Umbilical points of a non-Gaussian random field

TL;DR

This paper develops a geometric method to detect and quantify non-Gaussian features in random fields by analyzing the density of umbilical points, providing a new tool for identifying deviations from Gaussianity.

Contribution

It introduces a way to measure non-Gaussianity through changes in umbilical point densities caused by nonlinear perturbations.

Findings

Derived formulas for umbilical point density changes due to non-Gaussian perturbations

Demonstrated the method's ability to detect non-Gaussianity even when the field is not directly observable

Provided a framework for quantifying the size of non-Gaussian contributions

Abstract

Random fields in nature often have, to a good approximation, Gaussian characteristics. For such fields, the relative densities of umbilical points -- topological defects which can be classified into three types -- have certain fixed values. Phenomena described by nonlinear laws can however give rise to a non-Gaussian contribution, causing a deviation from these universal values. We consider a Gaussian field with a perturbation added to it, given by a nonlinear function of that field, and calculate the change in the relative density of umbilical points. This allows us not only to detect a perturbation, but to determine its size as well. This geometric approach offers an independent way of detecting non-Gaussianity, which even works in cases where the field itself cannot be probed directly.