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

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...

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Bibliographic Details
Published in:Stochastic processes and their applications 2022-08, Vol.150, p.411-445
Main Authors: Azaïs, Jean-Marc, Delmas, Céline
Format: Article
Language:English
Subjects:
Critical points
Gaussian fields
GOE matrices
Kac–Rice formula
Mathematics
Probability
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Summary: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.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2022.04.013