Journal of Applied Probability

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

Günter Last and Eva Ochsenreither

Abstract

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 ZW, 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.

Article information

Source
J. Appl. Probab., Volume 51A (2014), 311-332.

Dates
First available in Project Euclid: 2 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1417528483

Digital Object Identifier
doi:10.1239/jap/1417528483

Mathematical Reviews number (MathSciNet)
MR3317366

Zentralblatt MATH identifier
1314.60046

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Tessellation percolation Poisson-Voronoi tessellation intrinsic volume the Euler characteristic asymptotic mean asymptotic covariance

Citation

Last, Günter; Ochsenreither, Eva. Percolation on stationary tessellations: models, mean values, and second-order structure. J. Appl. Probab. 51A (2014), 311--332. doi:10.1239/jap/1417528483. https://projecteuclid.org/euclid.jap/1417528483


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References

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