Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields

We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of t...

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Bibliographic Details
Published inBernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability Vol. 24; no. 4B; p. 3422
Main Authors Cheng, Dan, Schwartzman, Armin
Format Journal Article
LanguageEnglish
Published England 01.11.2018
Subjects
GOI
Critical points
60G60
Gaussian random fields
Boundary
15B52
Sphere
Height density
Random matrices
Kac-Rice formula
Isotropic
GOE
60G15
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Summary:We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function only through a single parameter (Euclidean space) or two parameters (spheres), and include the special boundary case of random Laplacian eigenfunctions.
ISSN:1350-7265
DOI:10.3150/17-BEJ964