Critical point correlations in random gaussian fields

TL;DR

This paper analyzes the spatial correlations of critical points in random Gaussian fields, deriving asymptotic behaviors for various densities across different dimensions, with explicit formulas for 2D and 3D cases.

Contribution

It provides a comprehensive theoretical framework for understanding critical point correlations in Gaussian fields across arbitrary dimensions, including explicit formulas for 2D and 3D.

Findings

Asymptotic limits of two-point correlation functions are derived for various critical point densities.

Explicit formulas for critical point correlations are provided for two and three dimensions.

Numerical calculations verify the theoretical results.

Abstract

We consider fluctuations in the distribution of critical points - saddle points, minima and maxima - of random gaussian fields. We calculate the asymptotic limits of the two point correlation function for various critical point densities, for both long and short range. We perform the calculation for any dimension of the field, provide explicit formulae for two and three dimensions, and verify our results with numerical calculations.