# 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

## Abstract

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

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