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Limit theorems for the painting of graphs by clusters

Olivier Garet — 2001

ESAIM: Probability and Statistics

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 d 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 Theorems for the painting of graphs by clusters

Olivier Garet — 2010

ESAIM: Probability and Statistics

We consider a generalization of the so-called recently introduced by Häggström. We investigate the behavior of the magnetization in large boxes of the lattice d 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...

Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

Olivier GaretRégine Marchand — 2004

ESAIM: Probability and Statistics

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on d to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical...

Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

Olivier GaretRégine Marchand — 2010

ESAIM: Probability and Statistics

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on d to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical...

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