{"paper":{"title":"On pseudospectrum of inhomogeneous non-Hermitian random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Konstantin Tikhomirov","submitted_at":"2023-07-17T03:01:35Z","abstract_excerpt":"Let $A$ be an $n\\times n$ matrix with mutually independent centered Gaussian entries. Define \\begin{align*} \\sigma^*:=\\max\\limits_{i,j\\leq n}\\sqrt{{\\mathbb E}\\,|A_{i,j}|^2}, \\quad \\sigma:=\\max\\bigg(\\max\\limits_{j\\leq n}\\sqrt{{\\mathbb E}\\,\\|{\\rm col}_j(A)\\|_2^2}, \\max\\limits_{i\\leq n}\\sqrt{{\\mathbb E}\\,\\|{\\rm row}_i(A)\\|_2^2}\\bigg). \\end{align*} Assume that $\\sigma\\geq n^\\varepsilon\\,\\sigma^*$ for a constant $\\varepsilon>0$, and that a complex number $z$ satisfies $|z|=\\Omega(\\sigma)$. We prove that $$ s_{\\min}(A-z\\,{\\rm Id}) \\geq |z|\\,\\exp\\bigg(-n^{o(1)}\\,\\Big(\\frac{\\sqrt{n}\\,\\sigma^*}{\\sigma}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.08211","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/2307.08211/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"}