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

Part of: Geometric probability and stochastic geometry Stochastic processes Special processes

Published online by Cambridge University Press:  17 November 2017

Günter Last  and
Sebastian Ziesche
Günter Last*
Affiliation:
Karlsruhe Institute of Technology
Sebastian Ziesche*
Affiliation:
Karlsruhe Institute of Technology
*
* Postal address: Institute of Stochastics, Karlsruhe Institute of Technology, Englerstr. 2, D-76131 Karlsruhe, Germany.
* Postal address: Institute of Stochastics, Karlsruhe Institute of Technology, Englerstr. 2, D-76131 Karlsruhe, Germany.
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Abstract

In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. 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.

Keywords

Ornstein–Zernike equationGilbert graphrandom connection modelpair-connectedness functionPoisson processpercolationanalyticity

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017 , pp. 1260 - 1287
Copyright
Copyright © Applied Probability Trust 2017 

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