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

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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 Z2, 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.

Original languageEnglish (US)
Pages (from-to)565-580
Number of pages16
JournalAnnals of Applied Probability
Volume12
Issue number2
DOIs
StatePublished - May 2002

Keywords

  • Clusters
  • Coarsening
  • Percolation
  • Recurrence
  • Stochastic Ising model
  • Transience
  • Zero-temperature

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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