[Paper Review] Non-fixation of symmetric Activated Random Walk on the line for small sleep rate

Riddhipratim Basu, Shirshendu Ganguly+1 more|arXiv (Cornell University)|Aug 24, 2015

Stochastic processes and statistical mechanics3 citations

TL;DR

This paper investigates the Activated Random Walk (ARW) process on the integer line, showing that for small sleep rates $\lambda$, the critical mass density for fixation is strictly less than one and tends to zero as $\lambda \to 0$. It resolves two open questions by proving non-fixation occurs at arbitrarily low densities when $\lambda$ is sufficiently small, using coupling and duality techniques in particle system analysis.

ABSTRACT

We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on $\mathbb{Z},$ with initial mass density $\mu>0$ of active particles. Active particles do a symmetric random walk at rate one and fall asleep at rate $\lambda>0.$ Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process and show for small enough $\lambda$ the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as $\lambda$ tends to zero. This positively answers two open questions from Dickman, Rolla, Sidoravicius (J. Stat. Phys., 2010) and Rolla, Sidoravicius (Invent. Math., 2012).

Motivation & Objective

• To resolve open questions about fixation behavior in the Activated Random Walk model on the integer line.
• To determine whether the critical mass density for fixation remains below one when the sleep rate $\lambda$ is small.
• To analyze the asymptotic behavior of the critical density as $\lambda \to 0$.
• To establish that non-fixation occurs even at arbitrarily low particle densities when $\lambda$ is sufficiently small.

Proposed method

• Use of coupling techniques to compare ARW dynamics with other particle systems and control fixation behavior.
• Employment of duality methods to relate the ARW process to a dual branching process.
• Analysis of symmetric random walk dynamics with state-dependent transition rules (active vs. sleeping particles).
• Establishment of a lower bound on the non-fixation probability via stochastic domination arguments.
• Application of results from stochastic sandpile models and self-organized criticality theory.
• Use of symmetry and spatial homogeneity to simplify the analysis on $\mathbb{Z}$.

Experimental results

Research questions

• RQ1 Does the critical mass density for fixation in ARW on $\mathbb{Z}$ remain strictly less than one for small sleep rates $\lambda$?
• RQ2 What is the asymptotic behavior of the critical density as $\lambda \to 0$?
• RQ3 Can non-fixation be proven to occur at arbitrarily low mass densities when $\lambda$ is sufficiently small?
• RQ4 Does the ARW process exhibit non-fixation for all $\mu > 0$ when $\lambda$ is small enough?
• RQ5 Do the results confirm the conjectures posed in Dickman, Rolla, Sidoravicius (2010) and Rolla, Sidoravicius (2012)?

Key findings

• For sufficiently small sleep rate $\lambda$, the critical mass density for fixation in ARW on $\mathbb{Z}$ is strictly less than one.
• The critical density tends to zero as $\lambda \to 0$, confirming a conjecture on the asymptotic behavior of the threshold.
• Non-fixation occurs even at arbitrarily low particle densities when $\lambda$ is small enough, implying persistent activity.
• The results resolve two open questions from prior works on ARW and self-organized criticality.
• The analysis confirms that symmetric ARW on the line does not fixate for small $\lambda$, even with low initial mass density.