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The odometer in subcritical activated random walk

Tobias Johnson, Jacob Richey

TL;DR

The paper analyzes fixed-energy activated random walk on $\mathbb{Z}$ with i.i.d.-Bernoulli initial data in the subcritical regime $\rho<\rho_c(\lambda)$. It proves that the odometer $m(0)$ has a stretched-exponential tail, with $P_\rho(m(0)\ge n)$ decaying like $e^{-c\sqrt{n}}$, which implies finite $\mathbf{E}_\rho[m(0)]$. The authors leverage the sitewise abelian representation, the least-action principle, and a coupling to layer percolation via infection paths, deploying greedy-path constructions for upper bounds and large-deviation conditioning for lower bounds. These results illuminate fixation behavior and quantify odometer fluctuations below criticality, while underscoring challenges near criticality and inviting questions about universality and extensions to other initial conditions.

Abstract

We consider the activated random walk particle system, a model of self-organized criticality, on $\mathbb{Z}$ with i.i.d.-Bernoulli initial configuration. We show that at subcritical density, the system's odometer function, which counts the number of actions taken at each site, has a stretched exponential tail. It follows that the expected odometer at each site is finite.

The odometer in subcritical activated random walk

TL;DR

The paper analyzes fixed-energy activated random walk on with i.i.d.-Bernoulli initial data in the subcritical regime . It proves that the odometer has a stretched-exponential tail, with decaying like , which implies finite . The authors leverage the sitewise abelian representation, the least-action principle, and a coupling to layer percolation via infection paths, deploying greedy-path constructions for upper bounds and large-deviation conditioning for lower bounds. These results illuminate fixation behavior and quantify odometer fluctuations below criticality, while underscoring challenges near criticality and inviting questions about universality and extensions to other initial conditions.

Abstract

We consider the activated random walk particle system, a model of self-organized criticality, on with i.i.d.-Bernoulli initial configuration. We show that at subcritical density, the system's odometer function, which counts the number of actions taken at each site, has a stretched exponential tail. It follows that the expected odometer at each site is finite.

Paper Structure

This paper contains 6 sections, 6 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Setup
  4. Proofs
  5. Upper bound on the tail
  6. Lower bound on the tail

Key Result

Theorem 1

For any sleep rate $\lambda>0$ and $\rho<\rho_c(\lambda)$, there exist constants $c,\,c',\,C,\,C'>0$ depending only on $\lambda$ and $\rho$ such that for all $n$,

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 2
  • Conjecture 3
  • Proposition 4: hoffman2024density
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • proof : Proof of Theorem \ref{['thm:tailbound']}, upper bound
  • ...and 5 more