Introduction to the sandpile model

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

doi:10.1016/s0378-4371(98)00012-0

Cited by 65 publications

(67 citation statements)

References 29 publications

(62 reference statements)

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“…v) The 2 + 1 (2 space, 1 time) dimensional ASM has a dynamic exponent z = 2 [31,32] and the stationary 2-d probability distribution describes a critical system with correlation functions given by a c = −2 LCFT. At u = 1 the 1 + 1 RPM has a dynamic exponent z = 1 and the spectrum of the Hamiltonian is given by generic characters of a c = 0 LCFT.…”

Section: 3 16mentioning

confidence: 99%

The Raise and Peel Model of a Fluctuating Interface

Gier

Nienhuis

Pearce

et al. 2004

Journal of Statistical Physics

133

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We propose a one-dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter, the adsorption rate. At a special value of this parameter, the model has some interesting features. For example, the spectrum is given by conformal field theory, and the stationary non-equilibrium probability distribution is given by the two-dimensional equilibrium distribution of the ice model with domain wall type boundary conditions. This connection is used to find exact analytic expressions for several quantities of the stochastic model. Vice versa, some understanding of the ice model with domain wall type boundary conditions can be obtained by the study of the stochastic model. At the special point we study several properties of the model, such as the height fluctuations as well as cluster and avalanche distributions. The latter has a long tail which shows that the model exhibits self organized criticality. We also find in the stationary state a special phase transition without enhancement and with a crossover exponent φ = 2/3. Furthermore, we study the phase diagram of the model as a function of the adsorption rate and find two massive phases and a scale invariant phase where conformal invariance is broken.

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Section: 3 16mentioning

confidence: 99%

The Raise and Peel Model of a Fluctuating Interface

Gier

Nienhuis

Pearce

et al. 2004

Journal of Statistical Physics

133

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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%

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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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“…It will be convenient to order the topplings in so-called waves [9]. Suppose the addition of energy at time t takes place at site k and makes this site unstable.…”

Section: Model Definitionmentioning

confidence: 99%

A Probabilistic Approach to Zhang’s Sandpile Model

Boer

Meester

Quant

et al. 2008

Commun. Math. Phys.

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The current literature on sandpile models mainly deals with the abelian sandpile model (ASM) and its variants. We treat a less known -but equally interesting -model, namely Zhang's sandpile. This model differs in two aspects from the ASM. First, additions are not discrete, but random amounts with a uniform distribution on an interval [a, b]. Second, if a site topples -which happens if the amount at that site is larger than a threshold value E c (which is a model parameter), then it divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the infinite volume limit, this model tends to behave like the ASM in the sense that the stationary measure for the system in large volumes tends to be peaked narrowly around a finite set. This belief is supported by simulations, but so far not by analytical investigations.We study the stationary distribution of this model in one dimension, for several values of a and b. When there is only one site, exact computations are possible. Our main result concerns the limit as the number of sites tends to infinity, in the one-dimensional case. We find that the stationary distribution, in the case a ≥ E c /2, indeed tends to that of the ASM (up to a scaling factor), in agreement with Zhang's conjecture. For the case a = 0, b = 1 we provide strong evidence that the stationary expectation tends to 1/2.

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Introduction to the sandpile model

Cited by 65 publications

References 29 publications

The Raise and Peel Model of a Fluctuating Interface

The Raise and Peel Model of a Fluctuating Interface

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

A Probabilistic Approach to Zhang’s Sandpile Model

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