On the Ornstein-Zernike equation for stationary cluster processes and the random connection model

TL;DR

This paper analyzes the Ornstein-Zernike equation for stationary cluster processes, providing solutions under general conditions and exploring properties in the Poisson random connection model, advancing understanding in spatial stochastic models.

Contribution

It offers a general solution to the Ornstein-Zernike equation for stationary subcritical cluster models and examines its properties in the Poisson random connection framework.

Findings

Solved the Ornstein-Zernike equation under broad assumptions

Analyzed the properties of the OZE solution in the Poisson connection model

Extended the understanding of cluster process correlations

Abstract

In the first part of this paper we consider a general stationary subcritical cluster model in $\mathbb{R}^d$. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein-Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE-solution in the special case of a Poisson driven random connection model.