Diffusive bounds for the critical density of activated random walks

Abstract

We consider symmetric activated random walks on Z, and show that the critical densitywhere λ denotes the sleep rate.

Essential enhancements in Abelian networks: continuity and uniform strict monotonicity

“…It is known from [13] that µ c (λ) ≤ 1 for any λ ∈ [0, ∞) in wide generality, it was proved in [15,17] that µ c (λ) < 1 for any λ ∈ (0, ∞) and that µ c (λ) → 0 as λ → 0 in any vertex-transitive graph where the random walk is transient, extending previous results for biased jump distributions on Z d [11,16]. Moreover, it was proved in [1,4] that, on Z, µ c (λ) = O( √ λ) in the limit as λ → 0. It was proved in [15] that µ c (λ) ≥ λ 1+λ in any vertex-transitive graphs, generalising and extending a previous result of [14].…”

Active Phase for Activated Random Walk on $$\mathbb {Z}$$

“…As we mentioned, we have not tried to optimize these results. We suspect that by similar technique, we could achieve the optimal lower bound ρ * ≥ C √ λ as λ → 0 proven in [ARS22], and that we could show ρ * < 1 for all λ > 0 as proven in [HRR23].…”

“…The other models of ARW considered in this paper have been studied as well, though the literature on them is smaller. The fixed-energy model on a cycle of length n or a torus of width n has been shown to have two phases, with fixation occurring either in polynomial or exponential time [BGHR19,FG22,AFG22], but with no proof that the transition is sharp or that it coincides with ρ FE . The point-source model is studied only in [LS21], which relates the density of the model after stabilization to ρ FE and ρ DD , without proving existence of ρ DD or ρ PS .…”

How Far do Activated Random Walkers Spread from a Single Source?

“…We conjecture that clumps denser than the mean in the infinite-volume stationary state w ∼ π have exponentially small probability. Write The ideas introduced in [1,16] may be useful in proving incompressiblity for ζ sufficiently close to 1.…”

“…for the operation of stabilizing an ARW configuration η: running ARW dynamics until all particles fall asleep. 1 Consider S(nδ 0 ), the ARW aggregate formed by stabilizing n particles at the origin in Z d . We will rescale the aggregate and take a limit as n → ∞.…”