Tunneling of the Kawasaki dynamics at low temperatures in two dimensions

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature beta on a two dimensional torus Lambda(L) = {0,..., L -1}(2). We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are n(2) particles, n < L/2, and that the initial state is the configuration in which all sites of the square (0,..., n - 1)(2) are occupied. We show that in the time scale e(2 beta) the process evolves as a Markov process on Lambda(L) which jumps from any site x to any other site y not equal x at a strictly positive rate which can be expressed in terms of the hitting probabilities of simple Markovian dynamics.