Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version

In this paper we analyze metastability and nucleation in the context of the Kawasaki dynamics for the two-dimensional Ising lattice gas at very low temperature with periodic boundary conditions. Let $\beta>0$ be the inverse temperature and let $\Lambda\subset\Lambda^\beta\subset\mathbb{Z}^2$ be two boxes. We consider the asymptotic regime corresponding to the limit as $\beta\rightarrow\infty$ for finite volume $\Lambda$ and $\lim_{\beta\rightarrow\infty}\frac{1}{\beta}\log|\Lambda^\beta|=\infty$. We study the simplified model, in which particles perform independent random walks on $\Lambda^\beta\setminus\Lambda$ and inside $\Lambda$ particles perform simple exclusion, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $\Lambda^\beta$. The initial configuration is chosen such that $\Lambda$ is empty and $\rho|\Lambda^\beta|$ particles are distributed randomly over $\Lambda^\beta\setminus\Lambda$. Our results will use a deep analysis of a local model, i.e., particles perform Kawasaki dynamics inside $\Lambda$ and along each bond touching the boundary of $\Lambda$ from the outside to the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $\Delta>0$ is an activity parameter. Thus, in the local model the boundary of $\Lambda$ plays the role of an infinite gas reservoir with density $\rho$. We take $\Delta\in{(U_1,U_1+U_2)}$, so that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We investigate how the transition from empty to full takes place in the local model with particular attention to the critical configurations that asymptotically have to be crossed with probability 1.