Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamics

In this paper we analyze metastability and nucleation in the context of alocal version of the Kawasaki dynamics for the two-dimensional stronglyanisotropic Ising lattice gas at very low temperature. Let$Lambdasubsetmathbb{Z}^2$ be a finite box. Particles perform simpleexclusion on $Lambda$, but when they occupy neighboring sites they feel abinding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in thevertical one. Thus the Kawasaki dynamics is conservative inside the volume$Lambda$. Along each bond touching the boundary of $Lambda$ from the outsideto the inside, particles are created with rate $ho=e^{-Deltaeta}$, whilealong each bond from the inside to the outside, particles are annihilated withrate $1$, where $eta$ is the inverse temperature and $Delta>0$ is anactivity parameter. Thus, the boundary of $Lambda$ plays the role of aninfinite gas reservoir with density $ho$. We consider the parameter regime$U_1>2U_2$ also known as the strongly anisotropic regime. We take$Deltain{(U_1,U_1+U_2)}$ and we prove that the empty (respectively full)configuration is a metastable (respectively stable) configuration. We considerthe asymptotic regime corresponding to finite volume in the limit of largeinverse temperature $eta$. We investigate how the transition from empty tofull takes place. In particular, we estimate in probability, expectation anddistribution the asymptotic transition time from the metastable configurationto the stable configuration. Moreover, we identify the size of theemph{critical droplets}, as well as some of their properties. We observe verydifferent behavior in the weakly and strongly anisotropic regimes. We find thatthe emph{Wulff shape}, i.e., the shape minimizing the energy of a droplet atfixed volume, is not relevant for the nucleation pattern.