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

TL;DR

This paper derives formulas for the expected number of critical points in locally isotropic Gaussian fields using GOE matrices, extending previous work to non-isotropic cases and infinite dimensions.

Contribution

It introduces a Kac-Rice representation for non-isotropic Gaussian fields and confirms GOE matrices describe the limit as dimension tends to infinity.

Findings

Kac-Rice formula for non-isotropic fields established

Expected critical points linked to GOE matrices in high dimensions

Extension of isotropic results to more general Gaussian fields

Abstract

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 $N=\infty$, we show that such a representation can be always given by GOE matrices, as conjectured by Auffinger and Zeng in 2020.