Spread-out percolation on transitive graphs of polynomial growth

Let G be a vertex-transitive graph of superlinear polynomial growth. Given $r>0$, let Gᵣ be the graph on the same vertex set as G, with two vertices joined by an edge if and only if they are at graph distance at most r apart in G. We show that the critical probability pc(Gᵣ) for Bernoulli bond percolation on Gᵣ satisfies pc(Gᵣ) ~ 1/deg(Gᵣ) as r→∞. This extends work of Penrose and Bollob\'as-Janson-Riordan, who considered the case G=Zᵈ. Our result provides an important ingredient in parallel work of Georgakopoulos in which he introduces a new notion of dimension in groups. It also verifies a special case of a conjecture of Easo and Hutchcroft.