Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow
Hubert Lacoin
Université de Paris Dauphine, Paris, FranceFrançois Simenhaus
Université de Paris Dauphine, Paris, FranceFabio L. Toninelli
École Normale Supérieure de Lyon, France
![Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-jems-volume-16-issue-12.png&w=3840&q=90)
Abstract
Let be a simply connected, smooth enough domain of . For consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on with initial condition such that if and otherwise. It is conjectured [23] that, in the diffusive limit where space is rescaled by , time by and , the boundary of the droplet of "" spins follows a deterministic anisotropic curve-shortening flow, where the normal velocity at a point of its boundary is given by the local curvature times an explicit function of the local slope. The behavior should be similar at finite temperature , 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.
Cite this article
Hubert Lacoin, François Simenhaus, Fabio L. Toninelli, Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow. J. Eur. Math. Soc. 16 (2014), no. 12, pp. 2557–2615
DOI 10.4171/JEMS/493