Waves of topplings in an Abelian sandpile

Ivashkevich, E. V.; Ktitarev, D. V.; Priezzhev, V. B.

doi:10.1016/0378-4371(94)90188-0

Cited by 94 publications

(144 citation statements)

References 15 publications

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“…This process is continued until O becomes stable. Thus, an avalanche is broken into a sequence of waves of toppling [34]. It was shown by Priezzhev that the set of all waves is in one-to-one correspondence with all two-rooted spanning trees.…”

Section: Waves Of Topplingmentioning

confidence: 99%

The Abelian sandpile and related models

Dhar

1999

Physica A: Statistical Mechanics and its Applications

298

291

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The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality. This paper presents a brief review of known results about the model. The abelian group structure of the algebra of operators allows an exact calculation of many of its properties. In particular, when there is a preferred direction, one can calculate all the critical exponents characterizing the distribution of avalanche-sizes in all dimensions. For the undirected case, the model is related to q → 0 state Potts model. This enables exact calculation of some exponents in two dimensions, and there are some conjectures about others. We also discuss a generalization of the model to a network of communicating reactive processors. This includes sandpile models with stochastic toppling rules as a special case. We also consider a non-abelian stochastic variant, which lies in a different universality class, related to directed percolation.

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Section: Waves Of Topplingmentioning

confidence: 99%

The Abelian sandpile and related models

Dhar

1999

Physica A: Statistical Mechanics and its Applications

298

291

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“…This is done by a decomposition of avalanches into a sequence of waves (cf. [10,11]), and studying the almost sure finiteness of the waves. The latter can be achieved by a two-component spanning tree representation of waves, as introduced in [10,11].…”

Section: Infinite Volume: Basic Questions and Resultsmentioning

confidence: 99%

“…[10,11]), and studying the almost sure finiteness of the waves. The latter can be achieved by a two-component spanning tree representation of waves, as introduced in [10,11]. We then study the uniform two-component spanning tree in infinite volume and prove that the component containing the origin is almost surely finite.…”

Section: Infinite Volume: Basic Questions and Resultsmentioning

confidence: 99%

“…The proof is based on a correspondence with two-component spanning trees due to Ivashkevich, Ktitarev and Priezzhev [10], which is recalled below. The correspondence allows us to use some results on the uniform spanning forest that are stated separately as Proposition 7.11 below.…”

Section: Finiteness Of Wavesmentioning

confidence: 99%

“…The relation between recurrent configurations and spanning trees, introduced by Majumdar and Dhar [22], has been used by Priezzhev to compute the stationary height probabilities of the two-dimensional model in the thermodynamic limit [25]. Later on, Ivashkevich, Ktitarev and Priezzhev introduced the concept of "waves" to study the avalanche statistics, and made a connection between two-component spanning trees and waves [9,10]. In [26] this connection was used to argue that the critical dimension of the ASM is d = 4.…”

Section: Introductionmentioning

confidence: 99%

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Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3

Járai

Redig

2007

Probab. Theory Relat. Fields

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Abstract:We study the Abelian sandpile model on Z d . In d ≥ 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit µ of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure µ, and study ergodic properties of this process. The main techniques we use are a connection between the statistics of waves and uniform two-component spanning trees and results on the uniform spanning forest measure on Z d .

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Measures for Transient Configurations of the Sandpile-Model

Schulz¹

2006

Lecture Notes in Computer Science

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Waves of topplings in an Abelian sandpile

Cited by 94 publications

References 15 publications

The Abelian sandpile and related models

The Abelian sandpile and related models

Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3

Measures for Transient Configurations of the Sandpile-Model

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