Random Determinants, Mixed Volumes of Ellipsoids, and Zeros of Gaussian Random Fields

Abstract

Consider a d × d matrix M whose rows are independent, centered, nondegenerate Gaussian vectors ξ 1,…,ξ d with covariance matrices Σ1,…,Σ d . Denote by e i the dispersion ellipsoid of $ {\xi_{\mathrm{i}}}:{\varepsilon_i}=\left\{ {\mathbf{x}\in {{\mathbb{R}}^d}:{{\mathbf{x}}^{\top }}\sum\nolimits_i^{-1 } {\mathbf{x}\leq 1} } \right\} $ . We show that $$ \mathbb{E}\left| {\det M} \right|=\frac{d! }{{{{{\left( {2\pi } \right)}}^{{{d \left/ {2} \right.}}}}}}{V_d}\left( {{\varepsilon_1}\ldots {\varepsilon_d}} \right), $$ where V d (·,…,·) 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 ℝ d . As another application, we consider a smooth, centered, nondegenerate Gaussian random field X = (X 1,…,X k )⊤ : ℝ d → ℝ k . Using the Kac-Rice formula, we obtain a geometric interpretation of the intensity of zeros of X in terms of the mixed volume of dispersion ellipsoids of the gradients of $ {{{{X_i}}} \left/ {{\sqrt{{\mathbf{Var}{X_i}}}}} \right.} $ . 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 a typical system of algebraic equations.