热方程将伊辛液滴收缩为点
The heat equation shrinks Ising droplets to points
摘要：
设D是R^2的有界、足够光滑的整环。当L>0时，考虑（Z/L）^2（晶格间距为1/L的正方形晶格）上最近邻伊辛模型的连续时间零温热浴动力学，初始条件是当x \in D时σ x=-1，否则σ x=+.我们证明了H. Spohn：在扩散极限中，时间被L^2重新标度，并且L趋于无穷大，“-”自旋的液滴的边界遵循确定性各向异性曲线缩短流，使得法向速度由局部曲率乘以局部斜率的显函数给出。局部地，在合适的参考系中，液滴边界的演化遵循一维热方程。据我们所知，这是第一次证明平均曲率型液滴收缩的晶格模型与真正的微观动力学。一个重要的组成部分是我们最近的工作，其中凸D的情况得到解决。证明中的另一个关键点是获得确定性曲线缩短流的精确正则性估计。这是建立在几何和分析思想的格雷森，盖奇，汉密尔顿，盖奇，李，周，朱和其他人。
英文摘要：
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 σ_x=-1 if x\in D and σ_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.
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