On Long Range Percolation with Heavy Tails

Abstract

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