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May 2002 Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model
Federico Camia, Emilio De Santis, Charles M. Newman
Ann. Appl. Probab. 12(2): 565-580 (May 2002). DOI: 10.1214/aoap/1026915616


We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of $+1$ or $-1$ to each site in $\mathbf{Z}^2$, is the zero-temperature limit of the stochastic homogeneous Ising ferromagnet (with Glauber dynamics): the initial state is chosen uniformly at random and then each site, at rate 1, polls its four neighbors and makes sure it agrees with the majority, or tosses a fair coin in case of a tie. Among the main results (almost sure, with respect to both the process and initial state) are: clusters (maximal domains of constant sign) are finite for times $t< \infty$, but the cluster of a fixed site diverges (in diameter) as $t \to \infty$; each of the two constant states is (positive) recurrent. We also present other results and conjectures concerning positive and null recurrence and the role of absorbing states.


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Federico Camia. Emilio De Santis. Charles M. Newman. "Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model." Ann. Appl. Probab. 12 (2) 565 - 580, May 2002.


Published: May 2002
First available in Project Euclid: 17 July 2002

zbMATH: 1020.60094
MathSciNet: MR1910640
Digital Object Identifier: 10.1214/aoap/1026915616

Primary: 60J25 , 60K35
Secondary: 82C20 , 82C22

Keywords: Clusters , coarsening , percolation , recurrence , stochastic Ising model , transience , zero-temperature

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.12 • No. 2 • May 2002
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