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

Abstract

Let D be a simply connected, smooth enough domain of R2. For L > 0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on Z2 with initial condition such that σx = -1 if x ε LD and σx =+1 otherwise. It is conjectured [23] that, in the diffusive limit where space is rescaled by L, time by L2 and L ∞, 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 < Tc, with a different temperature-dependent anisotropy function. We prove this conjecture (at zero temperature) when 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.