$\theta (p_c,{\mathbb{Z}}^d)=0?$

Abstract

We discuss the famous conjecture {$\theta (p_c,{\mathbb{Z}}^d)=0$} for the Bernoulli site percolation model on ${\mathbb{Z}}^d$ with $d\ge 3$. We prove that, for $d\ge 3$, in the Bernoulli percolation model on ${\mathbb{Z}}^d$, at the critical point $p_c$, either there is no infinite cluster or the tail of the finite cluster distribution is not a stretched exponential.