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On the slow phase for fixed-energy Activated Random Walks

Bernardo N. B. de Lima, Leonardo T. Rolla, Célio Terra

TL;DR

The paper establishes the existence of a slow phase for fixed-energy Activated Random Walks on a one-dimensional ring in the high-density regime, valid for arbitrary sleep rate $\lambda>0$. It introduces a self-contained carpet-like toppling procedure that builds an on-the-fly environment to sustain activity and analyzes it via alternating modes, blocks, and flux balancing. A key technical contribution is a controlled exponential bound on the number of frozen particles, combining block-wise estimates, detailed sigma-algebra constructions, and hole-drift arguments. The results quantify long stabilization times and provide a rigorous pathway to understanding sustained activity in high-density ARWs with large sleep rates, with potential implications for related interacting particle systems.

Abstract

We study the Activated Random Walk model on the one-dimensional ring, in the high density regime. We develop a toppling procedure that gradually builds an environment that can be used to show that activity will be sustained for a long time. This yields a self-contained and relatively short proof of existence of a slow phase for arbitrarily large sleep rates.

On the slow phase for fixed-energy Activated Random Walks

TL;DR

The paper establishes the existence of a slow phase for fixed-energy Activated Random Walks on a one-dimensional ring in the high-density regime, valid for arbitrary sleep rate . It introduces a self-contained carpet-like toppling procedure that builds an on-the-fly environment to sustain activity and analyzes it via alternating modes, blocks, and flux balancing. A key technical contribution is a controlled exponential bound on the number of frozen particles, combining block-wise estimates, detailed sigma-algebra constructions, and hole-drift arguments. The results quantify long stabilization times and provide a rigorous pathway to understanding sustained activity in high-density ARWs with large sleep rates, with potential implications for related interacting particle systems.

Abstract

We study the Activated Random Walk model on the one-dimensional ring, in the high density regime. We develop a toppling procedure that gradually builds an environment that can be used to show that activity will be sustained for a long time. This yields a self-contained and relatively short proof of existence of a slow phase for arbitrarily large sleep rates.
Paper Structure (12 sections, 12 theorems, 29 equations, 1 figure)

This paper contains 12 sections, 12 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Description of the Model
  3. The toppling procedure
  4. Preliminary steps
  5. Attempted emission
  6. Properties preserved
  7. Alternating modes
  8. Upper bound for the number of frozen particles
  9. $\sigma$-algebras and flow of particles
  10. Bounding the exponential expectation
  11. One-block estimate
  12. Estimating the hole drift

Key Result

Theorem 1

Let $\mathcal{J}$ be the total number of jumps the particles do until the initial configuration is stabilized. For every $0< \lambda <+\infty$, there are constants $\delta, \delta '>0$, depending on $\lambda$, such that, if $\zeta$ is close enough to $1$, for every $N$ sufficiently large.

Figures (1)

  • Figure 1: Example of possible state of a piece of the ring when an attempted emission is about to begin. Filled circles are active carpet particles, empty circles are sleeping carpet particles. The squares represent free particles, one of which is frozen. The holes are indicated by underlining the site. There is a frozen particle at $iK+a$.

Theorems & Definitions (25)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof : Proof of Theorem \ref{['slow_phase']}
  • Proposition 3
  • Proposition 4
  • proof : Proof of Proposition \ref{['PropCondition']}
  • Proposition 5
  • proof
  • ...and 15 more