  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 ${\bf Z}^2$, 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 one, polls its 4 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.
