Asymptotic behaviors for percolation clusters with uncorrelated weights

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.