Activated Random Walks on ${\mathbb{Z}}^d$

Abstract

Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.

References in corpus (4)

• Activated Random Walks
• Sensitivity of the frog model to initial conditions
• Diffusive bounds for the critical density of activated random walks
• Essential enhancements in Abelian networks: continuity and uniform strict monotonicity

Cited by in corpus (7)

• Active Phase for Activated Random Walk on Z
• Active Phase for Activated Random Walks on the Lattice in all Dimensions
• How far do Activated Random Walkers spread from a single source?
• Uniform threshold for fixation of the stochastic sandpile model on the line
• Stochastic Sandpile on a Cycle
• The hockey-stick conjecture for activated random walk
• Macroscopic flow out of a segment for Activated Random Walks in dimension 1