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

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 ε _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.