Limit Theorems for the painting of graphs by clusters

Olivier Garet

ESAIM: Probability and Statistics (2010)

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

Abstract

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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 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.

How to cite

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Garet, 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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  7. O. Häggström, Coloring percolation clusters at random. Stoch. Proc. Appl. (to appear). Also available as preprint (2000).  URIhttp://www.math.chalmers.se/olleh/divide_and_color.ps
  8. H. Kesten and Yu. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab.18 (1990) 537-555.  
  9. C.M. Newman, Normal fluctuations and the FKG inequalities. Comm. Math. Phys.74 (1980) 119-128.  
  10. C.M. Newman and L.S. Schulman, Infinite clusters in percolation models. J. Statist. Phys.26 (1981) 613-628.  
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  12. Yu. Zhang, A martingale approach in the study of percolation clusters on the ℤ d lattice. J. Theor. Probab.14 (2001) 165-187.  

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