A note on truncated long-range percolation with heavy tails on oriented graphs

We consider oriented long-range percolation on a graph with vertex set $mathbb{Z}^d times mathbb{Z}_+$ and directed edges of the form $langle (x,t), (x+y,t+1)rangle$, for $x,y$ in $mathbb{Z}^d$ and $t in mathbb{Z}_+$. Any edge of this form is open with probability $p_y$, independently for all edges. Under the assumption that the values $p_y$ do not vanish at infinity, we show that there is percolation even if all edges of length more than $k$ are deleted, for $k$ large enough. We also state the analogous result for a long-range contact process on $mathbb{Z}^d$.