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Existence of phase transition of percolation on Sierpiński carpet lattices

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  14 July 2016

Masato Shinoda
Masato Shinoda*
Affiliation:
Nara Women's University
*
∗ Postal address: Department of Mathematics, Faculty of Science, Nara Women's University, Kita-Uoya Nishimachi Nara 630 8506, Japan. Email address: shinoda@cc.nara-wu.ac.jp
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Abstract

We study Bernoulli bond percolation on Sierpiński carpet lattices, which is a class of graphs corresponding to generalized Sierpiński carpets. In this paper we give a sufficient condition for the existence of a phase transition on the lattices. The proof is suitable for graphs which have self-similarity. We also discuss the relation between the existence of a phase transition and the isoperimetric dimension.

Keywords

PercolationSierpiński carpetphase transitionisoperimetric dimension

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B43: Percolation
Secondary: 82B26: Phase transitions (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 1 - 10
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1]. Benjamini, I., and Schramm, O. (1996). Percolation beyond ℤd, many questions and few answers. Electron. Commun. Prob. 1, 71–82.Google Scholar
[2]. Broadbent, S. R., and Hammersley, J. M. (1957). Percolation processes I. Crystals and mazes. Proc. Camb. Phil. Soc. 53, 629–641.Google Scholar
[3]. Chayes, J. T., Chayes, L., and Durrett, R. (1988). Connectivity properties of Mandelbrot's percolation process. Prob. Theory Relat. Fields 77, 307–324.Google Scholar
[4]. Dekking, M., and Meester, R. (1990). On the structure of Mandelbrot's percolation process and other random Cantor sets. J. Statist. Phys. 58, 1109–1126.Google Scholar
[5]. Falconer, K. J. (1985). The Geometry of Fractal Sets. Cambridge University Press.Google Scholar
[6]. Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
[7]. Grimmett, G., and Stacey, A. M. (1998). Critical probabilities for site and bond percolation model. Ann. Prob. 26, 1788–1812.CrossRefGoogle Scholar
[8]. Häggström, O. (2000). Markov random fields and percolation on general graphs. Adv. Appl. Prob. 32, 39–66.CrossRefGoogle Scholar
[9]. Kumagai, T. (1997). Percolation on pre-Sierpiński carpets. In New Trends in Stochastic Analysis (Proc. Taniguchi Internat. Workshop, 1994), eds Elworthy, K. D. et al., World Scientific, River Edge, NJ, pp. 288–304.Google Scholar
[10]. Murai, J. (1997). Percolation in high-dimensional Menger sponges. Kobe J. Math. 14, 49–61.Google Scholar
[11]. Osada, H. (1990). Isoperimetric constants and estimates of heat kernels of pre-Sierpiński carpets. Prob. Theory Relat. Fields 86, 469–490.CrossRefGoogle Scholar
[12]. Shinoda, M. (1996). Percolation on the pre-Sierpiński gasket. Osaka J. Math. 33, 533–554.Google Scholar