Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T21:07:38.567Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Alm, S. E. and Janson, S. (1990). Random self-avoiding walks on one-dimensional lattices. Commun. Statist. Stoch. Models 6, 169–212.Google Scholar
Axelson-Fisk, M. and Häggström, O. (2008). Biased random walk in a one-dimensional percolation model. Stoch. Process. Appl. 119, 3395–3415.Google Scholar
Berger, N., Gantert, N. and Peres, Y. (2003). The speed of biased random walk on percolation clusters. Prob. Theory Relat. Fields 126, 221–242.CrossRefGoogle Scholar
Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena, eds Domb, C. and Lebowitz, J. L., Academic Press, San Diego, CA, pp. 1–142.Google Scholar
Häggström, O. (1996). The random-cluster model on a homogeneous tree. Prob. Theory Relat. Fields 104, 231–253.Google Scholar
Járai, A. A. (2003). Incipient infinite percolation clusters in 2D. Ann. Prob. 31, 444–485.CrossRefGoogle Scholar
Kesten, H. (1986). The incipient infinite cluster in two-dimensional percolation. Prob. Theory Relat. Fields 73, 369–394.Google Scholar
Sznitman, A. S. (2003). On the anisotropic walk on the supercritical percolation cluster. Commun. Math. Phys. 240, 123–148.Google Scholar
Van den Berg, J. (1993). A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet. Commun. Math. Phys. 152, 161–166.Google Scholar