Decorrelation of a class of Gibbs particle processes and asymptotic properties ofU-statistics

Abstract

We study a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials of finite range. Using disagreement percolation, we prove exponential decay of the correlation functions, provided a dominating Boolean model is subcritical.

We also prove this property for the weighted moments of aU-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for suchU-statistics of the Gibbs particle process.

A by-product of our approach is a new uniqueness result for Gibbs particle processes.

Keywords

• Gibbs process
• particle process
• pair potential
• disagreement percolation
• correlation functions
• central limit theorem
• U-statistics