{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:EEJSPLQSWEGSV2YCBDBMR6ZLR5","short_pith_number":"pith:EEJSPLQS","canonical_record":{"source":{"id":"1012.2710","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-12-13T12:32:42Z","cross_cats_sorted":[],"title_canon_sha256":"75887265a29f955eec474a0981626deffeb7c2c7e86b8a89605185904e4d9519","abstract_canon_sha256":"ab658a0597ecdb97f16eac541bb4657b5e15bd66658b7fde1bea9dce4fed11b4"},"schema_version":"1.0"},"canonical_sha256":"211327ae12b10d2aeb0208c2c8fb2b8f788a55d1331f11789b527408c5c9222a","source":{"kind":"arxiv","id":"1012.2710","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.2710","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"arxiv_version","alias_value":"1012.2710v3","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2710","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"pith_short_12","alias_value":"EEJSPLQSWEGS","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EEJSPLQSWEGSV2YC","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EEJSPLQS","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:EEJSPLQSWEGSV2YCBDBMR6ZLR5","target":"record","payload":{"canonical_record":{"source":{"id":"1012.2710","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-12-13T12:32:42Z","cross_cats_sorted":[],"title_canon_sha256":"75887265a29f955eec474a0981626deffeb7c2c7e86b8a89605185904e4d9519","abstract_canon_sha256":"ab658a0597ecdb97f16eac541bb4657b5e15bd66658b7fde1bea9dce4fed11b4"},"schema_version":"1.0"},"canonical_sha256":"211327ae12b10d2aeb0208c2c8fb2b8f788a55d1331f11789b527408c5c9222a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:33.113485Z","signature_b64":"AlCxrPfWQ9fr/xcVDXEWMJjLIsK5AVz/XCFt3MbTs8jmN4ujfJsQtWs+h/xEpK36hA+ZTCFcfUWJ+3csvXX7DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"211327ae12b10d2aeb0208c2c8fb2b8f788a55d1331f11789b527408c5c9222a","last_reissued_at":"2026-05-18T04:23:33.113013Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:33.113013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.2710","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":"2026-05-18T04:23:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aQTcBv3/8u5a302iLnmSyEEM5J2pinoyZTezscM9xPedWPLtycdAcS0h0AXtgr1svTnvtbzyz1jnc0Zim03OBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T06:08:43.479309Z"},"content_sha256":"e510ff014d12c6494429cf58a7d2b6509e97c27e3a08331359bdbc5e9619c242","schema_version":"1.0","event_id":"sha256:e510ff014d12c6494429cf58a7d2b6509e97c27e3a08331359bdbc5e9619c242"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:EEJSPLQSWEGSV2YCBDBMR6ZLR5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Asymptotic Spectrum of Products of Independent Random Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Tikhomirov, Friedrich G\\\"otze","submitted_at":"2010-12-13T12:32:42Z","abstract_excerpt":"We consider products of independent random matrices with independent entries. The limit distribution of the expected empirical distribution of eigenvalues of such products is computed. Let $X^{(\\nu)}_{jk},{}1\\le j,r\\le n$, $\\nu=1,...,m$ be mutually independent complex random variables with $\\E X^{(\\nu)}_{jk}=0$ and $\\E {|X^{(\\nu)}_{jk}|}^2=1$. Let $\\mathbf X^{(\\nu)}$ denote an $n\\times n$ matrix with entries $[\\mathbf X^{(\\nu)}]_{jk}=\\frac1{\\sqrt{n}}X^{(\\nu)}_{jk}$, for $1\\le j,k\\le n$.\n  Denote by $\\lambda_1,...,\\lambda_n$ the eigenvalues of the random matrix $\\mathbf W:= \\prod_{\\nu=1}^m\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2710","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":"2026-05-18T04:23:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EtpfLTMKNQH/8w/xx81AuZ4MT5YfRzj8Ur4WX2G9fmx1Q3w1poPTC9QXzse5uLQXy8KAgmNHah1xJx1mnyfDCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T06:08:43.479707Z"},"content_sha256":"912fbdc5169a621d2149ea0a09e2315768d07226a0ae8db248ac4a6489b578fe","schema_version":"1.0","event_id":"sha256:912fbdc5169a621d2149ea0a09e2315768d07226a0ae8db248ac4a6489b578fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EEJSPLQSWEGSV2YCBDBMR6ZLR5/bundle.json","state_url":"https://pith.science/pith/EEJSPLQSWEGSV2YCBDBMR6ZLR5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EEJSPLQSWEGSV2YCBDBMR6ZLR5/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":"2026-05-21T06:08:43Z","links":{"resolver":"https://pith.science/pith/EEJSPLQSWEGSV2YCBDBMR6ZLR5","bundle":"https://pith.science/pith/EEJSPLQSWEGSV2YCBDBMR6ZLR5/bundle.json","state":"https://pith.science/pith/EEJSPLQSWEGSV2YCBDBMR6ZLR5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EEJSPLQSWEGSV2YCBDBMR6ZLR5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:EEJSPLQSWEGSV2YCBDBMR6ZLR5","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":"ab658a0597ecdb97f16eac541bb4657b5e15bd66658b7fde1bea9dce4fed11b4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-12-13T12:32:42Z","title_canon_sha256":"75887265a29f955eec474a0981626deffeb7c2c7e86b8a89605185904e4d9519"},"schema_version":"1.0","source":{"id":"1012.2710","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.2710","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"arxiv_version","alias_value":"1012.2710v3","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2710","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"pith_short_12","alias_value":"EEJSPLQSWEGS","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EEJSPLQSWEGSV2YC","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EEJSPLQS","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:912fbdc5169a621d2149ea0a09e2315768d07226a0ae8db248ac4a6489b578fe","target":"graph","created_at":"2026-05-18T04:23:33Z","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":"We consider products of independent random matrices with independent entries. The limit distribution of the expected empirical distribution of eigenvalues of such products is computed. Let $X^{(\\nu)}_{jk},{}1\\le j,r\\le n$, $\\nu=1,...,m$ be mutually independent complex random variables with $\\E X^{(\\nu)}_{jk}=0$ and $\\E {|X^{(\\nu)}_{jk}|}^2=1$. Let $\\mathbf X^{(\\nu)}$ denote an $n\\times n$ matrix with entries $[\\mathbf X^{(\\nu)}]_{jk}=\\frac1{\\sqrt{n}}X^{(\\nu)}_{jk}$, for $1\\le j,k\\le n$.\n  Denote by $\\lambda_1,...,\\lambda_n$ the eigenvalues of the random matrix $\\mathbf W:= \\prod_{\\nu=1}^m\\math","authors_text":"Alexander Tikhomirov, Friedrich G\\\"otze","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-12-13T12:32:42Z","title":"On the Asymptotic Spectrum of Products of Independent Random Matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2710","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:e510ff014d12c6494429cf58a7d2b6509e97c27e3a08331359bdbc5e9619c242","target":"record","created_at":"2026-05-18T04:23:33Z","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":"ab658a0597ecdb97f16eac541bb4657b5e15bd66658b7fde1bea9dce4fed11b4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-12-13T12:32:42Z","title_canon_sha256":"75887265a29f955eec474a0981626deffeb7c2c7e86b8a89605185904e4d9519"},"schema_version":"1.0","source":{"id":"1012.2710","kind":"arxiv","version":3}},"canonical_sha256":"211327ae12b10d2aeb0208c2c8fb2b8f788a55d1331f11789b527408c5c9222a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"211327ae12b10d2aeb0208c2c8fb2b8f788a55d1331f11789b527408c5c9222a","first_computed_at":"2026-05-18T04:23:33.113013Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:33.113013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AlCxrPfWQ9fr/xcVDXEWMJjLIsK5AVz/XCFt3MbTs8jmN4ujfJsQtWs+h/xEpK36hA+ZTCFcfUWJ+3csvXX7DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:33.113485Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.2710","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e510ff014d12c6494429cf58a7d2b6509e97c27e3a08331359bdbc5e9619c242","sha256:912fbdc5169a621d2149ea0a09e2315768d07226a0ae8db248ac4a6489b578fe"],"state_sha256":"9c23520cbe4dadadf5779e398df5fc07581d6ab977ef487b4f21e7b4a8a9001f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Sk1+znTRmP2xsdIqNxiTnGe1DSvVGGVNnhaj62C+/vtCfXRZ6dUlBnc0cvbg4WP3zYtUI/xhS46zqn7R+vVjCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T06:08:43.481709Z","bundle_sha256":"4eb78b033fb8e46a08afddc055d3121a3916ebb028fb16786db89b89e2f23554"}}