Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T11:37:22.425Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Baxter, R. J. (1970). Ornstein–Zernike relation and Percus–Yevick approximation for fluid mixtures. J. Chem. Phys. 52, 4559–4562. CrossRefGoogle Scholar
[2] Coniglio, A., De Angelis, U. and Forlani, A. (1977). Pair connectedness and cluster size. J. Phys. A 10, 1123–1139. CrossRefGoogle Scholar
[3] Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin. CrossRefGoogle Scholar
[4] Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York. Google Scholar
[5] Jörgens, K. (1970). Lineare Integraloperatoren. Teubner, Stuttgart. CrossRefGoogle Scholar
[6] Last, G. (2010). Modern random measures: Palm theory and related models. In New Perspectives in Stochastic Geometry, Oxford University Press, pp. 77–110. Google Scholar
[7] Last, G. (2014). Perturbation analysis of Poisson processes. Bernoulli 20, 486–513. CrossRefGoogle Scholar
[8] Last, G. and Penrose, M. (2017). Lectures on the Poisson Process. Cambridge University Press. CrossRefGoogle Scholar
[9] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press. CrossRefGoogle Scholar
[10] Molchanov, I. and Zuyev, S. (2000). Variational analysis of functionals of Poisson processes. Math. Operat. Res. 25, 485–508. CrossRefGoogle Scholar
[11] Ornstein, L. S. and Zernike, F. (1914). Accidental deviations of density and opalescence at the critical point of a single substance. In Proc. R. Netherlands Acad. Arts Sci., Vol. 17, pp. 793–806. Google Scholar
[12] Otter, R. (1949). The multiplicative process. Ann. Math. Statist. 20, 206–224. CrossRefGoogle Scholar
[13] Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation. Springer, Milan. CrossRefGoogle Scholar
[14] Penrose, M. (2003). Random Geometric Graphs. Oxford University Press. CrossRefGoogle Scholar
[15] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin. CrossRefGoogle Scholar
[16] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York. CrossRefGoogle Scholar
[17] Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York. CrossRefGoogle Scholar
[18] Torquato, S. (2012). Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. J. Chem. Phys. 136, 054106. CrossRefGoogle ScholarPubMed
[19] Ziesche, S. (2016). Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on R d . To appear in Ann. Inst. H. Poincaré Prob. Statist. Google Scholar