Sandpiles on the Square Lattice

Hough, Bob; Jerison, Dan; Levine, Lionel

doi:10.1007/s00220-019-03408-5

Cited by 14 publications

(30 citation statements)

References 43 publications

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“…See [18] for precise results and an excellent discussion on the above topic. A recent work of Hough, Jerison and Levine [20] prove among other things, an improved upper bound for the critical density for the fixed energy model on Z 2 . See also Levine [27] for an investigation of the relationship between the two notions of criticality.…”

Section: The Model Description and Main Resultsmentioning

confidence: 96%

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Non-fixation for Conservative Stochastic Dynamics on the Line

Basu

Ganguly

Hoffman

2017

Commun. Math. Phys.

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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...

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Section: The Model Description and Main Resultsmentioning

confidence: 96%

“…be the (i, ·) labeled sleepy particles inη x,∞ . 20 Fix a sequence of non negative integers, y = y(− 2r K + 1), . .…”

Section: )mentioning

confidence: 99%

Non-fixation for Conservative Stochastic Dynamics on the Line

Basu

Ganguly

Hoffman

2017

Commun. Math. Phys.

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“…The inequality (11) follows from a standard coupon collector bound; see, for example, [17, Prop. 2.4], which implies that for t > #V log #V + log(1/ǫ)#V we have…”

Section: 3mentioning

confidence: 99%

“…In contrast to (1), the abelian sandpile has an interval of critical means [9], and the problem of whether a sandpile on Z d stabilizes almost surely is not even known to be decidable [19]. The root cause of this nonuniversality is slow mixing: For example, the sandpile mixing time on both the ball B(0, n) ∩ Z d and on the torus Z d n is of order n d log n [11,12]. This extra log factor is responsible for the non-universality of the sandpile threshold state [18].…”

Section: Introduction: Activated Random Walkmentioning

confidence: 99%

Exact sampling and fast mixing of Activated Random Walk

Levine¹,

Li²

2021

Preprint

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Activated Random Walk (ARW) is an interacting particle system on the d-dimensional lattice Z d . On a finite subset V ⊂ Z d it defines a Markov chain on {0, 1} V . We prove that when V is a Euclidean ball intersected with Z d , the mixing time of the ARW Markov chain is at most 1 + o(1) times the volume of the ball. The proof uses an exact sampling algorithm for the stationary distribution, a coupling with internal DLA, and an upper bound on the time when internal DLA fills the entire ball. We conjecture cutoff at time ζ times the volume of the ball, where ζ < 1 is the limiting density of the stationary state.

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“…ARW is believed to manifest self organized criticality when run in a finite volume with carefully controlled driven diffusive dynamics. However, the rigorous study of ARW has so far been mostly restricted to the case of infinite volume limit where the counterpart of SOC is Absorbing state Phase Transition (APT) (although some recent results have called into question the exact relationship between these two notions [10,17,11]). Absorbing state Phase Transition was rigorously established for ARW on Z a few years ago in the fundamental work of Rolla and Sidoravicius [22].…”

Section: Introductionmentioning

confidence: 99%

Activated random walk on a cycle

Basu¹,

Ganguly²,

Hoffman³

et al. 2019

Ann. Inst. H. Poincaré Probab. Statist.

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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]

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Sandpiles on the Square Lattice

Cited by 14 publications

References 43 publications

Non-fixation for Conservative Stochastic Dynamics on the Line

Non-fixation for Conservative Stochastic Dynamics on the Line

Exact sampling and fast mixing of Activated Random Walk

Activated random walk on a cycle

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