We consider standard (Bernoulli) site percolation on $\mathbb{Z}^d$ with probability $p$ for each site to be occupied. $C$ denotes the occupied cluster of the origin and $|C|$ its cardinality. We show that for $p >$ (critical probability of the halfspace $\mathbb{Z}^{d - 1} \times \mathbb{Z}_+)$ one has $P_p\{|C| = n\} \leq \exp\{-C_1(p)n^{(d - 1)/d}\}$ for some constant $C_1(p) > 0$. This improves a recent result of Chayes, Chayes and Newman. The proof is based on a Peierls argument which shows exponential decay of the distribution of the size of an "exterior boundary" of $C$.
@article{1176990844,
author = {Kesten, Harry and Zhang, Yu},
title = {The Probability of a Large Finite Cluster in Supercritical Bernoulli Percolation},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 537-555},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990844}
}
Kesten, Harry; Zhang, Yu. The Probability of a Large Finite Cluster in Supercritical Bernoulli Percolation. Ann. Probab., Tome 18 (1990) no. 4, pp. 537-555. http://gdmltest.u-ga.fr/item/1176990844/