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

TL;DR

This paper studies the large-scale behavior of zero-temperature Ising droplets with positive magnetic field, revealing how the presence of magnetic fields alters the scaling limits and dynamics compared to the zero-field case.

Contribution

It extends previous results by analyzing the scaling limit of Ising droplets under positive magnetic field, contrasting it with the zero-field case and providing new insights into the dynamics.

Findings

Scaling limit differs with magnetic field presence

Comparison with zero-field anisotropic curvature motion

New techniques for analyzing droplet evolution

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 $L\in N$, 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.