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

TL;DR

This paper establishes a connection between the expected determinant of Gaussian matrices, the mixed volume of ellipsoids, and the zeros of Gaussian random fields, providing new geometric insights and formulas.

Contribution

It derives a formula linking the expected determinant of Gaussian matrices to mixed volumes of ellipsoids and extends this to Gaussian fields, offering a geometric interpretation of zero set intensities.

Findings

Expected determinant equals scaled mixed volume of ellipsoids.

Provides an explicit formula for mixed volume of arbitrary ellipsoids.

Relates zero set intensity of Gaussian fields to mixed volumes of gradients.

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 $\xi_i:\mathcal{E}_i={\mathbf{x}\in\mathbb{R}^d : \mathbf{x}^\top\Sigma_i^{-1} \mathbf{x}\leqslant1}$. 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 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.