Non-fixation for Conservative Stochastic Dynamics on the Line

Abstract

ABSTRACT. We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on Z with mass conservation. One starts with a mass density µ > 0 of initially active particles, each of which performs a 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. We investigate the question of fixation/non-fixation of the process and show for small enough λ the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as λ tends to zero. This settles a long standing open question. THE MODEL DESCRIPTION AND MAIN RESULTSelf organized criticality (SOC) is a universal property describing systems that have critical points as an attractor. In most of the examples of SOC, simple local moves give rise to complex global properties. These systems are driven under their natural evolution to the boundary between stable and unstable states without any fine-tuning of parameters. A canonical and widely studied example of such a model is the Abelian Sandpile model proposed by Bak, Tang and Weisenfeld [3]. In this model, in finite volume, particles are added at random, which dissipate across the boundary; the corresponding closed system in the infinite volume setting is a so-called fixed energy sandpile where the total number of particles is conserved.In a sequence of influential papers Dickman, Vespignani and their co-authors [14,15,42,43] developed a theory of SOC, which has since become widely accepted in statistical physics literature. Their theory predicts a specific relationship between driven dissipative systems and the corresponding systems where total number of particles is conserved; and in particular an absorbing state phase transition, exhibited by the latter system as the particle density is varied. In the finite state, even though there is no tuning parameter, they argued that the loss of particles through the sink(s) is balanced with the dynamical addition of new particles, driving the system to the edge of instability. However subsequently, there has been some controversy in the mathematics and physics community surrounding their predictions. We elaborate more on this in § 1.1.In the last fifteen years, there has also been significant progress in studying these systems via a combinatorial approach, in particular exploiting the so-called Abelian property. Roughly, the Abelian property in a distributed network stipulates that the system produces the same output regardless of the order in which a sequence of local moves are performed. A systematic mathematical treatment of such systems can be found in [19]. Even though this might seem a severe restriction, systems with Abelian property are known to produce intricate large-scale patterns starting with relatively simple local rules. In addition to the Abelian sandpile model and a number of cellular automata introduced by...

Diffusive bounds for the critical density of activated random walks

“…Our main result gives a diffusive upper bound for ζ c (λ) when d = 1, improving the recent result by Basu, Ganguly and Hoffman (Basu et al, 2018).…”

“…See Rolla (2020) for a detailed account. 1 For unbiased jumps in one dimension, it was shown in Basu et al (2018) that ζ c < 1 for λ small and ζ c → 0 as λ → 0. Proving that ζ c < 1 for some λ in d = 2 is still an open problem.…”

“…As in Basu et al (2018), in order to get the upper bound in (1.1) we prove a quantitative estimate for the finite-volume dynamics, defined as follows. For a finite interval V ⊆ Z, consider a dynamics such that walks are lost forever when they escape V .…”

“…(1.4) Theorem 1.3 follows from Theorem 1.2, see Basu et al (2019a, §4) or Rolla (2020, §6.1). The proof of Theorem 1.2 follows a general framework introduced in Basu et al (2018). The key idea is to decompose space into blocks with independent instructions, so that they interact only through the number of walks arriving at the center of each block after having been "emitted" from a neighboring box.…”

Deviation bounds for the first passage time in the frog model

“…Furthermore, we can also find the so-called activated random walk model. This is similar to the combustion model, but there are initially some active frogs and any active frog may fall back to sleep randomly; see [5], [8], [34], [35], [36], and the references given therein.…”