"Zero" temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion

Details "Zero" temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion "Zero" temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion

2010-07-21

arxiv.org/abs/1007.3599v2

Comm. Pure Appl. Math. 64 (2011), 778--831

Physics

Mathematical Physics

44 pages, 7 figures. v2: Theorem 1 improved to include a matching lower bound on tau_+

Scientific paper

We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an initial domain of linear size L of "-" spins disappears within a time \tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].

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