The scaling limit for zero-temperature planar ising droplets: With andwithout magnetic fields

Abstract

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 ∈ 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 by Lacoin et al. (J Eur Math Soc, in press). In that case, the time-scaling is diffusive and the scaling limit is given by anisotropic motion by curvature.