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{1026915616,
author = {Camia, Federico and De Santis, Emilio and Newman, Charles M.},
title = {Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model},
journal = {Ann. Appl. Probab.},
volume = {12},
number = {1},
year = {2002},
pages = { 565-580},
language = {en},
url = {http://dml.mathdoc.fr/item/1026915616}
}
Camia, Federico; De Santis, Emilio; Newman, Charles M. Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model. Ann. Appl. Probab., Tome 12 (2002) no. 1, pp. 565-580. http://gdmltest.u-ga.fr/item/1026915616/