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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.0371","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-06-02T12:24:01Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"9129fa4f335657dac9b949b76acb6b45588b2e8bcf01020a151010173782dc3d","abstract_canon_sha256":"ca395c78019806ffc66ebb2e194404b0a55c1fb681244064f68e0d7e5d59bea5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:25.132266Z","signature_b64":"R+fXmjwif9Z9nEas4FQ3pHpIaKEfSZPsVPR2SsUa+K4aG3i46EFHMNwToUq4pVS3TgnoknoEZcERk1+eeHNBDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cfe5b3fae5723917723e8f85fd0925d8bfac989b70df3cc6582b4877ec68586","last_reissued_at":"2026-05-18T03:54:25.131870Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:25.131870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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