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

Abstract

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