The scaling limit for zero temperature planar Ising droplets: with and without magnetic fields

We consider the continuous time, zero-temperature heat-bath dynamics for the nearest-neighbor Ising model on $Z^2$ with positive magnetic field. For a system of size $L\in N$, we start with initial condition $\sigma$ such that ${\sigma }_x=-1$ if $x\in [-L,L{]}^2$ and ${\sigma }_x=+1$ and investigate the scaling limit of the set of $-$ spins when both time and space are rescaled by $L$. We compare the obtained result and its proof with the case of zero-magnetic fields, for which a scaling result was proved in arXiv:1112.3160. In that case, the time-scaling is diffusive and the scaling limit is given by anisotropic motion by curvature.