[Paper Review] Active Phase for Activated Random Walk on Z

Christopher Hoffman, Jacob Richey+1 more|arXiv (Cornell University)|Sep 20, 2020

Stochastic processes and statistical mechanics20 references4 citations

TL;DR

This paper establishes the existence of a non-trivial active phase for the activated random walk model on the one-dimensional integer lattice, proving that for any finite sleep rate $\lambda$, there exists a density $\zeta < 1$ such that the system remains active almost surely. The authors extend prior results by developing a block-based toppling procedure under a 'carpet' initial condition, leveraging stochastic domination and energy-entropy arguments to control particle dynamics despite high sleep rates.

ABSTRACT

We consider the Activated Random Walk model on $\mathbb{Z}$. In this model, each particle performs a continuous-time simple symmetric random walk, and falls asleep at rate $λ$. A sleeping particle does not move but it is reactivated in the presence of another particle. We show that for any sleep rate $λ&lt; \infty$ if the density $ ζ$ is close enough to $1$ then the system stays active.

Motivation & Objective

• To establish the existence of a non-trivial active phase in the activated random walk model on $\mathbb{Z}$ for arbitrary finite sleep rates $\lambda$.
• To extend previous results that only held for small $\lambda$ by overcoming the challenge of high sleep rates in recurrent random walk settings.
• To develop a robust dynamical block procedure that enables energy-entropy type estimates even when $\lambda$ is large.
• To prove that the critical density remains strictly less than one, confirming universality and self-organized criticality in the one-dimensional case.

Proposed method

• Introduce a block-based toppling procedure with sources at intervals $K\mathbb{Z}$, where particles are dynamically attached to maintain activity.
• Use a stationary 'carpet' initial condition with most sites occupied by a single particle to provide a safe path for wandering particles.
• Apply stochastic domination techniques to bound the movement of holes (vacant sites) in the carpet, ensuring sufficient particle emission.
• Employ a conditional probability framework, conditioning on the hole position at each block, to control failure and emission events.
• Leverage classical gambler's ruin estimates to bound the probability of successful emission to neighboring blocks, ensuring leftward propagation with probability at least $1/3$.
• Use union bounds and exponential tail estimates (e.g., $e^{-a}$) to control rare events such as long excursions of holes or delayed emissions.

Experimental results

Research questions

• RQ1 Does the activated random walk on $\mathbb{Z}$ exhibit an active phase for arbitrarily large sleep rates $\lambda$?
• RQ2 Can the critical density $\zeta_c$ be strictly less than one for all finite $\lambda$?
• RQ3 Can the block-based toppling procedure be adapted to work under high sleep rates without assuming $\lambda$ is small?
• RQ4 Is the active phase robust under the presence of long-range space-time correlations due to particle conservation?

Key findings

• For every finite sleep rate $\lambda < \infty$, there exists a density $\zeta < 1$ such that the activated random walk on $\mathbb{Z}$ remains active almost surely.
• The critical density $\zeta_c$ is strictly less than one for all $\lambda < \infty$, confirming a non-trivial active phase beyond previous results limited to small $\lambda$.
• The block dynamics with a carpet initial condition successfully controls hole movement and emission events, even when $\lambda$ is large, via stochastic domination and exponential tail bounds.
• The probability that a hole reaches $a/2$ before returning to $a/3$ is bounded by $e^{-a}$, and the probability of delayed emission is bounded by $1/a$ for large $a$.
• The method ensures at least one successful emission per two attempted emissions, with each emission having at least a $1/3$ chance of moving left, enabling long-range propagation.