On the slow phase for fixed-energy Activated Random Walks

We study the Activated Random Walk model on the one-dimensional ring, in the high density regime. We develop a toppling procedure that gradually builds an environment that can be used to show that activity will be sustained for a long time. This yields a self-contained and relatively short proof of existence of a slow phase for arbitrarily large sleep rates.

💡 Research Summary

The paper investigates the Activated Random Walk (ARW) model on a one‑dimensional ring (the discrete torus ℤₙ) in the high‑density regime, i.e., when the particle density ζ is close to one. In ARW each particle is either active or sleeping. Active particles perform independent continuous‑time simple random walks at rate 1 and fall asleep at rate λ; sleeping particles remain immobile until an active particle lands on their site, at which point they become active again. Because the underlying graph is finite, the system eventually stabilises (all particles become sleeping), but the time to reach this absorbing state can be extremely large. The authors prove that for any finite sleep rate λ>0 there exists a “slow phase” in which the total number of jumps J made before stabilisation grows exponentially in the system size N with high probability.

The main result (Theorem 1) states that for every 0<λ<∞ there are constants δ,δ′>0 (depending only on λ) such that, if the density ζ is sufficiently close to 1, then for all sufficiently large N, \