{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:Q4FOXYSSWN34KQG2KCTSFC5MJP","short_pith_number":"pith:Q4FOXYSS","schema_version":"1.0","canonical_sha256":"870aebe252b377c540da50a7228bac4be6fd9e9f21bf1daf2eeaa79e618e1d0f","source":{"kind":"arxiv","id":"1010.5220","version":2},"attestation_state":"computed","paper":{"title":"Summing free unitary random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.MP","quant-ph"],"primary_cat":"math-ph","authors_text":"Andrzej Jarosz","submitted_at":"2010-10-25T19:02:01Z","abstract_excerpt":"I use quaternion free probability calculus - an extension of free probability to non-Hermitian matrices (which is introduced in a succinct but self-contained way) - to derive in the large-size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices, weighted by complex numbers. In the case of CUE summands, I write them in terms of two \"master equations,\" which I then solve and numerically test in four specific cases. I conjecture a finite-size extension of these results, exploiting the complementary error function. I prove a central limit "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1010.5220","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-10-25T19:02:01Z","cross_cats_sorted":["cond-mat.stat-mech","math.MP","quant-ph"],"title_canon_sha256":"2f1f7620e598943526733a40cd4f987e000271c5d3ee9f588d74841bced39704","abstract_canon_sha256":"ccb8c6f0695fc4a16e34bb28e4a49dbd3a6bc53caddee9ac88a66093ab19507e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:04:58.743991Z","signature_b64":"F4A3OcAjFFWaIfYq2sDJktnvW/e178v3zmyEsu2/qBCYE0HyvDe/9Mei3Eyq/pjicapPNqJVskcTqy0vegtUAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"870aebe252b377c540da50a7228bac4be6fd9e9f21bf1daf2eeaa79e618e1d0f","last_reissued_at":"2026-05-18T02:04:58.743376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:04:58.743376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Summing free unitary random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.MP","quant-ph"],"primary_cat":"math-ph","authors_text":"Andrzej Jarosz","submitted_at":"2010-10-25T19:02:01Z","abstract_excerpt":"I use quaternion free probability calculus - an extension of free probability to non-Hermitian matrices (which is introduced in a succinct but self-contained way) - to derive in the large-size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices, weighted by complex numbers. In the case of CUE summands, I write them in terms of two \"master equations,\" which I then solve and numerically test in four specific cases. I conjecture a finite-size extension of these results, exploiting the complementary error function. I prove a central limit "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.5220","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1010.5220","created_at":"2026-05-18T02:04:58.743477+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.5220v2","created_at":"2026-05-18T02:04:58.743477+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.5220","created_at":"2026-05-18T02:04:58.743477+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q4FOXYSSWN34","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q4FOXYSSWN34KQG2","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q4FOXYSS","created_at":"2026-05-18T12:26:12.377268+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP","json":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP.json","graph_json":"https://pith.science/api/pith-number/Q4FOXYSSWN34KQG2KCTSFC5MJP/graph.json","events_json":"https://pith.science/api/pith-number/Q4FOXYSSWN34KQG2KCTSFC5MJP/events.json","paper":"https://pith.science/paper/Q4FOXYSS"},"agent_actions":{"view_html":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP","download_json":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP.json","view_paper":"https://pith.science/paper/Q4FOXYSS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.5220&json=true","fetch_graph":"https://pith.science/api/pith-number/Q4FOXYSSWN34KQG2KCTSFC5MJP/graph.json","fetch_events":"https://pith.science/api/pith-number/Q4FOXYSSWN34KQG2KCTSFC5MJP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP/action/storage_attestation","attest_author":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP/action/author_attestation","sign_citation":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP/action/citation_signature","submit_replication":"https://pith.science/pith/Q4FOXYSSWN34KQG2KCTSFC5MJP/action/replication_record"}},"created_at":"2026-05-18T02:04:58.743477+00:00","updated_at":"2026-05-18T02:04:58.743477+00:00"}