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

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