Expected number and height distribution of critical points of smooth isotropic Gaussian random fields

Dan Cheng, Armin Schwartzman

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)3422-3446
Number of pages25
JournalBernoulli
Volume24
Issue number4B
DOIs
StatePublished - Nov 2018
Externally publishedYes

Fingerprint

Gaussian Random Field
Covariance Function
Euclidean space
Critical point
Gaussian Fields
Isotropy
Eigenfunctions
Two Parameters
Smoothness
Ensemble
Restriction
Invariant
Arbitrary
Family

Keywords

  • Boundary
  • Critical points
  • Gaussian random fields
  • GOE
  • GOI
  • Height density
  • Isotropic
  • Kac–Rice formula
  • Random matrices
  • Sphere

ASJC Scopus subject areas

  • Statistics and Probability

Cite this

Expected number and height distribution of critical points of smooth isotropic Gaussian random fields. / Cheng, Dan; Schwartzman, Armin.

In: Bernoulli, Vol. 24, No. 4B, 11.2018, p. 3422-3446.

Research output: Contribution to journalArticle

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