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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: [email protected]
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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: [email protected]

References

Aldous, D. and Diaconis, P. (2002). The asymmetric one-dimensional constrained Ising model: rigorous results. J. Statist. Phys. 107, 945–975.CrossRefGoogle Scholar
Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Prob. 9, 583–603.CrossRefGoogle Scholar
Fredrickson, G. H. and Andersen, H. C. (1984). Kinetic Ising models of the glass transition. Phys. Rev. Lett. 53, 1244–1247.CrossRefGoogle Scholar
Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.CrossRefGoogle Scholar
Pitts, S. J., Young, T. and Andersen, H. C. (2000). Facilitated spin models, mode coupling theory, and ergodic–nonergodic transitions. J. Chem. Phys. 113, 8671–8679.CrossRefGoogle Scholar
Richardson, D. (1973). Random growth in tessellation. Proc. Camb. Phil. Soc. 74, 515–528.CrossRefGoogle Scholar