JournalsjemsVol. 16, No. 12pp. 2557–2615

Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow

  • Hubert Lacoin

    Université de Paris Dauphine, Paris, France
  • François Simenhaus

    Université de Paris Dauphine, Paris, France
  • Fabio L. Toninelli

    École Normale Supérieure de Lyon, France
Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow cover
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Abstract

Let D\mathcal D be a simply connected, smooth enough domain of R2\mathbb R^2. For L>0L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on Z2\mathbb Z^2 with initial condition such that σx=1\sigma_x=-1 if xLDx\in L\mathcal D and σx=+1\sigma_x=+1 otherwise. It is conjectured [23] that, in the diffusive limit where space is rescaled by LL, time by L2L^2 and LL\to\infty, 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 T<TcT<T_c, with a different temperature-dependent anisotropy function.

We prove this conjecture (at zero temperature) when D\mathcal D 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