Stabilization and limit theorems for geometric functionals of Gibbs point processes

TL;DR

This paper establishes limit theorems for geometric functionals of Gibbs point processes, demonstrating convergence to Gaussian fields and laws of large numbers under certain conditions, with applications to various models.

Contribution

It introduces new limit laws for geometric functionals of Gibbs point processes, including the Strauss and area interaction models, under weak potential and localization assumptions.

Findings

Weak laws of large numbers for scaled measures

Weak convergence to Gaussian fields for centered measures

Applicability to diverse Gibbsian models and functionals

Abstract

Given a Gibbs point process $\P^{\Psi}$ on $\R^d$ having a weak enough potential $\Psi$, we consider the random measures $\mu_\la := \sum_{x \in \P^{\Psi} \cap Q_\la} \xi(x, \P^{\Psi} \cap Q_\la) \delta_{x/\la^{1/d}}$, where $Q_{\la} := [-\la^{1/d}/2,\la^{1/d}/2]^d$ is the volume $\la$ cube and where $\xi(\cdot,\cdot)$ is a translation invariant stabilizing functional. Subject to $\Psi$ satisfying a localization property and translation invariance, we establish weak laws of large numbers for $\la^{-1} \mu_\la(f)$, $f$ a bounded test function on $\R^d$, and weak convergence of $\la^{-1/2} \mu_\la(f),$ suitably centered, to a Gaussian field acting on bounded test functions. The result yields limit laws for geometric functionals on Gibbs point processes including the Strauss and area interaction point processes as well as more general point processes defined by the Widom-Rowlinson and hard-core model. We provide applications to random sequential packing on Gibbsian input, to functionals of Euclidean graphs, networks, and percolation models on Gibbsian input, and to quantization via Gibbsian input.