The heat equation shrinks Ising droplets to points
Lacoin, Hubert; Simenhaus, François; Toninelli, Fabio Lucio (2015), The heat equation shrinks Ising droplets to points, Communications on Pure and Applied Mathematics, 68, 9, p. 1640-1681. http://dx.doi.org/10.1002/cpa.21533
TypeArticle accepté pour publication ou publié
External document linkhttp://arxiv.org/abs/1306.4507v1
Journal nameCommunications on Pure and Applied Mathematics
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Abstract (EN)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.
Subjects / KeywordsAnisotropy; Ising model
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