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Conditional percolation on one-dimensional lattices

Part of: Equilibrium statistical mechanics Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Marina Axelson-Fisk  and
Olle Häggström
Marina Axelson-Fisk*
Affiliation:
Chalmers University of Technology
Olle Häggström*
Affiliation:
Chalmers University of Technology
*
∗ Postal address: Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden.
∗ Postal address: Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden.
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Abstract

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Conditioning independent and identically distributed bond percolation with retention parameter p on a one-dimensional periodic lattice on the event of having a bi-infinite path from -∞ to ∞ is shown to make sense, and the resulting model exhibits a Markovian structure that facilitates its analysis. Stochastic monotonicity in p turns out to fail in general for this model, but a weaker monotonicity property does hold: the average edge density is increasing in p.

Keywords

Conditional percolationstochastic dominationone-dimensional latticeMarkov chain

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B43: Percolation 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 1102 - 1122
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Research supported by the Swedish Research Council.

∗∗∗

Research supported by the Swedish Research Council and by the Göran Gustafsson Foundation for Research in the Natural Sciences and Medicine.

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