Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-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 a U-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for such U-statistics of the Gibbs particle process. A byproduct of our approach is a new uniqueness result for Gibbs particle processes.

References in corpus (7)

• Central limit theorems for $U$-statistics of Poisson point processes
• Perfect simulation for interacting point processes, loss networks and Ising models
• Conditional Intensity and Gibbsianness of Determinantal Point Processes
• Subcritical regimes in the Poisson Boolean model of continuum percolation
• Limit theory for geometric statistics of point processes having fast decay of correlations
• Cluster expansions for Gibbs point processes
• Perturbation analysis of Poisson processes

Cited by in corpus (6)

• Cluster expansions for Gibbs point processes
• On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential
• Revisiting Groeneveld's approach to the virial expansion
• Strong spatial mixing for repulsive point processes
• Structural properties of Gibbsian point processes in abstract spaces
• Hierarchical Cubes: Gibbs Measures and Decay of Correlations