  We study the coarsening model (zero-temperature Ising Glauber dynamics) on
$\mathbb{Z}^d$ (for $d \geq 2$) with an asymmetric tie-breaking rule. This is a
Markov process on the state space $\{-1,+1\}^{\mathbb{Z}^d}$ of "spin
configurations" in which each vertex updates its spin to agree with a majority
of its neighbors at the arrival times of a Poisson process. If a vertex has
equally many $+1$ and $-1$ neighbors, then it updates its spin value to $+1$
with probability $q \in [0,1]$ and to $-1$ with probability $1-q$. The initial
state of this Markov chain is distributed according to a product measure with
probability $p$ for a spin to be $+1$. In this paper, we show that for any
given $p>0$, there exist $q$ close enough to 1 such that a.s. every spin has a
limit of $+1$. This is of particular interest for small values of $p$, for
which it is known that if $q=1/2$, a.s. all spins have a limit of $-1$. For
dimension $d=2$, we also obtain near-exponential convergence rates for $q$
sufficiently large, and for general $d$, we obtain stretched exponential rates
independent of $d$. Two important ingredients in our proofs are refinements of
block arguments of Fontes-Schonmann-Sidoravicius and a novel exponential large
deviation bound for the Asymmetric Simple Exclusion Process.
