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

Levine, Lionel; Silvestri, Vittoria

doi:10.1007/s10955-021-02836-9

Cited by 7 publications

(6 citation statements)

References 41 publications

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“…because, conditioned on F, the remaining toppling instructions in τ are still distributed according to P λ,A and are independent of the instructions revealed to construct β (see proposition 4 in [LS21] for a formal statement of this strong Markov property). Combining this with equation (11), we obtain the claimed result.…”

Section: Starting With One Active Particle On Each Sleeping Sitementioning

confidence: 99%

Active Phase for Activated Random Walks on the Lattice in all Dimensions

Forien

Gaudillière

2022

Preprint

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We show that the critical density of the Activated Random Walk model on Z d is strictly less than one when the sleep rate λ is small enough, and tends to 0 when λ → 0, in any dimension d 1. As far as we know, the result is new for d = 2.We prove this by showing that, for high enough density and small enough sleep rate, the stabilization time of the model on the d-dimensional torus is exponentially large. To do so, we fix the the set of sites where the particles eventually fall asleep, which reduces the problem to a simpler model with density one. Taking advantage of the Abelian property of the model, we show that the stabilization time stochastically dominates the escape time of a one-dimensional random walk with a negative drift. We then check that this slow phase for the finite volume dynamics implies the existence of an active phase on the infinite lattice.

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Section: Starting With One Active Particle On Each Sleeping Sitementioning

confidence: 99%

Active Phase for Activated Random Walks on the Lattice in all Dimensions

Forien

Gaudillière

2022

Preprint

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“…The SS and ARW Markov chains still couple, provided suitable compatible toppling prescriptions are chosen to replace equations ( 10)-( 12). The general coupling can be used to derive results on mixing times and odometer bounds for SS on finite graphs (see, for example, [12,13]).…”

Section: Corollary 1 Consider the Ss Model Starting From An Arbitrary...mentioning

confidence: 99%

“…The SS Markov chain {s t } t∈N 0 takes values in the set S of all SS configurations. We let one Markov chain step correspond to one SS toppling, according to the toppling prescription given in (12).…”

Section: Couplingmentioning

confidence: 99%

“…Consider the coupled isomorphic Markov chains. For a fixed field of instructions, the number of times each site has SS-toppled is exactly two times the number of times each site has ARW-toppled up until the very last step, thanks to our compatible toppling prescriptions defined in ( 10)- (12). The last step can correspond to either one or two ARW topplings, giving the formula v(x) = u 2 .…”

Section: Lemma 6 (Coupling Lemma) Consider Arw With Sleep Rate λ(N)mentioning

confidence: 99%

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Stochastic sandpile on a cycle

Melchionna¹

2022

J. Phys. A: Math. Theor.

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In the stochastic sandpile model on a graph, particles interact pairwise as follows: if two particles occupy the same vertex, they must each take an independent random walk step with some probability $0<p<1$ of not moving. These interactions continue until each site has no more than one particle on it. We provide a formal coupling between the stochastic sandpile and the activated random walk models, and we use the coupling to show that for the stochastic sandpile with $n$ particles on the cycle graph $\mathbb{Z}_n,$ the system stabilizes in $O(n^3)$ time for all initial particle configurations, provided that $p(n)$ tends to $1$ sufficiently rapidly as $n \rightarrow \infty$.

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“…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 .…”

Section: Introductionmentioning

confidence: 99%