ZERO-TEMPERATURE 2D STOCHASTIC ISING MODEL AND ANISOTROPIC CURVE-SHORTENING FLOW

Hubert Lacoin (1) , François Simenhaus (1) , Fabio Lucio Toninelli (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Phys-ENS - Laboratoire de Physique de l'ENS Lyon

Let D be a simply connected, smooth enough domain of R2. For L > 0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on Z2 with initial condition such that σx = −1 if x ∈ LD and σx = +1 otherwise. It is conjectured [24] that, in the diffusive limit where space is rescaled by L, time by L2 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 < Tc, 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.

Mots clés: sing model; Glauber dynamics; Curve-shortening flow

Domaines: Probabilités [math.PR]

Hubert Lacoin, François Simenhaus, Fabio Lucio Toninelli. ZERO-TEMPERATURE 2D STOCHASTIC ISING MODEL AND ANISOTROPIC CURVE-SHORTENING FLOW. J. Eur. Math. Soc. (JEMS) 6, 2014, 16 (12). ⟨hal-00656387⟩