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Continuity of percolation probability in ∞ + 1 dimensions

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Yu Zhang
Yu Zhang*
Affiliation:
University of Colorado
*
∗Postal address: Department of Mathematics, University of Colorado, Colorado Springs, CO 80933, USA. Supported by a grant from the NSF.
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Abstract

We consider percolation on the graph of the product of a regular tree T with degree d and the line ℤ, in which each tree edge is open with probability 1 – exp(–JTß) and each line edge is open with probability . Let C(o, 0) be the open cluster for . Denote by θ (β) the percolation probability. Here we show that θ (β) is continuous when ß > ßc, where ßc = sup{ß : θ(β) = 0}.

Keywords

PERCOLATIONTREES

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 427 - 433
Copyright
Copyright © Applied Probability Trust 1996 

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References

van den Berg, J. and Keane, M. (1984) On the continuity of the percolation probability function. Ann. Prob. 19, 1520–1536.Google Scholar
Grimmett, G. (1989) Percolation. Springer, Berlin.CrossRefGoogle Scholar
Grimmett, G. and Marstrand, J. (1991) The supercritical phase of percolation is well behaved. Proc. R. Soc. London. A 430, 439–457.Google Scholar
Grimmett, G. and Newman, C. (1989) Percolation in 8 + 1 dimensions. In Disorder in Physical Systems. ed. Grimmett, G. and Welsh, D., pp. 167–190. Oxford University Press, Oxford.Google Scholar
Pemantle, R. (1990) The contact process on trees. Ann. Prob. 20, 2089–2116.Google Scholar
Russo, L. (1978) A note on percolation. Z. Wahrscheinlichkeitsth. 43, 39–48.CrossRefGoogle Scholar
Wu, C. (1993) Critical behavior of percolation and Markov fields on branching planes. J. Appl. Prob. 30, 538–547.CrossRefGoogle Scholar