{"paper":{"title":"Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"math.OA","authors_text":"Ping Zhong","submitted_at":"2021-08-22T21:10:11Z","abstract_excerpt":"Given an $n\\times n$ random matrix $X_n$ with i.i.d. entries of unit variance, the circular law says that the empirical spectral distribution (ESD) of $X_n/\\sqrt{n}$ converges to the uniform measure on the unit disk. Let $M_n$ be a deterministic matrix that converges in $*$-moments to an operator ${x}$. It is known from the work by \\'{S}niady and Tao--Vu that the ESD of $X_n/\\sqrt{n}+M_n$ converges to the Brown measure of ${x}+c$, where $c$ is Voiculescu's circular operator. We obtain a formula for the Brown measure of ${x}+c$ which provides a description of the limit distribution. This answer"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.09844","kind":"arxiv","version":5},"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/2108.09844/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"}