  Let $G$ be a vertex-transitive graph of superlinear polynomial growth. Given
$r>0$, let $G_r$ 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 $p_c(G_r)$ for Bernoulli bond
percolation on $G_r$ satisfies $p_c(G_r) \sim 1/\mathrm{deg}(G_r)$ as
$r\to\infty$. This extends work of Penrose and Bollob\'as-Janson-Riordan, who
considered the case $G=\mathbb{Z}^d$.
  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.
