  It has been conjectured that the critical density of the Activated Random
Walk model is strictly less than one for any value of the sleeping rate. We
prove this conjecture on $\mathbb{Z}^d$ when $d \geq 3$ and, more generally, on
graphs where the random walk is transient. Moreover, we establish the
occurrence of a phase transition on non-amenable graphs, extending previous
results which require that the graph is amenable or a regular tree.
