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

Hao Xu; Haoran Yang; Qiang Zeng

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arXiv · 2307.12281

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

Hao Xu·Haoran Yang·Qiang Zeng

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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.

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BibTeX RIS

Hao Xu, Haoran Yang, Qiang Zeng. 2023-07-23. On the expected number of critical points of locally isotropic Gaussian random fields. https://arxiv.org/abs/2307.12281

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