We show that the critical density of the Activated Random Walk model on Z d is strictly less than one when the sleep rate λ is small enough, and tends to 0 when λ → 0, in any dimension d 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.

We consider the planar Ising model in a finite square box and we replace the temperature parameter with a function depending on the magnetization. This creates a feedback from the spin configuration onto the parameter, which drives the system towards the critical point. Using the finite-size scaling results of [CM11], we show that, when the size of the box grows to infinity, the temperature concentrates around the critical temperature of the planar Ising model on the square lattice.