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Our main result is the following convergence result: if the law of $P_n$ converges to the law of $p$ and the law of $Q_n$ converges to the law of $q$, then the empirical spectral distributions of the $X_n$ converges to the Brown measure of $X = p + i q$. To prove this, we use the Hermit"},"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":"2411.17159","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2024-11-26T07:01:16Z","cross_cats_sorted":[],"title_canon_sha256":"cc2e04396b16b073081fada93eece55f0a1d108a4a3192bc4e69632c8fc7f9ce","abstract_canon_sha256":"920e6a9a94f574986aea8d4b5ae7f555c18cfe29fbea36c4ac85c7069e243b01"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:56:02.620184Z","signature_b64":"WmoPjsWga/KPKR82yn3bLld7ez82lDecTd/VX9D2Ytpu/3BUMRrFe+vacKkiIF/O4czw3JKZ6OiL9CaSHDR7AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b4f717748fdcf0057f789a2282681884f84f7a7e41ccd1e554888840ca4246f","last_reissued_at":"2026-07-05T09:56:02.619772Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:56:02.619772Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence of the Laws of Non-Hermitian Sums of Projections","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Max Sun Zhou","submitted_at":"2024-11-26T07:01:16Z","abstract_excerpt":"We consider the random matrix model $X_n = P_n + i Q_n$, where $P_n$ and $Q_n$ are independently Haar-unitary rotated Hermitian matrices with at most $2$ atoms in their spectra. Let $(M, \\tau)$ be a tracial von Neumann algebra and let $p, q \\in (M, \\tau)$, where $p$ and $q$ are Hermitian and freely independent. Our main result is the following convergence result: if the law of $P_n$ converges to the law of $p$ and the law of $Q_n$ converges to the law of $q$, then the empirical spectral distributions of the $X_n$ converges to the Brown measure of $X = p + i q$. To prove this, we use the Hermit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.17159","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.17159/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2411.17159","created_at":"2026-07-05T09:56:02.619833+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.17159v2","created_at":"2026-07-05T09:56:02.619833+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.17159","created_at":"2026-07-05T09:56:02.619833+00:00"},{"alias_kind":"pith_short_12","alias_value":"PNHXC52I7XHQ","created_at":"2026-07-05T09:56:02.619833+00:00"},{"alias_kind":"pith_short_16","alias_value":"PNHXC52I7XHQAV7X","created_at":"2026-07-05T09:56:02.619833+00:00"},{"alias_kind":"pith_short_8","alias_value":"PNHXC52I","created_at":"2026-07-05T09:56:02.619833+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2411.17166","citing_title":"Quaternionic Green's Function and the Brown Measure of Atomic Operators","ref_index":2024,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB","json":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB.json","graph_json":"https://pith.science/api/pith-number/PNHXC52I7XHQAV7XRGRCQJUBRB/graph.json","events_json":"https://pith.science/api/pith-number/PNHXC52I7XHQAV7XRGRCQJUBRB/events.json","paper":"https://pith.science/paper/PNHXC52I"},"agent_actions":{"view_html":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB","download_json":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB.json","view_paper":"https://pith.science/paper/PNHXC52I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.17159&json=true","fetch_graph":"https://pith.science/api/pith-number/PNHXC52I7XHQAV7XRGRCQJUBRB/graph.json","fetch_events":"https://pith.science/api/pith-number/PNHXC52I7XHQAV7XRGRCQJUBRB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB/action/storage_attestation","attest_author":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB/action/author_attestation","sign_citation":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB/action/citation_signature","submit_replication":"https://pith.science/pith/PNHXC52I7XHQAV7XRGRCQJUBRB/action/replication_record"}},"created_at":"2026-07-05T09:56:02.619833+00:00","updated_at":"2026-07-05T09:56:02.619833+00:00"}