Normal approximation for Gibbs processes via disagreement couplings

TL;DR

This paper advances central limit theorems for Gibbs processes by relaxing stabilization assumptions, extending interaction ranges, and providing quantitative normal approximation bounds through disagreement couplings.

Contribution

It introduces a CLT for weakly stabilizing functionals, extends the interaction range up to the percolation threshold, and develops disagreement couplings for multiple spatial locations.

Findings

CLT for weakly stabilizing functionals established

Interaction range extended to percolation threshold

Quantitative Kolmogorov bounds for normal approximation provided

Abstract

This work improves the existing central limit theorems (CLTs) for geometric functionals of Gibbs processes in three aspects. First, we derive a CLT for weakly stabilizing functionals, thereby improving on the previously used assumption of exponential stabilization. Second, we show that this CLT holds for interaction ranges up to the percolation threshold of the dominating Poisson process. This avoids imprecise branching bounds from graphical construction. Third, by constructing simultaneous couplings of several Palm processes for Gibbs functionals, we provide a quantitative CLT in terms of Kolmogorov bounds for normal approximation. An important conceptual ingredient in these advances is the extension of disagreement coupling adapted to unbounded windows and to the comparison at multiple spatial locations.