Continuity of the critical value for long-range percolation

We show that for long-range percolation with polynomially decaying connection probabilities in dimension $d\geq 2$, the critical value depends continuously on the precise specifications of the model. Among other things, we use this result to show transience of the infinite supercritical long-range percolation cluster in dimension $d\geq 3$ and to prove a shape theorem for super-critical long-range percolation in the strong decay regime.