Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields

Abstract

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 ${\xi }_i:{\mathcal{E}}_i=\mathbf{x}\in {\mathbb{R}}^d:{\mathbf{x}}^{\top }{\Sigma }_i^{-1}\mathbf{x}⩽1$. 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 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 $X=(X_1,...,X_k{)}^{\top }:{\mathbb{R}}^d\to {\mathbb{R}}^k$. Using 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 zero sets of equations to mixed volumes in a way which is reminiscent of the well-known Bernstein theorem about the number of solutions of the typical system of algebraic equations.