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

2012/06/02 by Kabluchko, Zakhar, Zaporozhets, Dmitry
#52A39 #53C65 #Differential Geometry (math.DG) #FOS: Mathematics #Primary 60B20 #Probability (math.PR) #Secondary 60G15

paper · doi:10.48550/arxiv.1206.0371

Abstract

Consider a d× d matrix M whose rows are independent centered non-degenerate Gaussian vectors ξ₁,...,ξ_d with covariance matrices Σ₁,...,Σ_d. Denote by E_i the location-dispersion ellipsoid of ξ_i:E_i=x∈ℝ^d : x^\topΣ_i^-1 x\leqslant1. We show that 𝔼 |det M|=\fracd!(2π)^d/2V_d(E₁,...,E_d), where V_d(⋅,...,⋅) 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 ℝ^d. As another application, we consider a smooth centered non-degenerate Gaussian random field X=(X₁,...,X_k)^\top:ℝ^d→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/√(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.

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