  We study the dynamics of two conservative lattice gas models on the infinite
d-dimensional hypercubic lattice: the Activated Random Walks (ARW) and the
Stochastic Sandpiles Model (SSM), introduced in the physics literature in the
early nineties. Theoretical arguments and numerical analysis predicted that the
ARW and SSM undergo a phase transition between an absorbing phase and an active
phase as the initial density crosses a critical threshold. However a rigorous
proof of the existence of an absorbing phase was known only for one-dimensional
systems. In the present work we establish the existence of such phase
transition in any dimension. Moreover, we obtain several quantitative bounds
for how fast the activity ceases at a given site or on a finite system. The
multi-scale analysis developed here can be extended to other contexts providing
an efficient tool to study non-equilibrium phase transitions.
