Limit theorems for the painting of graphs by clusters

Abstract

We consider a generalization of the so-called divide and color model recently introduced by Haggstrom. We investigate the behaviour of the magnetization in large boxes and its fluctuations. Thus, laws of large numbers and Central Limit theorems are proved, both quenched and annealed. We show that the properties of the underlying percolation process roughly influence the behaviour of the colorying model. In the subcritical case, the limit magnetization is deterministic and the Central Limit Theorem admits a Gaussian limit. A contrario, the limit magnetization is not deterministic in the supercritical case and the limit of the Central Limit Theorem is not Gaussian, except in the particular model with exactly two colors which are equally probable.