The Probability of a Large Finite Cluster in Supercritical Bernoulli Percolation

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 $C$.