Critical Droplets and Sharp Asymptotics for Kawasaki Dynamics with Strongly 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. Let $\Lambda\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites feel binding energy $-U_1<0$ horizontal direction $-U_2<0$ vertical one. Thus is conservative inside volume $\Lambda$. Along each bond touching boundary $\Lambda$ from outside to inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along outside, annihilated $1$, where $\beta>0$ inverse temperature $\Delta>0$ an activity parameter. Thus, plays role infinite reservoir density $\rho$. We consider parameter regime $U_1>2U_2$ also known as strongly anisotropic regime. take $\Delta\in{(U_1,U_1+U_2)}$, so that empty (respectively full) configuration metastable stable) configuration. investigate how transition full takes place particular attention critical configurations asymptotically have crossed probability 1. The derivation some geometrical properties saddles allows us identify geometry minimal gates their boundaries case. observe different behaviors case respect isotropic ($U_1=U_2$) weakly ($U_1<2U_2$) ones. Moreover, derive mixing time, spectral gap sharp estimates asymptotic time