## 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 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 < ∞, 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 language | English (US) |
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Pages (from-to) | 565-580 |

Number of pages | 16 |

Journal | Annals of Applied Probability |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - May 2002 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty