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. Longrange 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.

Exact sampling and fast mixing of Activated Random Walk

“…Now we adapt the proof to handle the case of central driving. Note driving enters the proof only in equation (24). In the case of central driving, we have instead EM = (N + N α )P(τ z 0 ≤ τ r 0 ).…”

“…and 0 otherwise; • S is the stabilization operator for activated random walk with sleep rate λ and base chain P , which we now define. Following [24], consider the total ordering on N ∪ {s} for all n = 0. In particular, s…”

Stochastic Sandpile on a Cycle

“…Lemma 2 (Abelian Property, [4,7]). Fix a stack of instructions for SS, and let α and β both be legal and stabilizing sequences.…”

“…Our main result will compare SS to a similar system: the activated random walk model (abbreviated ARW) (see [4]). We briefly describe ARW as a continuous-time process here, and give a discrete-time construction in the Section 2.…”

How far do Activated Random Walkers spread from a single source?

“…The first rigorous results about Activated Random Walk were proved by Hoffman and Sidoravicius (unpublished) in the case of totally asymmetric walks, and by Rolla and Sidoravicius [16] in the case of symmetric walks; we refer to [15] for a complete history. Surprisingly, many basic questions about this simple one-dimensional particle system remain open!…”