Geodesic
Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 18, Tome 408 (2012), pp. 187-196 Cet article a éte moissonné depuis la source Math-Net.Ru

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Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $\xi_1,\ldots,\xi_d$ with covariance matrices $\Sigma_1,\dots,\Sigma_d$. Denote by $\mathcal E_i$ the location-dispersion ellipsoid of $\xi_i$: $\mathcal E_i=\{\mathbf x\in\mathbb R^d\colon\mathbf x^\top\Sigma_i^{-1} \mathbf x\leqslant1\}$. We show that $$ \mathbb E\,|\det M|=\frac{d!}{(2\pi)^{d/2}}V_d(\mathcal{E}_1,\dots,\mathcal E_d), $$ where $V_d(\cdot,\dots,\cdot)$ denotes the mixed volume. We also generalize this result to the case of rectangular matrices. As a direct corollary we get an analytic expression for the mixed volume of $d$ arbitrary ellipsoids in $\mathbb R^d$. As another application, we consider a smooth centered non-degenerate Gaussian random field $X=(X_1,\dots,X_k)^\top\colon\mathbb R^d\to\mathbb R^k$. Using the Kac–Rice formula, we obtain the geometric interpretation of the intensity of zeros of $X$ in terms of the mixed volume of location-dispersion ellipsoids of the gradients of $X_i/\sqrt{\mathbf{Var}X_i}$. This relates the zero sets of equations to the mixed volumes in a way which resembles the well-known Bernstein theorem on the number of solutions of a typical system of algebraic equations. 
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     author = {D. N. Zaporozhets and Z. Kabluchko},
     title = {Random determinants, mixed volumes of ellipsoids, and zeros of {Gaussian} random fields},
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}
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D. N. Zaporozhets; Z. Kabluchko. Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 18, Tome 408 (2012), pp. 187-196. http://geodesic.mathdoc.fr/item/ZNSL_2012_408_a11/

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