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 beta 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 beta. We show that in a proper time scale L^2, theta_beta particles form almost always a square and that this square evolves as a Brownian motion when the temperature vanishes.