The Heat Equation Shrinks Ising Droplets to Points

Abstract

Let be a bounded, smooth enough domain of ℝ2. For L > 0 consider the continuous‐time, zero‐temperature heat bath stochastic dynamics for the nearest‐neighbor Ising model on (ℤ/L)2 (the square lattice with lattice spacing 1/L) with initial condition such that σx =−1 if x ∊ and σx = + 1 otherwise. We prove the following classical conjecture due to H. Spohn: In the diffusive limit where time is rescaled by L2 and L → ∞, the boundary of the droplet of “‐” spins follows a deterministic anisotropic curve‐shortening flow such that the normal velocity is given by the local curvature times an explicit function of the local slope. Locally, in a suitable reference frame, the evolution of the droplet boundary follows the one‐dimensional heat equation.To our knowledge, this is the first proof of mean‐curvature‐type droplet shrinking for a lattice model with genuine microscopic dynamics.An important ingredient is in our forthcoming work, where the case of convex was solved. The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curve‐shortening flow. This builds on geometric and analytic ideas of Grayson, Gage and Hamilton, Gage and Li, Chou and Zhu, and others.© 2015 Wiley Periodicals, Inc.