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

Annals of Applied Probability, ISSN: 1050-5164, Vol: 12, Issue: 2, Page: 565-580
2002
  • 13
    Citations
  • 0
    Usage
  • 3
    Captures
  • 0
    Mentions
  • 0
    Social Media
Metric Options:   Counts1 Year3 Year

Metrics Details

  • Citations
    13
    • Citation Indexes
      13
  • Captures
    3

Article Description

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 Z, 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 < ∞, but the cluster of a fixed site diverges (in diameter) as t → ∞; 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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