329fb04fc5e69e7d78880b2bf9","abstract_canon_sha256":"a56d17e644677e4f2d02442b784975da25a1c3b9bbb36883e0cb4d93d77a3861"},"schema_version":"1.0"},"canonical_sha256":"027ada68c20c57c1292aa7ce786c255db2d3b025380de5cbff37937164a8cb09","source":{"kind":"arxiv","id":"1905.12027","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.12027","created_at":"2026-05-17T23:44:49Z"},{"alias_kind":"arxiv_version","alias_value":"1905.12027v1","created_at":"2026-05-17T23:44:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.12027","created_at":"2026-05-17T23:44:49Z"},{"alias_kind":"pith_short_12","alias_value":"AJ5NU2GCBRL4","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"AJ5NU2GCBRL4CKJK","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"AJ5NU2GC","created_at":"2026-05-18T12:33:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:AJ5NU2GCBRL4CKJKU7HHQ3BFLW","target":"record","payload":{"canonical_record":{"source":{"id":"1905.12027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-05-28T18:46:54Z","cross_cats_sorted":[],"title_canon_sha256":"25162deaeb26e403985ad82f0016a9a925fdfa329fb04fc5e69e7d78880b2bf9","abstract_canon_sha256":"a56d17e644677e4f2d02442b784975da25a1c3b9bbb36883e0cb4d93d77a3861"},"schema_version":"1.0"},"canonical_sha256":"027ada68c20c57c1292aa7ce786c255db2d3b025380de5cbff37937164a8cb09","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:49.192054Z","signature_b64":"z3qBfhCfh7wQFvVmfKO3hza6RGQBknHtiRVZ41VlXSPG9zC8MfG9HupDyKTWs42UIx++izCCzyGnHvfHOJg4Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"027ada68c20c57c1292aa7ce786c255db2d3b025380de5cbff37937164a8cb09","last_reissued_at":"2026-05-17T23:44:49.191449Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:49.191449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1905.12027","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:44:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I4ULsxAHCtMGzDFUgFZwUaD7AMHGCCnpLBXh2M/MdXhKMFkyBlTwudACBzPJQvJYLpKJcgwjgtdYRfeSR7mUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T00:15:13.738245Z"},"content_sha256":"66bf0ce3a397206d9dd36197a93eb2ff8c883cfa43a618cb5b2b6b541220fed5","schema_version":"1.0","event_id":"sha256:66bf0ce3a397206d9dd36197a93eb2ff8c883cfa43a618cb5b2b6b541220fed5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:AJ5NU2GCBRL4CKJKU7HHQ3BFLW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the regularity of complex multiplicative chaos","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Eero Saksman, Janne Junnila, Lauri Viitasaari","submitted_at":"2019-05-28T18:46:54Z","abstract_excerpt":"Denote by $\\mu_\\beta=\"\\exp(\\beta X)\"$ the Gaussian multiplicative chaos which is defined using a log-correlated Gaussian field $X$ on a domain $U\\subset\\mathbb{R}^d$. The case $\\beta\\in\\mathbb{R}$ has been studied quite intensively, and then $\\mu_\\beta$ is a random measure on $U$. It is known that $\\mu_\\beta$ can also be defined for complex values $\\beta$ lying in certain subdomain of $\\mathbb{C}$, and then the realizations of $\\mu_\\beta$ are random generalized functions on $U$. In this note we complement the results of Junnila et al. (where the case of purely imaginary $\\beta$ was considered)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.12027","kind":"arxiv","version":1},"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":"2026-05-17T23:44:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uDjtjhXlwYLbfNfDKL0kbC/5nytwka4yKh46KM77YEeuLe849R+hUSlyXMQ7DC1D/TQPql/MMCPHxKiKcz08Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T00:15:13.739516Z"},"content_sha256":"25871551557731d456a986e7ed302e003467e5e3b7b7c971d78da9546879e3ee","schema_version":"1.0","event_id":"sha256:25871551557731d456a986e7ed302e003467e5e3b7b7c971d78da9546879e3ee"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AJ5NU2GCBRL4CKJKU7HHQ3BFLW/bundle.json","state_url":"https://pith.science/pith/AJ5NU2GCBRL4CKJKU7HHQ3BFLW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AJ5NU2GCBRL4CKJKU7HHQ3BFLW/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":"2026-08-23T00:15:13Z","links":{"resolver":"https://pith.science/pith/AJ5NU2GCBRL4CKJKU7HHQ3BFLW","bundle":"https://pith.science/pith/AJ5NU2GCBRL4CKJKU7HHQ3BFLW/bundle.json","state":"https://pith.science/pith/AJ5NU2GCBRL4CKJKU7HHQ3BFLW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AJ5NU2GCBRL4CKJKU7HHQ3BFLW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:AJ5NU2GCBRL4CKJKU7HHQ3BFLW","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":"a56d17e644677e4f2d02442b784975da25a1c3b9bbb36883e0cb4d93d77a3861","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-05-28T18:46:54Z","title_canon_sha256":"25162deaeb26e403985ad82f0016a9a925fdfa329fb04fc5e69e7d78880b2bf9"},"schema_version":"1.0","source":{"id":"1905.12027","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.12027","created_at":"2026-05-17T23:44:49Z"},{"alias_kind":"arxiv_version","alias_value":"1905.12027v1","created_at":"2026-05-17T23:44:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.12027","created_at":"2026-05-17T23:44:49Z"},{"alias_kind":"pith_short_12","alias_value":"AJ5NU2GCBRL4","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"AJ5NU2GCBRL4CKJK","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"AJ5NU2GC","created_at":"2026-05-18T12:33:12Z"}],"graph_snapshots":[{"event_id":"sha256:25871551557731d456a986e7ed302e003467e5e3b7b7c971d78da9546879e3ee","target":"graph","created_at":"2026-05-17T23:44:49Z","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":"Denote by $\\mu_\\beta=\"\\exp(\\beta X)\"$ the Gaussian multiplicative chaos which is defined using a log-correlated Gaussian field $X$ on a domain $U\\subset\\mathbb{R}^d$. The case $\\beta\\in\\mathbb{R}$ has been studied quite intensively, and then $\\mu_\\beta$ is a random measure on $U$. It is known that $\\mu_\\beta$ can also be defined for complex values $\\beta$ lying in certain subdomain of $\\mathbb{C}$, and then the realizations of $\\mu_\\beta$ are random generalized functions on $U$. In this note we complement the results of Junnila et al. (where the case of purely imaginary $\\beta$ was considered)","authors_text":"Eero Saksman, Janne Junnila, Lauri Viitasaari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-05-28T18:46:54Z","title":"On the regularity of complex multiplicative chaos"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.12027","kind":"arxiv","version":1},"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":"sha256:66bf0ce3a397206d9dd36197a93eb2ff8c883cfa43a618cb5b2b6b541220fed5","target":"record","created_at":"2026-05-17T23:44:49Z","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":"a56d17e644677e4f2d02442b784975da25a1c3b9bbb36883e0cb4d93d77a3861","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-05-28T18:46:54Z","title_canon_sha256":"25162deaeb26e403985ad82f0016a9a925fdfa329fb04fc5e69e7d78880b2bf9"},"schema_version":"1.0","source":{"id":"1905.12027","kind":"arxiv","version":1}},"canonical_sha256":"027ada68c20c57c1292aa7ce786c255db2d3b025380de5cbff37937164a8cb09","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"027ada68c20c57c1292aa7ce786c255db2d3b025380de5cbff37937164a8cb09","first_computed_at":"2026-05-17T23:44:49.191449Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:49.191449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z3qBfhCfh7wQFvVmfKO3hza6RGQBknHtiRVZ41VlXSPG9zC8MfG9HupDyKTWs42UIx++izCCzyGnHvfHOJg4Dg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:49.192054Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.12027","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:66bf0ce3a397206d9dd36197a93eb2ff8c883cfa43a618cb5b2b6b541220fed5","sha256:25871551557731d456a986e7ed302e003467e5e3b7b7c971d78da9546879e3ee"],"state_sha256":"05d9d44f13ebc561ced752e867267efa2cb089920f9fa425b73463371626c370"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bdNWkfEbPQb4oCPtFy3hd9yb13Jgy/XEbKEEAmPWQ3OuMhgkUy9/MhmQ/A5L8Z2l0BS1su1FhYkPX+oa9mZ4DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T00:15:13.745190Z","bundle_sha256":"0dcd7ebb50b57a2baf694d643a14851ebd3d8efd7fc600696c8fb7a55ae0e680"}}