Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices

Jean-Marc Azaïs and Céline Delmas

Stochastic Processes and their Applications, 2022, vol. 150, issue C, 411-445

Abstract: Let X={X(t):t∈RN} be an isotropic Gaussian random field with real values. The first part studies the mean number of critical points of X with index k using random matrices tools. An exact expression for the probability density of the kth eigenvalue of a N-GOE matrix is obtained. We deduce some exact expressions for the mean number of critical points with a given index. A second part studies the attraction or repulsion between these critical points. A measure is the correlation function. We prove attraction between critical points when N>2, neutrality for N=2 and repulsion for N=1. The attraction between critical points that occurs when the dimension is greater than two is due to critical points with adjacent indexes. A strong repulsion between maxima and minima is proved. The correlation function between maxima (or minima) depends on the dimension of the ambient space.

Keywords: Critical points; Gaussian fields; Kac–Rice formula; GOE matrices (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2022.04.013

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