# Limit Theorems for the painting of graphs by clusters

ESAIM: Probability and Statistics (2010)

- Volume: 5, page 105-118
- ISSN: 1292-8100

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topGaret, Olivier. "Limit Theorems for the painting of graphs by clusters." ESAIM: Probability and Statistics 5 (2010): 105-118. <http://eudml.org/doc/197748>.

@article{Garet2010,

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 $\mathbb\{Z\}^d$
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 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.
},

author = {Garet, Olivier},

journal = {ESAIM: Probability and Statistics},

keywords = {Percolation; coloring
model; Law of Large Number; Central Limit Theorem.; laws of large numbers; percolation; coloring model; central limit theorem},

language = {eng},

month = {3},

pages = {105-118},

publisher = {EDP Sciences},

title = {Limit Theorems for the painting of graphs by clusters},

url = {http://eudml.org/doc/197748},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Garet, Olivier

TI - Limit Theorems for the painting of graphs by clusters

JO - ESAIM: Probability and Statistics

DA - 2010/3//

PB - EDP Sciences

VL - 5

SP - 105

EP - 118

AB -
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 $\mathbb{Z}^d$
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 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.

LA - eng

KW - Percolation; coloring
model; Law of Large Number; Central Limit Theorem.; laws of large numbers; percolation; coloring model; central limit theorem

UR - http://eudml.org/doc/197748

ER -

## References

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