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Percolation on stationary tessellations: models, mean values, and second-order structure

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

Published online by Cambridge University Press:  30 March 2016

Günter Last  and
Eva Ochsenreither
Günter Last
Affiliation:
Institut für Stochastik, Karlsruher Institut für Technologie, 76128 Karlsruhe, Germany. Email address: guenter.last@kit.edu.
Eva Ochsenreither
Affiliation:
Institut für Stochastik, Karlsruher Institut für Technologie, 76128 Karlsruhe, Germany. Email address: e.ochsenreither@kit.edu.
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Abstract

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We consider a stationary face-to-face tessellation X of Rd and introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability p and white otherwise. We are interested in geometric properties of the union Z of black faces. Under natural integrability assumptions, we first express asymptotic mean values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes of Z ∩ W, where the observation window W is assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes of X. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.

Keywords

TessellationpercolationPoisson-Voronoi tessellationintrinsic volumethe Euler characteristicasymptotic meanasymptotic covariance

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Part 7. Stochastic geometry
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 311 - 332
Copyright
Copyright © Applied Probability Trust 2014 

References

Baryshnikov, Y., and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Prob. 15, 213–253.Google Scholar
Baumstark, V., and Last, G. (2007). Some distributional results for Poisson-Voronoi tessellations. Adv. Appl. Prob. 39, 16–40.Google Scholar
Bollobás, B., and Riordan, O. (2006). The critical probability for random Voronoi percolation in the plane is 1/2. Prob. Theory Relat. Fields 136, 417–468.CrossRefGoogle Scholar
Groemer, H. (1972). Eulersche charakteristik, projektionen und quermassintegrale. Math. Ann. 198, 23–56.CrossRefGoogle Scholar
Heinrich, L., and Muche, L. (2008). Second-order properties of the point process of nodes in a stationary Voronoi tessellation. Math. Nachr. 281, 350–375.Google Scholar
Hug, D., and Schneider, R. (2007). Asymptotic shapes of large cells in random tessellations. Geom. Funct. Anal. 17, 156–191.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Last, G. (2010). Modern random measures: Palm theory and related models. In New Perspectives in Stochastic Geometry, eds Kendall, W. and Molchanov, I., Oxford University Press, pp. 77–110.Google Scholar
Last, G., and Thorisson, H. (2009). Invariant transports of stationary random measures and mass-stationarity. Ann. Prob. 37, 790–813.Google Scholar
Neher, R. A., Mecke, K., and Wagner, H. (2008). Topological estimation of percolation thresholds. J. Statist. Mech. Theory Exp. 2008, 14 pp.CrossRefGoogle Scholar
Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Prob. 12, 989–1035.Google Scholar
Schneider, R., and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Wu, L. (2000). A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Prob. Theory Relat. Fields 118, 427–438.CrossRefGoogle Scholar
Zuyev, S. (1999). Stopping sets: gamma-type results and hitting properties. Adv. Appl. Prob. 31, 355–366.Google Scholar