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 β 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 β. 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.