Local repulsion of planar Gaussian critical points

TL;DR

This paper investigates the local interactions between critical points of planar Gaussian fields, revealing conditions under which they exhibit soft or hard repulsion or attraction, with precise quantification in specific models.

Contribution

It provides a detailed analysis of the local repulsion and attraction phenomena among critical points in Gaussian fields, including explicit quantification of these effects.

Findings

Critical points can experience soft repulsion or attraction depending on the model.

In the random planar wave model, repulsion is maximal.

For specific critical point types, the paper quantifies the magnitude of local repulsion.

Abstract

We study the local repulsion between critical points of a stationary isotropic smooth planar Gaussian field. We show that the critical points can experience a soft repulsion which is maximal in the case of the random planar wave model, or a soft attraction of arbitrary high order. If the type of critical points is specified (extremum, saddle point), the points experience a hard local repulsion, that we quantify with the precise magnitude of the second factorial moment of the number of points in a small ball.