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

Abstract

We discuss the famous conjecture {$\theta(p_c,{\mathbb Z}^d)=0$} for the Bernoulli site percolation model on $\Z^d$ with $d\geq 3$. We prove that, for $d\geq 3$, 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.