1112.3160 (H. Lacoin et al.)

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

H. Lacoin, F. Simenhaus, F. L. Toninelli

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 $TView original: http://arxiv.org/abs/1112.3160