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

Abstract

Let $\DD$ be a simply connected, smooth enough domain of $\bbR^2$. 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 $x\in L\DD$ and $\sigma_x=+1$ otherwise. It is conjectured \cite{cf:Spohn} 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 \emph{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 $\DD$ is convex. Existence and regularity of the solution of the deterministic curve-shortening flow is not obvious \textit{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.