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

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We consider the continuous time, zero-temperature heat-bath dynamics for the nearest-neighbor Ising model on Z² with positive magnetic field. For a system of size L∈ N, we start with initial condition σ such that σₓ=-1 if x∈[-L,L]² and σₓ=+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.