Non-equilibrium Phase Transitions: Activated Random Walks at Criticality

TL;DR

This paper rigorously studies the critical behavior of Activated Random Walks, proving non-fixation at criticality for certain cases and identifying scaling limits in one dimension, advancing understanding of non-equilibrium phase transitions.

Contribution

The paper provides rigorous results on the critical behavior of Activated Random Walks, including proofs of non-fixation at criticality for specific cases and the identification of scaling limits in one dimension.

Findings

System at critical point does not reach an absorbing state for infinite sleep rate.

Identified the scaling limit of flow through the origin in one-dimensional asymmetric system.

Proved no fixation at criticality for the one-dimensional totally-asymmetric case.

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.