On critical points of Gaussian random fields under diffeomorphic transformations

2019/11/19 by Dan Cheng, Cheng, Dan, Armin Schwartzman +1
Mathematics · #15B52 #60G15 #60G60 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR) #Statistics Theory (math.ST) #math.PR #math.ST #msc:15B52 #msc:60G15 #msc:60G60 #stat.TH

paper · pdf · doi:10.48550/arxiv.1911.08100

arxiv created 2019/11/19 · openalex publication_date 2019/11/19 · arxiv updated 2019/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \X(t), t∈ M\ and \Z(t'), t'∈ M'\ be smooth Gaussian random fields parameterized on Riemannian manifolds M and M', respectively, such that X(t) = Z(f(t)), where f: M → M' is a diffeomorphic transformation. We study the expected number and height distribution of the critical points of X in connection with those of Z. As an important case, when X is an anisotropic Gaussian random field, then we show that its expected number of critical points becomes proportional to that of an isotropic field Z, while the height distribution remains the same as that of Z.

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