Stochastic geometry and topology of non-Gaussian fields

TL;DR

This paper explores how geometric and topological features of fields are altered by non-Gaussian perturbations, providing new methods to detect non-Gaussianity beyond traditional correlation measurements.

Contribution

It introduces a novel approach to identify non-Gaussianity by analyzing geometric and topological properties of fields, applicable to both local and nonlocal mechanisms.

Findings

Non-Gaussian perturbations modify extrema statistics of Gaussian fields.

Discrepancies in geometric/topological measures serve as indicators of non-Gaussianity.

Method applicable to static and dynamic non-Gaussian field generation.

Abstract

Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.