The Annals of Applied Probability

Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model

Federico Camia, Emilio De Santis, and Charles M. Newman

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 12, Number 2 (2002), 565-580.

Dates
First available in Project Euclid: 17 July 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1026915616

Digital Object Identifier
doi:10.1214/aoap/1026915616

Mathematical Reviews number (MathSciNet)
MR1910640

Zentralblatt MATH identifier
1020.60094

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J25: Continuous-time Markov processes on general state spaces
Secondary: 82C22: Interacting particle systems [See also 60K35] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

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

Citation

Camia, Federico; De Santis, Emilio; Newman, Charles M. Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model. Ann. Appl. Probab. 12 (2002), no. 2, 565--580. doi:10.1214/aoap/1026915616. https://projecteuclid.org/euclid.aoap/1026915616


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  • NEW YORK, NY 10003 E-MAIL: federico.camia@physics.ny u.edu E. DE SANTIS DIPARTIMENTO DI MATEMATICA "GUIDO CASTELNUOVO" UNIVERSITÀ DI ROMA "LA SAPIENZA" PIAZZALE ALDO MORO 2 00185 ROMA ITALY E-MAIL: desantis@mat.uniroma1.it C. M. NEWMAN COURANT INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY
  • NEW YORK, NY 10012 E-MAIL: newman@courant.ny u.edu