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The direct-connectedness function in the random connection model

Part of: Geometric probability and stochastic geometry Equilibrium statistical mechanics Stochastic processes Special processes

Published online by Cambridge University Press:  24 August 2022

Sabine Jansen Open the ORCID record for Sabine Jansen [Opens in a new window] ,
Leonid Kolesnikov Open the ORCID record for Leonid Kolesnikov [Opens in a new window]  and
Kilian Matzke
Sabine Jansen*
Affiliation:
Ludwig-Maximilians-Universität München
Leonid Kolesnikov*
Affiliation:
Ludwig-Maximilians-Universität München
Kilian Matzke*
Affiliation:
Ludwig-Maximilians-Universität München
*
*Postal address: Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333 München, Germany.
*Postal address: Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333 München, Germany.
*Postal address: Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333 München, Germany.
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Abstract

We investigate expansions for connectedness functions in the random connection model of continuum percolation in powers of the intensity. Precisely, we study the pair-connectedness and the direct-connectedness functions, related to each other via the Ornstein–Zernike equation. We exhibit the fact that the coefficients of the expansions consist of sums over connected and 2-connected graphs. In the physics literature, this is known to be the case more generally for percolation models based on Gibbs point processes and stands in analogy to the formalism developed for correlation functions in liquid-state statistical mechanics.

We find a representation of the direct-connectedness function and bounds on the intensity which allow us to pass to the thermodynamic limit. In some cases (e.g., in high dimensions), the results are valid in almost the entire subcritical regime. Moreover, we relate these expansions to the physics literature and we show how they coincide with the expression provided by the lace expansion.

Keywords

Ornstein–Zernike equationrandom connection modelconnectedness functionsPoisson processpercolationgraphical expansionslace expansion

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60G55: Point processes 82B43: Percolation 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 179 - 222
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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