Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow

Abstract

Let \mathcal D be a simply connected, smooth enough domain of \mathbb R^2 . For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on \mathbb Z^2 with initial condition such that \sigma_x=-1 if x\in L\mathcal D and \sigma_x=+1 otherwise. It is conjectured [23] that, in the diffusive limit where space is rescaled by L , time by L^2 and L\to\infty , the boundary of the droplet of " - " spins follows a deterministic anisotropic curve-shortening flow, where the normal velocity at a point of its boundary is given by the local curvature times an explicit function of the local slope. The behavior should be similar at finite temperature T<T_c , with a different temperature-dependent anisotropy function. We prove this conjecture (at zero temperature) when \mathcal D is convex. Existence and regularity of the solution of the deterministic curve-shortening flow is not obvious a priori and is part of our result. To our knowledge, this is the first proof of mean curvature-type droplet shrinking for a model with genuine microscopic dynamics.