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

Hao Xu, Haoran Yang, Qiang Zeng

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.

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

Abstract

We consider locally isotropic Gaussian random fields on the -dimensional Euclidean space for fixed . 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 , we show that such a representation can be always given by GOE matrices, as conjectured by Auffinger and Zeng in 2020.
Paper Structure (7 sections, 118 equations)

This paper contains 7 sections, 118 equations.

Table of Contents

  1. Introduction
  2. Representations for non-isotropic Gaussian fields on $\mathbb{R}^N$
  3. A perturbed GOI matrix model
  4. The nondegeneracy condition
  5. Representation with GOI matrices
  6. Representation with GOE matrices
  7. Representation for fields with structure functions in $\mathcal{D}_\infty$

Theorems & Definitions (8)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • Remark 3.1