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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 $β$ on a two dimensional torus $Λ_L=\{0,..., L-1\}^2$ . We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are $n^2\ll L$ particles and that the initial state is the configuration in which all sites of the square $\mb x + \{0,..., n-1\}^2$ are occupied. We show that in the time scale $e^{2β}$ the process is close to a Markov process on $Λ_L$ which jumps from any site $\mb x$ to any other site $\mb y\not =\mb x$ at a strictly positive rate which can be expressed in terms of the jump rates of simple random walks.