A nucleation-and-growth model

Abstract

We consider the following simple nucleation-and-growth model. On the lattice ℤd, starting with all sites unoccupied, a site becomes occupied at rate e-β1 if it has no occupied neighbors, at rate ε = e-βγ if it has 1 occupied neighbor, and at rate 1 if it has 2 or more occupied neighbors. Occupied sites remain occupied forever. The parameters Γ ≧ γ are fixed, and we are interested in the behavior of the system as β → ∞. We show that the relaxation time of this system scales as eβκc, where κc = max {γ, (Γ + γ)/(d + 1)}.