No repulsion between critical points for planar Gaussian random fields

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.

This version of the paper contains the text of the paper accepted for publication in 'Electronic Communications in Probability' and three appendices which show some technical computations. arXiv admin note: text overlap with arXiv:1704.04943