Local behavior of critical points of isotropic Gaussian random fields

2023/10/19 by Paul Marriott, Marriott, Paul, Weinan Qi +3
Mathematics · #60G60(primary) 60G70(secondary) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2310.12843

openalex publication_date 2023/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we examine isotropic Gaussian random fields defined on \mathbb R^N satisfying certain conditions. Specifically, we investigate the type of a critical point situated within a small vicinity of another critical point, with both points surpassing a given threshold. It is shown that the Hessian of the random field at such a critical point is equally likely to have a positive or negative determinant. Furthermore, as the threshold tends to infinity, almost all the critical points above the threshold are local maxima and the saddle points with index N-1. Consequently, we conclude that the closely paired critical points above a high threshold must comprise one local maximum and one saddle point with index N-1.

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