Signed zeros of Gaussian vector fields–density, correlation functions and curvature

Signed zeroes of Gaussian vector fields–Density, correlation functions and curvature

Abstract. We calculate correlation functions of the (charge) density of zeroes of Gaussian distributed vector fields. We are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract Riemann Cartan or Riemannian manifold. As an application, we discuss one and two point functions. The zeroes of a two dimensional Gaussian vector field model the di...

Correlations and screening of topological charges in gaussian random fields

2-point topological charge correlation functions of several types of geometric singularity in gaussian random fields are calculated explicitly, using a general scheme: zeros of n-dimensional random vectors, signed by the sign of their jacobian determinant; critical points (gradient zeros) of real scalars in two dimensions signed by the hessian; and umbilic points of real scalars in two dimensio...

The distribution of extremal points of Gaussian scalar fields

We consider the signed density of extremal point of (twodimensional) scalar fields with a Gaussian distribution. We assign a positive unit-charge to the maxima and minima of the function and a negative one to its saddles. At first, we compute the average density for a field in half-space with Dirichlet boundary conditions. Then we calculate the charge-charge correlation function (without bounda...

Symmetric curvature tensor

Recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. Using this machinery, we have defined the concept of symmetric curvature. This concept is natural and is related to the notions divergence and Laplacian of vector fields. This concept is also related to the derivations on the algebra of symmetric forms which has been discu...

Expected Number of Local Maxima of Some Gaussian Random Polnomials