We consider random processes occurring on bond percolation clusters and represented as a generalization of the "divide and color model" introduced by Häggström in 2001. We investigate the asymptotic behaviors for bond percolation clusters with uncorrelated weights. For subcritical and supercritical phases, we prove the law of large numbers and central limit theorems in the models corresponding to the so-called quenched and annealed probabilities.
In this paper, we study the asymptotic distributions of the heterogeneous random coagulation-fragmentation processes (HCFP) which model the coagulation, fragmentation and diffusion of clusters of particles on a lattice. Based on a closed form of the stationary distribution for the HCFP, we prove that the mutually dependent clusters with a finite size (finite particles) in the sub-critical stage will become independent in the critical and super-critical stages, and the asymptotic distributions of the number of clusters may converge to Gaussian or Poisson distribution according to its size, to be small or large in the critical and super-critical stages.
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