Asymptotic behavior of the critical density of activated random walk

2025/11/30 by Henry Kaufman, Kaufman, Harley, Meisel, Josh
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2512.00720

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 ∞. 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.

Citations

• Mean-field limit for activated random walk on the integer lattice
• Universality Conjectures for Activated Random Walk
• Essential enhancements in Abelian networks: continuity and uniform strict monotonicity
• Diffusive bounds for the critical density of activated random walks
• Critical density of activated random walks on transitive graphs
• Non-fixation for conservative stochastic dynamics on the line
• Absorbing-state phase transition in biased activated random walk
• On Fixation of Activated Random Walks
• Nonfixation for Activated Random Walks

Related

• Diffusive bounds for the critical density of activated random walks
• Mean-field limit for activated random walk on the integer lattice
• Active Phase for Activated Random Walk on \mathbb Z²
• Active phase for activated random walks on ℤ^d, d ≥ 3, with density less than one and arbitrary sleeping rate
• On the slow phase for fixed-energy Activated Random Walks