# Limit theorems for the painting of graphs by clusters

ESAIM: Probability and Statistics (2001)

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

## Access Full Article

top## Abstract

top## How to cite

topGaret, Olivier. "Limit theorems for the painting of graphs by clusters." ESAIM: Probability and Statistics 5 (2001): 105-118. <http://eudml.org/doc/104268>.

@article{Garet2001,

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},

language = {eng},

pages = {105-118},

publisher = {EDP-Sciences},

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

url = {http://eudml.org/doc/104268},

volume = {5},

year = {2001},

}

TY - JOUR

AU - Garet, Olivier

TI - Limit theorems for the painting of graphs by clusters

JO - ESAIM: Probability and Statistics

PY - 2001

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

UR - http://eudml.org/doc/104268

ER -

## References

top- [1] J.T. Chayes, L. Chayes, G.R. Grimmett, H. Kesten and R.H. Schonmann, The correlation length for the high-density phase of Bernoulli percolation. Ann. Probab. 17 (1989) 1277-1302. Zbl0696.60094MR1048927
- [2] J.T. Chayes, L. Chayes and C.M. Newman, Bernoulli percolation above threshold: An invasion percolation analysis. Ann. Probab. 15 (1987) 1272-1287. Zbl0627.60099MR905331
- [3] H.-O. Georgii, Spontaneous magnetization of randomly dilute ferromagnets. J. Statist. Phys. 25 (1981) 369-396. MR630351
- [4] G. Grimmett, Percolation. Springer-Verlag, Berlin, 2nd Edition (1999). Zbl0926.60004MR1707339
- [5] O. Häggström, Positive correlations in the fuzzy Potts model. Ann. Appl. Probab. 9 (1999) 1149-1159. Zbl0957.60099MR1728557
- [6] O. Häggström, R.H. Schonmann and J.E. Steif, The Ising model on diluted graphs and strong amenability. Ann. Probab. 28 (2000) 1111-1137. Zbl1023.60085MR1797305
- [7] O. Häggström, Coloring percolation clusters at random. Stoch. Proc. Appl. (to appear). Also available as preprint http://www.math.chalmers.se/olleh/divide_and_color.ps (2000). Zbl1058.60090MR1865356
- [8] H. Kesten and Yu. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 (1990) 537-555. Zbl0705.60092MR1055419
- [9] C.M. Newman, Normal fluctuations and the FKG inequalities. Comm. Math. Phys. 74 (1980) 119-128. Zbl0429.60096MR576267
- [10] C.M. Newman and L.S. Schulman, Infinite clusters in percolation models. J. Statist. Phys. 26 (1981) 613-628. Zbl0509.60095MR648202
- [11] C.M. Newman and L.S. Schulman, Number and density of percolating clusters. J. Phys. A 14 (1981) 1735-1743. MR620606
- [12] Yu. Zhang, A martingale approach in the study of percolation clusters on the ${\mathbb{Z}}^{d}$ lattice. J. Theor. Probab. 14 (2001) 165-187. Zbl0974.60098MR1822899

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.