{"paper":{"title":"Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Zakhar Kabluchko","submitted_at":"2012-06-02T12:24:01Z","abstract_excerpt":"Consider a $d\\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $\\xi_1,...,\\xi_d$ with covariance matrices $\\Sigma_1,...,\\Sigma_d$. Denote by $\\mathcal{E}_i$ the location-dispersion ellipsoid of $\\xi_i:\\mathcal{E}_i={\\mathbf{x}\\in\\mathbb{R}^d : \\mathbf{x}^\\top\\Sigma_i^{-1} \\mathbf{x}\\leqslant1}$. We show that $$ \\mathbb{E}\\,|\\det M|=\\frac{d!}{(2\\pi)^{d/2}}V_d(\\mathcal{E}_1,...,\\mathcal{E}_d), $$ where $V_d(\\cdot,...,\\cdot)$ denotes the {\\it mixed volume}. We also generalize this result to the case of rectangular matrices. As a direct corollary we get an an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0371","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}