Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus

We consider the Kawasaki dynamics at inverse temperature . for the Ising lattice gas on a two-dimensional square of length 2L+1 with periodic boundary conditions. We assume that initially the particles form a square of length n, which may increase, as well as L, with .. We show that in a proper time scale the particles form almost always a square and that the center of mass of the square evolves as a Brownian motion when the temperature vanishes.