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We prove that the empirical spectral distribution of $P^N=\\mathfrak{p}(X_1^N,\\cdots,X_n^N)$ converges as $N$ tends to infinity to the so-called Brown measure of $\\mathfrak{p}$ evaluated at free circular variables. For polynomials of degree at most 2, the convergence was proven by Cook, Guionnet, and Husson \\cite{cook2022spectrum}, and we prove that the convergence in fact holds for polynomials $\\mathfrak{p}$ of "},"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":"2606.01664","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-06-01T04:18:29Z","cross_cats_sorted":[],"title_canon_sha256":"0515287d6a701c85f99d9ba9978993daa3f1946ee2696675a88c0cd5651328b1","abstract_canon_sha256":"f36452e883a94f2815379818ed45799c3ef9222d69a68357eb63fa157c71f2e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:39.558763Z","signature_b64":"/+Dw8vVehUDKG0irfWpp/45DeeyQn8bX4FQbCECTC/smaZiB0+ZZmXnCWzsf+GP3Bbrq6hKHXQD9NxJucZ2BCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"436d2cfed3add977428641d7c21b86b49bff07edadf49df9442708359fe77799","last_reissued_at":"2026-06-02T02:04:39.558422Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:39.558422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Brown measure convergence for the spectrum of polynomials in Ginibre matrices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yi Han","submitted_at":"2026-06-01T04:18:29Z","abstract_excerpt":"Fix a multivariate polynomial $\\mathfrak{p}$ in $n$ non-commuting variables of arbitrary degree, and consider $n$ independent $N\\times N$ complex Ginibre matrices $X_1^N,\\cdots,X_n^N$. We prove that the empirical spectral distribution of $P^N=\\mathfrak{p}(X_1^N,\\cdots,X_n^N)$ converges as $N$ tends to infinity to the so-called Brown measure of $\\mathfrak{p}$ evaluated at free circular variables. 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