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We prove that, under suitable assumptions, we have the $d\\to\\infty$ eigenvalue distribution formula $\\delta m\\tilde{W}\\sim\\pi_{mn\\rho}\\boxtimes\\nu$, where $\\rho$ is the law of $\\varphi$, viewed as a square matrix, $\\pi$ is the free Poisson law, $\\nu$ is the law of $D=\\varphi(1)$, and $\\delta=tr(D)$."},"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":"1201.4792","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-23T18:13:21Z","cross_cats_sorted":[],"title_canon_sha256":"f5bbe691492bb1eaaf228b72a49b9c938cbda8bdf228623bd99926e3033835bc","abstract_canon_sha256":"da521d24cffbe6899e94e14e525baff8ffe6c603309adbf6f5700ad29c6bf480"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:21.294880Z","signature_b64":"8JRZGdFIVHLtpgDWFbousY4QGtfr+BFufLGiax2PNOAzR4xxsZ/uur0UsH7Tyc4dh5LZVs7ZQ497QN2d/0OkBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0be5c3ce5dac3f4d91ff250e6a2df2ddfd6f8fceaac34c40f957e794ba32dd2","last_reissued_at":"2026-05-18T02:28:21.294260Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:21.294260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Block-modified Wishart matrices and free Poisson laws","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ion Nechita, Teodor Banica","submitted_at":"2012-01-23T18:13:21Z","abstract_excerpt":"We study the random matrices of type $\\tilde{W}=(id\\otimes\\varphi)W$, where $W$ is a complex Wishart matrix of parameters $(dn,dm)$, and $\\varphi:M_n(\\mathbb C)\\to M_n(\\mathbb C)$ is a self-adjoint linear map. 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