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

Dan Cheng; Armin Schwartzman

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arXiv · 1511.06835

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

Dan Cheng·Armin Schwartzman

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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 (sphere), and include the special boundary case of random Laplacian eigenfunctions.

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Dan Cheng, Armin Schwartzman. 2016-09-19. Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields. https://arxiv.org/abs/1511.06835

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