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Kozma, Gady

Percolation, perimetry, planarity. (English) Zbl 1131.60087

Rev. Mat. Iberoam. 23, No. 2, 671-676 (2007).
Summary: Let \(G\) be a planar graph with polynomial growth and isoperimetric dimension bigger than 1. Then the critical \(p\) for Bernoulli percolation on \(G\) satisfies \(p_{c} < 1\).

Cited in 5 Documents

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
05C10 Planar graphs; geometric and topological aspects of graph theory

Keywords:

percolation; isoperimetric dimension; planar graph; duality
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\textit{G. Kozma}, Rev. Mat. Iberoam. 23, No. 2, 671--676 (2007; Zbl 1131.60087)
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References:

[1] Benjamini, I. and Schramm, O.: Percolation beyond \(\mathbbZ^d\), many questions and a few answers. Electron. Comm. Probab. 1 (1996), no. 8, 71-82. · Zbl 0890.60091
[2] Bruhn, H. and Diestel, R.: Duality in infinite graphs. Combin. Probab. Comput. 15 (2006), no. 1-2, 75-90. · Zbl 1082.05028
[3] Grimmett, G.: Percolation . Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321 . Springer-Verlag, Berlin, 1999.
[4] Procacci, A. and Scoppola, B.: Infinite graphs with a nontrivial bond percolation threshold: some sufficient conditions. J. Statist. Phys. 115 (2004), no. 3-4, 1113-1127. · Zbl 1052.82014
[5] Thomassen, C.: Straight line representations of infinite planar graphs. J. London Math. Soc. (2) 16 (1977), no. 3, 411-423. · Zbl 0373.05032
[6] Thomassen, C.: Planarity and duality of finite and infinite graphs. J. Combin. Theory Ser. B 29 (1980), no. 2, 244-271. · Zbl 0441.05023
[7] Thomassen, C.: Duality of infinite graphs. J. Combin. Theory Ser. B 33 (1982), no. 2, 137-160. · Zbl 0501.05054
[8] Wagner, K.: Fastplättbare Graphen. (German). J. Combinatorial Theory 3 (1967), 326-365. · Zbl 0153.54101
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