Non-equilibrium Phase Transitions: Activated Random Walks at Criticality

Abstract

In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including ${\mathbb{Z}}^d$, and under general initial conditions, the system at the critical point does not reach an absorbing state. We prove this for the case where the sleep rate $\lambda$ is infinite. Moreover, for the one-dimensional asymmetric system, we identify the scaling limit of the flow through the origin at criticality. The case $\lambda <+\infty$ remains largely open, with the exception of the one-dimensional totally-asymmetric case, for which it is known that there is no fixation at criticality.

References in corpus (4)

• Fixed-Energy Sandpiles Belong Generically to Directed Percolation
• Activated Random Walkers: Facts, Conjectures and Challenges
• Absorbing-State Phase Transition for Driven-Dissipative Stochastic Dynamics on $Z$
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• Active Phase for Activated Random Walk on Z
• Universality and Sharpness in Absorbing-State Phase Transitions
• Active Phase for Activated Random Walks on the Lattice in all Dimensions
• Uniform threshold for fixation of the stochastic sandpile model on the line
• The golf model on $\mathbb{Z}/n\mathbb{Z}$ and on $\mathbb{Z}$
• Macroscopic flow out of a segment for Activated Random Walks in dimension 1
• Stretched exponential decay for subcritical parking times on ${\mathbb{Z}}^d$