On critical points of Gaussian random fields under diffeomorphic transformations

Abstract

Let $\{X(t),t\in M\}$ and $\{Z(t^{\prime }),t^{\prime }\in M^{\prime }\}$ be smooth Gaussian random fields parameterized on Riemannian manifolds $M$ and $M^{\prime }$, respectively, such that $X(t)=Z(f(t))$, where $f:M\to M^{\prime }$ 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 $Z$.