On Long Range Percolation with Heavy Tails

Abstract

Consider independent long range percolation on ${\mathbf{Z}}^2$, where horizontal and vertical edges of length $n$ are open with probability $p_n$. We show that if ${\mathrm{limsup}}_{n\to \infty }{p}_n>0,$ then there exists an integer $N$ such that $P_N(0\leftrightarrow \infty )>0$, where $P_N$ is the truncated measure obtained by taking $p_{N,n}=p_n$ for $n\le N$ and $p_{N,n}=0$ for all $n>N$.