No repulsion between critical points for planar Gaussian random fields

TL;DR

This paper investigates the spatial distribution of critical points in isotropic stationary Gaussian fields, revealing that, generally, these points neither repel nor attract each other, and analyzing how their interactions depend on their index.

Contribution

It provides the first detailed asymptotic analysis of the two-point correlation function of critical points in Gaussian fields, showing their neutral interaction behavior.

Findings

Critical points do not exhibit repulsion or attraction in generic Gaussian fields.

The short-range behavior of critical points depends on their index.

The main term of the two-point correlation function near the diagonal is computed.

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.