Correlations and screening of topological charges in gaussian random fields

TL;DR

This paper explicitly calculates the correlation functions of topological charges in Gaussian random fields for various geometric singularities, revealing their dependence on the field's spatial correlation and their adherence to the Stillinger-Lovett sum rule.

Contribution

It introduces a general scheme for calculating topological charge correlations in Gaussian fields for different singularities, connecting topological properties with statistical physics.

Findings

Correlation functions depend on the underlying spatial correlation.

Topological charge correlations obey the Stillinger-Lovett sum rule.

Explicit formulas are derived for zeros, critical points, and umbilic points.

Abstract

2-point topological charge correlation functions of several types of geometric singularity in gaussian random fields are calculated explicitly, using a general scheme: zeros of $n$-dimensional random vectors, signed by the sign of their jacobian determinant; critical points (gradient zeros) of real scalars in two dimensions signed by the hessian; and umbilic points of real scalars in two dimensions, signed by their index. The functions in each case depend on the underlying spatial correlation function of the field. These topological charge correlation functions are found to obey the first Stillinger-Lovett sum rule for ionic fluids.