Asymptotic behavior of the critical density of activated random walk

TL;DR

The paper analyzes the asymptotic behavior of the critical density for activated random walk as the sleep rate $\lambda$ tends to 0 or $\infty$, deriving new sharp lower bounds and first-order asymptotics in key regimes. Employing a refined weak–strong stabilization framework and a site-wise (Diaconis–Fulton) representation, it establishes precise dimension-specific rates, including $\rho_c(\mathbb{Z}^2, \lambda) = 1 - O((\lambda\log\lambda)^{-1})$ for high sleep rates and $\rho_c(\mathbb{Z}^2, \lambda) = \Omega(\lambda\log(\lambda^{-1}))$ in low sleep rates, plus a superpolynomial convergence in 1D. The work also provides first-order approximations for transient walks, clarifies behavior on unimodular graphs, and introduces the carpet procedure to couple stabilization dynamics with escape probabilities, enabling tight control of fixation probabilities. Collectively, these results advance the quantitative understanding of self-organized criticality in ARW and illuminate the interplay between sleep-activation dynamics, Green’s function, and geometric stabilization procedures.

Abstract

We study the asymptotic behavior of the critical density of the activated random walk model as the sleep rate $λ$ tends to $0$ and $\infty$. For large $λ$, we prove new lower bounds in dimensions 1 and 2, showing that in one dimension the critical density approaches $1$ superpolynomially fast. For small $λ$, we prove a new lower bound in two dimensions for how fast the critical density vanishes. We also obtain the first-order approximation for transient walks in both regimes.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5: Abelian property
• Lemma 6: Equivalence of particle and site fixation
• Proposition 1: Conservation of mass
• Proposition 2
• proof
• Proposition 3