{"paper":{"title":"Eigenvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Non-Hermitian matrix Brownian motion admits scale-invariant SDEs coupling eigenvalues to eigenvector overlaps via a regularized Fuglede-Kadison determinant.","cross_cats":["cond-mat.stat-mech","math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Makoto Katori, Satoshi Yabuoku, Syota Esaki","submitted_at":"2023-06-01T02:39:19Z","abstract_excerpt":"The non-Hermitian matrix-valued Brownian motion is the stochastic process of a random matrix whose entries are given by independent complex Brownian motions. The bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for their scale transformations with an invariant eigenvalue process. The eigenvector-overlap process is a Hermitian matrix-valued process, each element of which is given by a product of an overlap of right eigenvectors and that of left eigenvectors. We derive a set of stochastic differential equations (SDEs) for the coupled system "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We derive a set of stochastic differential equations (SDEs) for the coupled system of the eigenvalue process and the eigenvector-overlap process and prove the scale-transformation invariance of the obtained SDE system.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for their scale transformations with an invariant eigenvalue process.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives coupled SDEs for eigenvalue and eigenvector-overlap processes in non-Hermitian Brownian motion and SPDEs for the regularized FK determinant random field.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Non-Hermitian matrix Brownian motion admits scale-invariant SDEs coupling eigenvalues to eigenvector overlaps via a regularized Fuglede-Kadison determinant.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fb96d1eeff09584cd584bb1ddbdfcd723bba55eecb1e8b402ea6c1c85722e7f1"},"source":{"id":"2306.00300","kind":"arxiv","version":4},"verdict":{"id":"9d3e9b0a-1a9d-44eb-8e53-e48db82dc33d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T08:40:05.220151Z","strongest_claim":"We derive a set of stochastic differential equations (SDEs) for the coupled system of the eigenvalue process and the eigenvector-overlap process and prove the scale-transformation invariance of the obtained SDE system.","one_line_summary":"Derives coupled SDEs for eigenvalue and eigenvector-overlap processes in non-Hermitian Brownian motion and SPDEs for the regularized FK determinant random field.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for their scale transformations with an invariant eigenvalue process.","pith_extraction_headline":"Non-Hermitian matrix Brownian motion admits scale-invariant SDEs coupling eigenvalues to eigenvector overlaps via a regularized Fuglede-Kadison determinant."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2306.00300/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":42,"sample":[{"doi":"","year":2000,"title":"Random Ma trices: Theory and Applications 9 (4), 2050015 (2000)","work_id":"07e89d82-30ef-4206-b547-3bed9cebeca2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"D.: Circular law","work_id":"0d59eb9c-b25f-430f-b485-1abe40166076","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"A., Speicher, R., Tarnowski, W.: Squared eig envalue condition numbers and eigenvector correlations from the single ring theorem","work_id":"6a57751c-86fb-4a0d-bc73-fab82a75afb4","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"T., ´Sniady, P., Speicher, R.: Eigenvalues of non-Hermitian random matric es and Brown measure of non-normal operators: Hermitian reductio n and linearization method","work_id":"5316de0a-f89e-431b-b0c6-a8902f3558a1","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"R.: The Cauchy Transform, Potential Theory and Confor mal Mapping","work_id":"e3e435d2-5c31-43a1-8620-adeb58a00a33","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":42,"snapshot_sha256":"49008eb04c30b9f7666b4af32fd7eeacbe2994136b8a0805b82c8f4d0109ad34","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"}