  We show that the critical density of the Activated Random Walk model on
$\mathbb{Z}^d$ is strictly less than one when the sleep rate $\lambda$ is small
enough, and tends to $0$ when $\lambda\to 0$, in any dimension $d\geqslant 1$.
As far as we know, the result is new for $d=2$.
  We prove this by showing that, for high enough density and small enough sleep
rate, the stabilization time of the model on the $d$-dimensional torus is
exponentially large. To do so, we fix the the set of sites where the particles
eventually fall asleep, which reduces the problem to a simpler model with
density one. Taking advantage of the Abelian property of the model, we show
that the stabilization time stochastically dominates the escape time of a
one-dimensional random walk with a negative drift. We then check that this slow
phase for the finite volume dynamics implies the existence of an active phase
on the infinite lattice.
