On random surface area

TL;DR

This paper derives a formula for the expected surface area of the zero set of a smooth Gaussian field over a compact domain, with applications and auxiliary results for non-random fields.

Contribution

It introduces a new formula for the average surface area of Gaussian field zero sets and provides integral expressions for non-random fields.

Findings

Derived a formula for expected surface area of Gaussian zero sets

Provided integral expressions for surface area of non-random fields

Applied results to specific cases and examples

Abstract

Consider a random smooth Gaussian field $G(x):F\to\mathbb{R}$, where $F$ is a compact in $\mathbb{R}^d$. We derive a formula for average area of a surface generated by the equation $G(x)=0$ and give some applications. As an auxiliary result we obtain an integral expression for area of a surface induced by zeros of a \emph{non-random} smooth field.