Activated Random Walk in a Supercritical Configuration

The Activated Random Walk (ARW) is one example of self-organized criticality, which describes stochastic processes that tend to a critical point without outside influence. One important aspect of the ARW is the existence of a critical density, a parameter that tells us immediately whether the system will eventually stabilize, with all particles falling asleep, or if it will keep running forever. This critical density, however, has primarily been studied in the context of all particles starting active. This talk will discuss ongoing work to show that an ARW system starting with a supercritical density and just one active particle has a probability of stabilizing strictly between 0 and 1.