On critical points of Gaussian random fields under diffeomorphic transformations

TL;DR

This paper investigates how diffeomorphic transformations affect the critical points of Gaussian random fields on manifolds, revealing proportionality in critical point counts and invariance in height distribution under certain conditions.

Contribution

It provides a theoretical analysis of the impact of diffeomorphic transformations on the critical points and height distribution of Gaussian random fields, especially in the anisotropic case.

Findings

Expected number of critical points scales proportionally under transformations.

Height distribution remains unchanged for anisotropic fields.

Results connect properties of original and transformed Gaussian fields.

Abstract

Let $\{X(t), t\in M\}$ and $\{Z(t'), t'\in M'\}$ be smooth Gaussian random fields parameterized on Riemannian manifolds $M$ and $M'$, respectively, such that $X(t) = Z(f(t))$, where $f: M \to M'$ is a diffeomorphic transformation. We study the expected number and height distribution of the critical points of $X$ in connection with those of $Z$. As an important case, when $X$ is an anisotropic Gaussian random field, then we show that its expected number of critical points becomes proportional to that of an isotropic field $Z$, while the height distribution remains the same as that of $Z$.