{"paper":{"title":"On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Aaron Potechin, Wenjun Cai","submitted_at":"2022-06-05T17:37:59Z","abstract_excerpt":"In this paper, we introduce and analyze a new operation $\\circ_{R}$ which mixes two distributions $\\Omega$ and $\\Omega'$ via a random orthogonal matrix. In particular, we take $\\Omega \\circ_R \\Omega'$ to be the limit as $n \\to \\infty$ of the distribution of singular values of $DRD'$ where $D$ and $D'$ are $n \\times n$ diagonal matrices whose diagonal entries have distributions $\\Omega$ and $\\Omega'$ respectively and $R$ is a random $n \\times n$ orthogonal matrix. We show that $\\circ_R$ has several nice properties. We first observe that $\\circ_R$ is commutative and associative and compute the m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.02224","kind":"arxiv","version":2},"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/2206.02224/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"}