Probability Distribution of Curvatures of Isosurfaces in Gaussian Random Fields

TL;DR

This paper derives the joint probability distribution of principal curvatures on isosurfaces within isotropic Gaussian random fields, revealing symmetry properties similar to Gaussian orthogonal ensembles.

Contribution

It provides a novel analytical expression for the curvature distribution in Gaussian random fields, advancing understanding of their geometric properties.

Findings

Derived the joint probability distribution of principal curvatures.

Identified symmetry properties akin to Gaussian orthogonal ensembles.

Enhanced the theoretical framework for analyzing isosurfaces in Gaussian fields.

Abstract

An expression for the joint probability distribution of the principal curvatures at an arbitrary point in the ensemble of isosurfaces defined on isotropic Gaussian random fields on Rn is derived. The result is obtained by deriving symmetry properties of the ensemble of second derivative matrices of isotropic Gaussian random fields akin to those of the Gaussian orthogonal ensemble.