Abstract
We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a ‘generic’ field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.
Highlights
1.1 Two-point correlation function for critical points of planar random fieldsThe number of critical points of a function and their positions are its important qualitative descriptor, and their study is an actively pursued field of research within a wide range of disciplines, such as classical analysis, probability (e.g. [9, 10]), mathematical and theoretical physics ([13]), spectral geometry (e.g. [16, 14]), and cosmology and the study of Cosmic Microwave Background (CMB) radiation (e.g. [11])
We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields
We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal
Summary
The number of critical points of a function and their positions are its important qualitative descriptor, and their study is an actively pursued field of research within a wide range of disciplines, such as classical analysis (see e.g. [8]), probability (e.g. [9, 10]), mathematical and theoretical physics ([13]), spectral geometry (e.g. [16, 14]), and cosmology and the study of Cosmic Microwave Background (CMB) radiation (e.g. [11]). The first relevant result [4] was obtained in 2017 when we analysed the asymptotic behaviour of K2 for a particular Gaussian field: the random monochromatic isotropic plane waves, referred to as “Berry’s Random Wave Model” (RWM). The work [4] allowed for the separation of the critical points into maxima, minima and saddles, and studied the effect of such a separation on the corresponding 2-point correlation function, resulting in some cases in qualitatively different behaviour to (1.3). It is natural to inquire about the analogous question for other Gaussian random fields, i.e. for the asymptotic law of the 2-point-correlation function around the diagonal for other Gaussian random fields Whether it is true, that for a generic stationary field, the critical points nether attract nor repel each other. In the isotropic case there are no mysterious cancellations, suggesting the same for the asymptotic behaviour in the generic case
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