Truncation of long-range percolation with non-summable interactions in dimensions $d\ge 3$

Consider independent long-range percolation on ${\mathbb{Z}}^d$ for $d\ge 3$. Assuming that the expected degree of the origin is infinite, we show that there exists an $N\in \mathbb{N}$ such that an infinite open cluster remains after deleting all edges of length at least $N$. For the isotropic case in dimensions $d\ge 3$, we show that if the expected degree of the origin is at least ${10}^{400}$, then there exists an infinite open cluster almost surely. We also use these results to prove corresponding statements for the long-range $q$-states Potts model.