"Zero" Temperature Stochastic 3D Ising Model and Dimer Covering Fluctuations: A First Step Towards Interface Mean Curvature Motion

"We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at a temperature that 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 factors, is the first rigorous confirmation of the Lifshitz law tau(+) similar or equal to const x L(2), conjectured on heuristic grounds [8, 13]. In dimension d = 2, tau(+) can be shown to be of order L(2) without logarithmic corrections: the upper bound was proven in [6], 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 [2]. (C) 2010 Wiley Periodicals, Inc."