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

Dan Cheng, Armin Schwartzman

Research output: Contribution to journalArticle

9 Scopus citations

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

    Fingerprint

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability

Cite this