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

H. Lacoin, F. Simenhaus, F. L. Toninelli
2013 arXiv   pre-print
Let be a simply connected, smooth enough domain of ^2. For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on Z^2 with initial condition such that σ_x=-1 if x∈ L and σ_x=+1 otherwise. It is conjectured cf:Spohn that, in the diffusive limit where space is rescaled by L, time by L^2 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
more » ... ry is given by the local curvature times an explicit function of the local slope. The behavior should be similar at finite temperature T<T_c, 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.
arXiv:1112.3160v2 fatcat:s53txnxjw5cpjlbi3vl7dchazq