Continuity of the critical value and a shape theorem for long-range percolation

Consider supercritical long-range percolation on ${\mathbb{Z}}^d$ where two vertices $x,y\in {\mathbb{Z}}^d$ are connected with probability asymptotic to $\mathrm{\parallel }x-y{\mathrm{\parallel }}^{-s}$ for some $s>2d$. Conditioned that the origin is in the infinite cluster, we prove a shape theorem for the set of points that can be reached within $n$ steps from the origin. As part of the proof, we show that for long-range percolation with polynomially decaying connection probabilities in dimensions $d\ge 2$, the critical value depends continuously on the precise specifications of the model.