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Extreme paths in oriented two-dimensional percolation

Part of: Markov processes Special processes

Published online by Cambridge University Press:  21 June 2016

E. D. Andjel  and
L. F. Gray
E. D. Andjel*
Affiliation:
Université d'Aix-Marseille
L. F. Gray*
Affiliation:
University of Minnesota
*
* Postal address: Université d'Aix-Marseille, 39 Rue Joliot Curie, 13453 Marseille, France. Email address: andjel@impa.br
** Postal address: School of Mathematics, University of Minnesota, Minneapolis, MN 55455-0488, USA. Email address: gray@math.umn.edu
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Abstract

A useful result about leftmost and rightmost paths in two-dimensional bond percolation is proved. This result was introduced without proof in Gray (1991) in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete-time contact process and two-dimensional site percolation. Among the consequences are a natural monotonicity in the probability of percolation between different sites and a somewhat counter-intuitive correlation inequality.

Keywords

Oriented percolationextreme pathsinequalities

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 369 - 380
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1]Andjel, E. D. and Gray, L. F. (2015).Extreme paths in oriented 2D percolation. Preprint. Available at http://arxiv.org/abs/1411.0956.Google Scholar
[2]Andjel, E. D. and Sued, M. (2008).An inequality for oriented 2-D percolation. In In and Out of Equilibrium (Progr. Prob.60), Birkhäuser, Basel, pp.21–30.Google Scholar
[3]Durrett, R. (1984).Oriented percolation in two dimensions.Ann. Prob. 12, 999–1040.Google Scholar
[4]Gray, L. F. (1991).Is the contact process dead? In Mathematics of Random Media (Lectures Appl. Math.27), American Mathematical Society, Providence, RI, pp.19–29.Google Scholar
[5]Van den Berg, J., Häggström, O. and Kahn, J. (2006).Proof of a conjecture of N. Konno for 1D contact process. In Dynamics & Stochastics (IMS Lecture Notes Monogr. Ser.48), Institute of Statistical Mathematics, Beachwood, OH, pp.16–23.CrossRefGoogle Scholar