# 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 $D$ be a simply connected, 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 $Z_{2}$ with initial condition such that $σ_{x}=−1$ if $x∈LD$ and $σ_{x}=+1$ otherwise. It is conjectured [23] that, in the diffusive limit where space is rescaled by $L$, time by $L_{2}$ and $L→∞$, 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 $D$ 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