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

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 β > 0 be the inverse temperature and let Λ ⊂ Λβ ⊂ Z2 be two boxes. We consider the asymptotic regime corresponding to the limit as β → ∞ for finite volume Λ and limβ→∞ β1 log|Λβ| = ∞. We study the simplified model, in which particlesperform independent random walks on Λβ Λ and inside Λ particles perform simple exclusion, but when they occupy neighboring sites they feel a binding energy −U1 < 0 in the horizontal direction and −U2 < 0 in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume Λβ. The initial configuration is chosen such that Λ is empty and ρ|Λβ| particles are distributed randomly over Λβ Λ. Our results will use a deep analysis of a local model, i.e., particles perform Kawasaki dynamics inside Λ and along each bond touching the boundary of Λ from the outside to the inside, particles are created with rate ρ = e−∆β, while along each bond from the inside to the outside, particles are annihilated with rate 1, where ∆ > 0 is an activity parameter. Thus, in the local model the boundary of Λ plays the role of an infinite gas reservoir with density ρ. We take ∆ ∈ (U1 , U1 + U2 ), 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. To this end, we provide a model-independent strategy to identify some unessential saddles (that are not in the union of minimal gates) for the transition from the metastable (or stable) to the stable states and we apply this method to the local model. The derivation of further geometrical properties of the saddles allows us to use this strategy and to identify the union of all the minimal gates for the nucleation in the isotropic (U1 = U2) and weakly anisotropic (U1 < 2U2) cases. More precisely, in the weakly anisotropic case we are able to identify the full geometry of the minimal gates and their boundaries. We observe very different behaviors for these two cases with respect to the strongly anisotropic one (U1 > 2U2). Moreover, we investigate the asymptotics, mixing time and spectral gap for isotropic and weakly anisotropic cases.