Non-fixation of symmetric Activated Random Walk on the line for small sleep rate

We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on $mathbb{Z},$ with initial mass density $muu003e0$ of active particles. Active particles do a symmetric random walk at rate one and fall asleep at rate $lambdau003e0.$ Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process and show for small enough $lambda$ the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as $lambda$ tends to zero. This positively answers two open questions from Dickman, Rolla, Sidoravicius (J. Stat. Phys., 2010) and Rolla, Sidoravicius (Invent. Math., 2012).