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Decorrelation of a class of Gibbs particle processes and asymptotic properties of \(U\)-statistics. (English) Zbl 1457.60066
Summary: 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 by-product of our approach is a new uniqueness result for Gibbs particle processes.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
60D05 Geometric probability and stochastic geometry
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