Limit Theorems for the painting of graphs by clusters

Laboratoire de Mathématiques, Applications et Physique Mathématique d'Orléans, UMR 6628, Université d'Orléans, BP. 6759, 45067 Orléans Cedex 2, France; Olivier.Garet@labomath.univ-orleans.fr.

Received: 15 May 2001
Revised: 2 July 2001
Revised: 3 October 2001

Abstract

We consider a generalization of the so-called divide and color model recently introduced by Häggström. We investigate the behavior of the magnetization in large boxes of the lattice 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 deeply influence the behavior of the coloring model. In the subcritical case, the limit magnetization is deterministic and the Central Limit Theorem admits a Gaussian limit. Conversely, 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. We also prove a Central Limit Theorem for the size of the intersection of the infinite cluster with large boxes in supercritical bond percolation.