The heat equation shrinks Ising droplets to points
Résumé
Let D be a bounded, smooth enough domain of R^2. For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on (Z/L)^2 (the square lattice with lattice spacing 1/L) with initial condition such that \sigma_x=-1 if x\in D and \sigma_x=+ otherwise. We prove the following classical conjecture due to H. Spohn: In the diffusive limit where time is rescaled by L^2 and L tends to infinity, 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 our recent work, where the case of convex D 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-Hamilton, Gage-Li, Chou-Zhu and others.