• Bernoulli
  • Volume 24, Number 4B (2018), 3422-3446.

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

Dan Cheng and Armin Schwartzman

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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.

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Bernoulli, Volume 24, Number 4B (2018), 3422-3446.

Received: January 2017
Revised: June 2017
First available in Project Euclid: 18 April 2018

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boundary critical points Gaussian random fields GOE GOI height density isotropic Kac–Rice formula random matrices sphere


Cheng, Dan; Schwartzman, Armin. Expected number and height distribution of critical points of smooth isotropic Gaussian random fields. Bernoulli 24 (2018), no. 4B, 3422--3446. doi:10.3150/17-BEJ964.

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