Let be a simply connected, smooth enough domain of . For consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on with initial condition such that if and otherwise. It is conjectured  that, in the diffusive limit where space is rescaled by , time by and , 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 , with a different temperature-dependent anisotropy function.
We prove this conjecture (at zero temperature) when 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.
Cite this article
Hubert Lacoin, François Simenhaus, Fabio L. Toninelli, Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow. J. Eur. Math. Soc. 16 (2014), no. 12, pp. 2557–2615