Activated random walk on a cycle

Abstract

We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle Z /n Z. One starts with a mass density µ > 0 of initially active particles, each of which performs a simple symmetric random walk at rate one and falls asleep at rate λ > 0. Sleepy particles become active on coming in contact with other active particles. There have been several recent results concerning fixation/non-fixation of the ARW dynamics on infinite systems depending on the parameters µ and λ. On the finite graph Z /n Z, unless there are more than n particles, the process fixates (reaches an absorbing state) almost surely in finite time. In a first rigorous result for a finite system, establishing well known beliefs in the statistical physics literature, we show that the number of steps the process takes to fixate is linear in n (up to poly-logarithmic terms), when the density is sufficiently low compared to the sleep rate, and exponential in n when the sleep rate is sufficiently small compared to the density, reflecting the fixation/non-fixation phase transition in the corresponding infinite system as established in [22]. arXiv:1709.09163v2 [math.PR]

Diffusive bounds for the critical density of activated random walks

“…Proving that ζ c < 1 for some λ in d = 2 is still an open problem. Existence of slow stabilization phase and a fast stabilization phase for a conservative finite-volume dynamics was studied in Basu et al (2019a).…”

“…See alsoCabezas and Rolla (2021);Basu et al (2019b);Podder and Rolla (2021);Taggi (2020) for more recent developments.…”

Activated Random Walks on $\mathbb{Z}^{d}$

“…For the unbiased case, it was shown in [BGH18] that ζ c → 0 as λ → 0, and the proof given in §5 is adapted from [BGH18]. Existence of slow and fast regimes for the fixed-energy model on Z n was proved in [BGHR19]. The proof given in §6 is adapted from [BGHR19], whose analysis is substantially based on results and arguments from [BGH18, Jan18, KV03, RS12].…”

“…Existence of slow and fast regimes for the fixed-energy model on Z n was proved in [BGHR19]. The proof given in §6 is adapted from [BGHR19], whose analysis is substantially based on results and arguments from [BGH18, Jan18, KV03, RS12].…”

Stochastic Sandpile on a Cycle

“…As we develop below, the ARW model couples nicely to the SS model, has dynamics on the cycle graph which are well-studied [8,9], and can alleviate some of the aforementioned difficulties in analyzing SS. Aside from its own mathematical interest, the ARW model is useful to proofs of fast stabilization of the stochastic sandpile model for two main reasons.…”