Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow
Toninelli, Fabio Lucio; Simenhaus, François; Lacoin, Hubert (2014), Zero-temperature 2D stochastic Ising model and anisotropic curve-shortening flow, Journal of the European Mathematical Society, 16, 12, p. 2557-2615. http://dx.doi.org/10.4171/JEMS/493
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00656387/fr/
Journal nameJournal of the European Mathematical Society
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Abstract (EN)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  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.
Subjects / KeywordsCurve-shortening flow; Glauber dynamics; sing model
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