Probability distribution for the Gaussian curvature of the zero level surface of a random function

TL;DR

This paper derives the probability distribution of Gaussian curvature at a random point on the zero level surface of a stationary, isotropic Gaussian random function, providing explicit formulas and insights into the surface's geometric properties.

Contribution

It presents an explicit algebraic formula for the Gaussian curvature distribution on the nodal surface of a Gaussian random function, extending understanding of random surface geometry.

Findings

Explicit algebraic distribution for Gaussian curvature at nodal surface points

Average Gaussian curvature matches known theoretical values

Distribution for non-zero level surfaces expressed as an integral

Abstract

A rather natural construction for a smooth random surface in space is the level surface of value zero, or 'nodal' surface f(x,y,z)=0, of a (real) random function f; the interface between positive and negative regions of the function. A physically significant local attribute at a point of a curved surface is its Gaussian curvature (the product of its principal curvatures) because, when integrated over the surface it gives the Euler characteristic. Here the probability distribution for the Gaussian curvature at a random point on the nodal surface f=0 is calculated for a statistically homogeneous ('stationary') and isotropic zero mean Gaussian random function f. Capitalizing on the isotropy, a 'fixer' device for axes supplies the probability distribution directly as a multiple integral. Its evaluation yields an explicit algebraic function with a simple average. Indeed, this average Gaussian curvature has long been known. For a non-zero level surface instead of the nodal one, the probability distribution is not fully tractable, but is supplied as an integral expression.