Limit theorems for the painting of graphs by clusters
ESAIM: Probability and Statistics, Tome 5 (2001), pp. 105-118.

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 ℤd 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.

Classification : 60K35, 82B20, 82B43
Mots-clés : percolation, coloring model, law of large number, central limit theorem 
@article{PS_2001__5__105_0,
     author = {Garet, Olivier},
     title = {Limit theorems for the painting of graphs by clusters},
     journal = {ESAIM: Probability and Statistics},
     pages = {105--118},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2001},
     mrnumber = {1875666},
     zbl = {0992.60090},
     language = {en},
     url = {https://www.numdam.org/item/PS_2001__5__105_0/}
}
TY  - JOUR
AU  - Garet, Olivier
TI  - Limit theorems for the painting of graphs by clusters
JO  - ESAIM: Probability and Statistics
PY  - 2001
SP  - 105
EP  - 118
VL  - 5
PB  - EDP-Sciences
UR  - https://www.numdam.org/item/PS_2001__5__105_0/
LA  - en
ID  - PS_2001__5__105_0
ER  - 
%0 Journal Article
%A Garet, Olivier
%T Limit theorems for the painting of graphs by clusters
%J ESAIM: Probability and Statistics
%D 2001
%P 105-118
%V 5
%I EDP-Sciences
%U https://www.numdam.org/item/PS_2001__5__105_0/
%G en
%F PS_2001__5__105_0
Garet, Olivier. Limit theorems for the painting of graphs by clusters. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 105-118. https://www.numdam.org/item/PS_2001__5__105_0/

[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. | MR | Zbl

[2] J.T. Chayes, L. Chayes and C.M. Newman, Bernoulli percolation above threshold: An invasion percolation analysis. Ann. Probab. 15 (1987) 1272-1287. | MR | Zbl

[3] H.-O. Georgii, Spontaneous magnetization of randomly dilute ferromagnets. J. Statist. Phys. 25 (1981) 369-396. | MR

[4] G. Grimmett, Percolation. Springer-Verlag, Berlin, 2nd Edition (1999). | MR | Zbl

[5] O. Häggström, Positive correlations in the fuzzy Potts model. Ann. Appl. Probab. 9 (1999) 1149-1159. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[8] H. Kesten and Yu. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 (1990) 537-555. | MR | Zbl

[9] C.M. Newman, Normal fluctuations and the FKG inequalities. Comm. Math. Phys. 74 (1980) 119-128. | MR | Zbl

[10] C.M. Newman and L.S. Schulman, Infinite clusters in percolation models. J. Statist. Phys. 26 (1981) 613-628. | MR | Zbl

[11] C.M. Newman and L.S. Schulman, Number and density of percolating clusters. J. Phys. A 14 (1981) 1735-1743. | MR

[12] Yu. Zhang, A martingale approach in the study of percolation clusters on the ℤd lattice. J. Theor. Probab. 14 (2001) 165-187. | MR | Zbl