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Approximate Lifshitz law for the zero-temperature stochastic Ising model in any dimension

We study the Glauber dynamics for the zero-temperature Ising model in dimension d=4 with "plus" boundary condition.Let T+ be the time needed for an hypercube of size L entirely filled with "minus" spins to become entirely "plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called "Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key point of our proof is to use the detail knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces at equilibrium and mixing time for monotone interfaces dynamics extracted from [2], to get the result in higher dimension.