Expected number density of critical points of smooth Gaussian random fields in arbitrary dimensions

2026/07/27 by Satoshi Kuriki, Takahiko Matsubara, Satoshi Iso
#math.ST #astro-ph.CO #math.PR #stat.TH

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Abstract

We obtain explicit formulas for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields on ℝ^d. The expected number density formula is expressed in terms of at most one-dimensional integrals, regardless of the dimension d. To obtain the formulas, we provide a variant of de Bruijn's theorem, as well as Weierstrass' convolution formula with a Gaussian random variable and its inversion.

Citations

• Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields
• Statistical Topology and the Random Interstellar Medium
• Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields
• Random Matrices and complexity of Spin Glasses

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