{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:K42A6JSL7BT7ELE52SOD5P7GTJ","short_pith_number":"pith:K42A6JSL","canonical_record":{"source":{"id":"1508.03111","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-13T03:42:31Z","cross_cats_sorted":[],"title_canon_sha256":"cc36aa3ed42f6454429477d29aec06989a25458a719ba03377d51dbf49b3f9fa","abstract_canon_sha256":"23902320a39aee1361823129dd35b5700219dfb31af8005a234ffdf9300b7b07"},"schema_version":"1.0"},"canonical_sha256":"57340f264bf867f22c9dd49c3ebfe69a46a7e2a575208facfba49f5c13cd50fd","source":{"kind":"arxiv","id":"1508.03111","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.03111","created_at":"2026-05-18T00:29:33Z"},{"alias_kind":"arxiv_version","alias_value":"1508.03111v2","created_at":"2026-05-18T00:29:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.03111","created_at":"2026-05-18T00:29:33Z"},{"alias_kind":"pith_short_12","alias_value":"K42A6JSL7BT7","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"K42A6JSL7BT7ELE5","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"K42A6JSL","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:K42A6JSL7BT7ELE52SOD5P7GTJ","target":"record","payload":{"canonical_record":{"source":{"id":"1508.03111","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-13T03:42:31Z","cross_cats_sorted":[],"title_canon_sha256":"cc36aa3ed42f6454429477d29aec06989a25458a719ba03377d51dbf49b3f9fa","abstract_canon_sha256":"23902320a39aee1361823129dd35b5700219dfb31af8005a234ffdf9300b7b07"},"schema_version":"1.0"},"canonical_sha256":"57340f264bf867f22c9dd49c3ebfe69a46a7e2a575208facfba49f5c13cd50fd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:33.804123Z","signature_b64":"3X6XheNzVWpd/NsePdNEGpoBX+hb5JyB/7S9V82HbmWVTc6JZ9pC7z/16KijSaztH9JF9v/OQRX63SS0CdZSAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"57340f264bf867f22c9dd49c3ebfe69a46a7e2a575208facfba49f5c13cd50fd","last_reissued_at":"2026-05-18T00:29:33.803510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:33.803510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.03111","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":"2026-05-18T00:29:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HwQn0WOjWndTTnthAUwH676ZPIe4TPigR8Y5ElIt/9kGfBAYcCxkgyhwkQiB8latddvR6tt5o4k8Y7HyLu0LBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T02:44:33.823549Z"},"content_sha256":"64e592b60ee60e610d7e1e8758ac012d98172ca11ace14d31d17c635f57ac720","schema_version":"1.0","event_id":"sha256:64e592b60ee60e610d7e1e8758ac012d98172ca11ace14d31d17c635f57ac720"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:K42A6JSL7BT7ELE52SOD5P7GTJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Empirical Distributions of Eigenvalues of Product Ensembles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Tiefeng Jiang, Yongcheng Qi","submitted_at":"2015-08-13T03:42:31Z","abstract_excerpt":"Assume a finite set of complex random variables form a determinantal point process, we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to %We study the limits of the empirical distributions of the eigenvalues of two types of $n$ by $n$ random matrices as $n$ goes to infinity. The first one is the product of $m$ i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of $m$ independent Haar unitary matrices with sizes $n_j\\times n_j$ for $1\\leq j \\leq m$. Assuming $m$ depends on $n$, by using the special s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03111","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":""},"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":"2026-05-18T00:29:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3F9vVQKHbaSiHQ44UpID5+ZzUxq9CIZ4Wyi83nNDVDy0ZSovsjQJMbeR/dyzG4NOIP7U5zrRW9d9nvo1ExuKCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T02:44:33.824200Z"},"content_sha256":"fd81685af277e1baf64fb083c926085b64c3bbb7882cff94304b60cd75f7c2e3","schema_version":"1.0","event_id":"sha256:fd81685af277e1baf64fb083c926085b64c3bbb7882cff94304b60cd75f7c2e3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K42A6JSL7BT7ELE52SOD5P7GTJ/bundle.json","state_url":"https://pith.science/pith/K42A6JSL7BT7ELE52SOD5P7GTJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K42A6JSL7BT7ELE52SOD5P7GTJ/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":"2026-05-30T02:44:33Z","links":{"resolver":"https://pith.science/pith/K42A6JSL7BT7ELE52SOD5P7GTJ","bundle":"https://pith.science/pith/K42A6JSL7BT7ELE52SOD5P7GTJ/bundle.json","state":"https://pith.science/pith/K42A6JSL7BT7ELE52SOD5P7GTJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K42A6JSL7BT7ELE52SOD5P7GTJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:K42A6JSL7BT7ELE52SOD5P7GTJ","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":"23902320a39aee1361823129dd35b5700219dfb31af8005a234ffdf9300b7b07","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-13T03:42:31Z","title_canon_sha256":"cc36aa3ed42f6454429477d29aec06989a25458a719ba03377d51dbf49b3f9fa"},"schema_version":"1.0","source":{"id":"1508.03111","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.03111","created_at":"2026-05-18T00:29:33Z"},{"alias_kind":"arxiv_version","alias_value":"1508.03111v2","created_at":"2026-05-18T00:29:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.03111","created_at":"2026-05-18T00:29:33Z"},{"alias_kind":"pith_short_12","alias_value":"K42A6JSL7BT7","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"K42A6JSL7BT7ELE5","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"K42A6JSL","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:fd81685af277e1baf64fb083c926085b64c3bbb7882cff94304b60cd75f7c2e3","target":"graph","created_at":"2026-05-18T00:29:33Z","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":"Assume a finite set of complex random variables form a determinantal point process, we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to %We study the limits of the empirical distributions of the eigenvalues of two types of $n$ by $n$ random matrices as $n$ goes to infinity. The first one is the product of $m$ i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of $m$ independent Haar unitary matrices with sizes $n_j\\times n_j$ for $1\\leq j \\leq m$. Assuming $m$ depends on $n$, by using the special s","authors_text":"Tiefeng Jiang, Yongcheng Qi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-13T03:42:31Z","title":"Empirical Distributions of Eigenvalues of Product Ensembles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03111","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":"sha256:64e592b60ee60e610d7e1e8758ac012d98172ca11ace14d31d17c635f57ac720","target":"record","created_at":"2026-05-18T00:29:33Z","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":"23902320a39aee1361823129dd35b5700219dfb31af8005a234ffdf9300b7b07","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-13T03:42:31Z","title_canon_sha256":"cc36aa3ed42f6454429477d29aec06989a25458a719ba03377d51dbf49b3f9fa"},"schema_version":"1.0","source":{"id":"1508.03111","kind":"arxiv","version":2}},"canonical_sha256":"57340f264bf867f22c9dd49c3ebfe69a46a7e2a575208facfba49f5c13cd50fd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"57340f264bf867f22c9dd49c3ebfe69a46a7e2a575208facfba49f5c13cd50fd","first_computed_at":"2026-05-18T00:29:33.803510Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:33.803510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3X6XheNzVWpd/NsePdNEGpoBX+hb5JyB/7S9V82HbmWVTc6JZ9pC7z/16KijSaztH9JF9v/OQRX63SS0CdZSAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:33.804123Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.03111","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:64e592b60ee60e610d7e1e8758ac012d98172ca11ace14d31d17c635f57ac720","sha256:fd81685af277e1baf64fb083c926085b64c3bbb7882cff94304b60cd75f7c2e3"],"state_sha256":"d6bdf99359953a6727061037654524761bfc10574ba6df3ac7444aa7f6b90f62"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8noMKaAaB4V8FeupkGGZ5VR060cv2boX4tFXthB1cdDr7oXpJkei1bq3SU7Jtt2IgBO/TBgBrlilZ5beyNowBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T02:44:33.827271Z","bundle_sha256":"cc058505cb18d9c40f5ac1b20c1cd2265d8f8a2cadf2d85503b1086db157ec2f"}}