The distribution of extremal points of Gaussian scalar fields

TL;DR

This paper analyzes the distribution and correlations of extremal points in Gaussian scalar fields, providing analytical results for their density, charge correlations, and applications to random waves and surfaces.

Contribution

It introduces a comprehensive framework for computing extremal point densities and correlations in Gaussian fields, including boundary effects and a novel generating functional.

Findings

Derived the average density of extremal points in half-space with boundary conditions.

Calculated charge-charge correlation functions for Gaussian scalar fields.

Connected the two-point function generating functional to scalar curvature in Riemannian geometry.

Abstract

We consider the signed density of the extremal points of (two-dimensional) scalar fields with a Gaussian distribution. We assign a positive unit charge to the maxima and minima of the function and a negative one to its saddles. At first, we compute the average density for a field in half-space with Dirichlet boundary conditions. Then we calculate the charge-charge correlation function (without boundary). We apply the general results to random waves and random surfaces. Furthermore, we find a generating functional for the two-point function. Its Legendre transform is the integral over the scalar curvature of a 4-dimensional Riemannian manifold.