On the expected number of critical points of locally isotropic Gaussian random fields

We consider locally isotropic Gaussian random fields on the $N$-dimensional Euclidean space for fixed $N$. Using the so called Gaussian Orthogonally Invariant matrices first studied by Mallows in 1961 which include the celebrated Gaussian Orthogonal Ensemble (GOE), we establish the Kac–Rice representation of expected number of critical points of non-isotropic Gaussian fields, complementing the isotropic case obtained by Cheng and Schwartzman in 2018. In the limit $N=\mathrm{\infty }$, we show that such a representation can be always given by GOE matrices, as conjectured by Auffinger and Zeng in 2020.