[Paper Review] Universality and Sharpness in Absorbing-State Phase Transitions

Leonardo T. Rolla, Vladas Sidoravičius+1 more|arXiv (Cornell University)|Jul 19, 2017

Theoretical and Computational Physics19 references3 citations

TL;DR

This paper establishes the existence of a well-defined critical density ζc in the Activated Random Walk (ARW) model on Z^d, regardless of initial particle configuration or distribution, proving that configurations with density ζ < ζc are almost surely stabilizable, while those with ζ > ζc are almost surely explosive. The result resolves a key issue in self-organized criticality by showing universality and sharpness in ARW, independent of initial state assumptions, and supports the model's superior mixing properties compared to the Abelian Sandpile Model.

ABSTRACT

We consider the Activated Random Walk model in any dimension with any sleep rate and jump distribution and ergodic initial state. We show that the stabilization properties depend only on the average density of particles, regardless of how they are initially located on the lattice.

Motivation & Objective

• To establish the existence of a well-defined critical density ζc in the Activated Random Walk model on Z^d, independent of initial particle distribution.
• To demonstrate that stabilization properties in ARW depend solely on average particle density, not on initial spatial configuration or ergodicity assumptions.
• To show that the phase transition at ζc is sharp: all ergodic states with ζ < ζc are a.s. stabilizable, and all with ζ > ζc are a.s. explosive.
• To remove the need for restrictive initial state assumptions (e.g., independence, light tails) in prior analytical techniques for ARW.
• To support the conjecture that ARW exhibits better mixing and universality than the Abelian Sandpile Model, particularly regarding the density conjecture.

Proposed method

• Uses a two-stage stabilization procedure: first, a partial stabilization using a shifted instruction field to generate a subcritical configuration η₀′ ≤ ξ₀.
• Applies the local Abelian property and Least Action Principle to ensure order-independence of toppling sequences in finite volumes.
• Introduces a modified instruction field 𝒥̃ by deleting the first h₀′(x) instructions at each site x, ensuring independence from the initial state ξ₀.
• Leverages the i.i.d. nature of the instruction field and the independence of 𝒥̃ from ξ₀ and h₀′ to preserve stabilization probability under field shift.
• Constructs a finite odometer h₁′(x) for the residual configuration η₀′ using the shifted field, ensuring finite stabilization in all finite sets.
• Combines the two stages via monotonicity and the Least Action Principle to bound the total odometer m_{η₀;𝒥}(x) ≤ h₀′(x) + h₁′(x) < ∞ for all x ∈ Z^d.

Experimental results

Research questions

• RQ1 Does the critical density ζc in the Activated Random Walk model depend on the initial particle configuration or distribution?
• RQ2 Can the stabilization behavior of ARW be universally characterized by a single critical threshold ζc, regardless of initial ergodic state?
• RQ3 Is the phase transition in ARW sharp, such that all configurations with ζ < ζc are almost surely stabilizable and all with ζ > ζc are almost surely explosive?
• RQ4 Can prior analytical techniques for ARW be generalized by removing assumptions on initial state independence or tail behavior?
• RQ5 Does ARW exhibit stronger mixing properties than the Abelian Sandpile Model, as suggested by the universality of ζc?

Key findings

• The critical density ζc is well-defined and universal in the ARW model on Z^d, depending only on the average particle density, not on initial configuration or distribution.
• For any spatially ergodic initial state with density ζ < ζc, the system is almost surely stabilizable, regardless of particle clustering or initial structure.
• For any spatially ergodic initial state with density ζ > ζc, the system is almost surely explosive, meaning activity persists indefinitely.
• The sharp phase transition at ζc holds universally across all ergodic initial states, confirming a universal and non-trivial critical behavior.
• The proof technique eliminates the need for restrictive assumptions (e.g., independence, light-tailed distributions) on the initial state, generalizing prior results.
• The result supports the conjecture that ARW has better mixing properties than the Abelian Sandpile Model, as it is insensitive to initial state details.