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Ergodicity and mixing properties of the northeast model

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

George Kordzakhia  and
Steven P. Lalley
George Kordzakhia
Affiliation:
University of California, Berkeley
Steven P. Lalley*
Affiliation:
University of Chicago
*
∗∗Postal address: Department of Statistics, University of Chicago, 5734 University Avenue, Chicago, IL 60637, USA. Email address: lalley@galton.uchicago.edu
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Abstract

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The northeast model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbors have spin 1, it may reset its own spin by tossing a p-coin; at all other times, its spin remains frozen. It is proved that the northeast model has a phase transition at pc = 1 - βc, where βc is the critical parameter for oriented percolation. For p < pc, the trivial measure, δ0, that puts mass one on the configuration with all spins set at 0 is the unique ergodic, translation-invariant, stationary measure. For p ≥ pc, the product Bernoulli-p measure on configuration space is the unique nontrivial, ergodic, translation-invariant, stationary measure for the system, and it is mixing. For p > ⅔, it is shown that there is exponential decay of correlations.

Keywords

Northeast modelfacilitated spin-flip systemoriented percolationexponential mixing

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 782 - 792
Copyright
© Applied Probability Trust 2006 

Footnotes

∗

Current address: Food and Drug Administration, Center for Drug Evaluation and Research, Division of Biometrics 1, Building 22, Room 4235, 10903 New Hampshire Avenue, Silver Spring, MD 20993-0002, USA. Email address: kordzakh@stat.berkeley.edu

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