Limit theorems for the painting of graphs by clusters

Olivier Garet

ESAIM: Probability and Statistics (2001)

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

Abstract

top
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

top

Garet, 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. [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. [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. [3] H.-O. Georgii, Spontaneous magnetization of randomly dilute ferromagnets. J. Statist. Phys. 25 (1981) 369-396. MR630351
  4. [4] G. Grimmett, Percolation. Springer-Verlag, Berlin, 2nd Edition (1999). Zbl0926.60004MR1707339
  5. [5] O. Häggström, Positive correlations in the fuzzy Potts model. Ann. Appl. Probab. 9 (1999) 1149-1159. Zbl0957.60099MR1728557
  6. [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. [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. [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. [9] C.M. Newman, Normal fluctuations and the FKG inequalities. Comm. Math. Phys. 74 (1980) 119-128. Zbl0429.60096MR576267
  10. [10] C.M. Newman and L.S. Schulman, Infinite clusters in percolation models. J. Statist. Phys. 26 (1981) 613-628. Zbl0509.60095MR648202
  11. [11] C.M. Newman and L.S. Schulman, Number and density of percolating clusters. J. Phys. A 14 (1981) 1735-1743. MR620606
  12. [12] Yu. Zhang, A martingale approach in the study of percolation clusters on the ℤ d lattice. J. Theor. Probab. 14 (2001) 165-187. Zbl0974.60098MR1822899

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.