A covariance formula for the number of excursion set components of Gaussian fields and applications

TL;DR

This paper develops a covariance formula for counting excursion set components of Gaussian fields, enabling precise variance analysis for large domains and different correlation decay behaviors, with applications to understanding the geometry of Gaussian fields.

Contribution

It introduces a new covariance formula for the number of excursion set components of Gaussian fields and applies it to derive variance bounds based on correlation decay rates.

Findings

Variance of component count is proportional to volume for integrable correlations.

Provides an upper bound on variance for slowly decaying correlations.

Improves bounds for oscillating correlation cases like monochromatic random waves.

Abstract

We derive a covariance formula for the number of excursion or level set components of a smooth stationary Gaussian field on $\mathbb{R}^d$ contained in compact domains. We also present two applications of this formula: (1) for fields whose correlations are integrable we prove that the variance of the component count in large domains is of volume order and give an expression for the leading constant, and (2) for fields with slower decay of correlation we give an upper bound on the variance which is of optimal order if correlations are regularly varying, and improves on best-known bounds if correlations are oscillating (e.g.\ monochromatic random waves).