Model of Cluster Growth and Phase Separation: Exact Results in One   Dimension
  We present exact results for a lattice model of cluster growth in 1D. The growth mechanism involves interface hopping and pairwise annihilation supplemented by spontaneous creation of the stable-phase, +1, regions by overturning the unstable-phase, -1, spins with probability p. For cluster coarsening at phase coexistence, p=0, the conventional structure-factor scaling applies. In this limit our model falls in the class of diffusion-limited reactions A+A->inert. The +1 cluster size grows diffusively, ~t**(1/2), and the two-point correlation function obeys scaling. However, for p>0, i.e., for the dynamics of formation of stable phase from unstable phase, we find that structure-factor scaling breaks down; the length scale associated with the size of the growing +1 clusters reflects only the short-distance properties of the two-point correlations. 
Vladimir Privman