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  1. Home
  3. On Critical Points Of Gaussian Random Fields Under Diffeomorphic Transformations.
  1. Home
  3. On Critical Points Of Gaussian Random Fields Under Diffeomorphic Transformations.

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On critical points of Gaussian random fields under diffeomorphic transformations.

Dan Cheng1, Armin Schwartzman2

  • 1Arizona State University.

Statistics & Probability Letters
|December 19, 2024

View abstract on PubMed

Summary
This summary is machine-generated.

Researchers studied critical points of Gaussian random fields on Riemannian manifolds. They found that for anisotropic fields, the number of critical points scales with isotropic fields, while height distributions remain unchanged.

Keywords:
15B5260G1560G60AnisotropicCritical pointsDiffeomorphic transformationGaussian random fieldsHeight distributionIsotropic

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Area of Science:

  • Differential Geometry
  • Probability Theory
  • Stochastic Processes

Background:

  • Gaussian random fields (GRFs) are fundamental in various scientific domains.
  • Analyzing critical points of GRFs on manifolds is crucial for understanding their topological and geometric properties.
  • Diffeomorphic transformations preserve the topological structure of manifolds.

Purpose of the Study:

  • To investigate the relationship between the critical points of two smooth Gaussian random fields on Riemannian manifolds.
  • To determine how diffeomorphic transformations affect the expected number and height distribution of critical points.
  • To analyze the specific case of anisotropic Gaussian random fields.

Main Methods:

  • Utilizing concepts from differential geometry and stochastic calculus.
  • Applying integral geometry formulas to compute expected Betti numbers.
  • Analyzing the properties of Gaussian distributions under diffeomorphic mappings.

    • Main Results:

    • Established a connection between the critical points of a GRF and those of another GRF under a diffeomorphic transformation.
    • Demonstrated that for an anisotropic GRF, the expected number of critical points is proportional to that of an associated isotropic GRF.
    • Showed that the height distribution of critical points for an anisotropic GRF is identical to that of the corresponding isotropic GRF.

    Conclusions:

    • The study provides new insights into the geometric properties of Gaussian random fields on manifolds.
    • The findings simplify the analysis of critical points for anisotropic GRFs by relating them to simpler isotropic cases.
    • This research has implications for fields utilizing random field theory, such as cosmology and statistical physics.