On critical points of Gaussian random fields under diffeomorphic transformations

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.