{"paper":{"title":"R-transforms for non-Hermitian matrices: a spherical integral approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"R-transforms for non-Hermitian random matrices originate from a single scalar function of two variables.","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"cond-mat.dis-nn","authors_text":"Marc Potters, Pierre Bousseyroux","submitted_at":"2026-01-14T10:40:27Z","abstract_excerpt":"In this paper, we establish a connection between the formalism of $\\mathcal{R}$-transforms for non-Hermitian random matrices and the framework of spherical integrals, using the replica method. This connection was previously proved in the Hermitian setting and in the case of bi-invariant random matrices. We show that the $\\mathcal{R}$-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables. This provides a new and transparent way to compute $\\mathcal{R}$-transforms, which until now had been known only in restricted cases such as bi-invariant"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that the R-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The replica method can be rigorously applied to establish the connection between non-Hermitian R-transforms and spherical integrals for general ensembles.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"R-transforms for non-Hermitian matrices derive from one scalar function of two variables via spherical integrals and the replica method.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"R-transforms for non-Hermitian random matrices originate from a single scalar function of two variables.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8ea04b34b84207cdf6e47b3d023fcb6dc53325534f7caea2856daa9167e8e283"},"source":{"id":"2601.09360","kind":"arxiv","version":2},"verdict":{"id":"4b56d0d0-abcf-402c-9762-bfa80a655b0c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T14:30:49.808732Z","strongest_claim":"We show that the R-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables.","one_line_summary":"R-transforms for non-Hermitian matrices derive from one scalar function of two variables via spherical integrals and the replica method.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The replica method can be rigorously applied to establish the connection between non-Hermitian R-transforms and spherical integrals for general ensembles.","pith_extraction_headline":"R-transforms for non-Hermitian random matrices originate from a single scalar function of two variables."},"references":{"count":55,"sample":[{"doi":"","year":2005,"title":"A Fourier view on the R-transform and related asymptotics of spherical integrals.Journal of functional analysis, 222(2):435–490, 2005","work_id":"ddb0e65c-5308-4e96-9b8c-0166c3629bbf","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"Rectangular R-transform as the limit of rectangular spherical integrals","work_id":"fd5456c9-0336-4ab9-b786-fc7c1007c4ae","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1992,"title":"F. Haake, F. Izrailev, N. Lehmann, et al. Statistics of complex levels of random matrices for open quantum systems.Z. Phys. B, 88:359–370, 1992","work_id":"8bea926d-2424-4b0f-83f0-02dc44544aae","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1996,"title":"Y. V. Fyodorov and H.-J. Sommers. Statistics of S-matrix poles in few-channel chaotic scattering. JETP Letters, 63:1026–1030, 1996","work_id":"7b6f82b6-392a-4b4f-bf61-71b953503313","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1918,"title":"Y. V. Fyodorov and H.-J. Sommers. Statistics of resonance poles, phases, and time delays in quantum chaotic scattering.J. Math. Phys., 38:1918–1981, 1997","work_id":"16d92496-b291-4c7e-8e9c-9e1817e51850","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":55,"snapshot_sha256":"5f90b8164eb7a9a595c168f0d6582d4c9b2f6d22a38fea60d90bf49e0029df1a","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"af047aa4dddd54f6fb5bc1392d557d8634ce0d29a27f5fe92054b63e04617115"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}