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 ${\mathbb{Z}}^d$.