TY - JOUR

T1 - Random Determinants, Mixed Volumes of Ellipsoids, and Zeros of Gaussian Random Fields

AU - Zaporozhets, D.

AU - Kabluchko, Z.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Consider a d × d matrix M whose rows are independent, centered, nondegenerate Gaussian vectors ξ1,...,ξd with covariance matrices Σ1,...,Σd. Denote by εi the dispersion ellipsoid of ξi:εi}= {x ∈ ℝ}d}: x⊤ ∑i-1} . We show that (Formula presented.) where Vd (·,...,·) 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 ℝd. As another application, we consider a smooth, centered, nondegenerate Gaussian random field X = (X1,...,Xk)⊤ : ℝd → ℝk. 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 Xi/√VarXi. 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. © 2014 Springer Science+Business Media New York.

AB - Consider a d × d matrix M whose rows are independent, centered, nondegenerate Gaussian vectors ξ1,...,ξd with covariance matrices Σ1,...,Σd. Denote by εi the dispersion ellipsoid of ξi:εi}= {x ∈ ℝ}d}: x⊤ ∑i-1} . We show that (Formula presented.) where Vd (·,...,·) 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 ℝd. As another application, we consider a smooth, centered, nondegenerate Gaussian random field X = (X1,...,Xk)⊤ : ℝd → ℝk. 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 Xi/√VarXi. 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. © 2014 Springer Science+Business Media New York.

UR - http://www.scopus.com/inward/record.url?scp=84902275899&partnerID=8YFLogxK

U2 - 10.1007/s10958-014-1844-9

DO - 10.1007/s10958-014-1844-9

M3 - Article

AN - SCOPUS:84902275899

VL - 199

SP - 168

EP - 173

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -