Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian
  random fields   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.

16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat