Asymptotic behaviors for percolation clusters with uncorrelated weights

Abstract

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.

Infinite clusters in locally dependent percolation on ℤ d

“…Remark 2 Apparently, we can consider the asymptotic behaviors for locally dependent percolation clusters or out-clusters, by the arguments analogous to the case of independent percolation [5] .…”

Confidence intervals for the critical value in the divide and color model

“…Our object of study in this paper is the critical value function in Häggström's divide and color (DaC) model [10]. This is a stochastic model that was originally motivated by physical considerations (see [10,6]), but it has since then been used for biological modelling in [9] as well and inspired several generalizations (see, e.g., [11,4,8]). Our results concerning the location of the phase transition give a clear picture of the behavior of the DaC model on two important lattices and lead to intriguing open questions.…”