Electronic Journal of Probability

Uniqueness of the stationary distribution and stabilizability in Zhang's sandpile model

Ronald Meester, Anne Fey-den Boer, and Haiyan Liu

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Abstract

We show that Zhang's sandpile model $(N, [a, b])$ on $N$ sites and with uniform additions on $[a,b]$ has a unique stationary measure for all $0\leq a < b\leq 1$. This generalizes earlier results of cite{anne} where this was shown in some special cases. We define the infinite volume Zhang's sandpile model in dimension $d\geq1$, in which topplings occur according to a Markov toppling process, and we study the stabilizability of initial configurations chosen according to some measure $mu$. We show that for a stationary ergodic measure $\mu$ with density $\rho$, for all $\rho < \frac{1}{2}$, $\mu$ is stabilizable; for all $\rho\geq 1$, $\mu$ is not stabilizable; for $\frac{1}{2}\leq \rho<1$, when $\rho$ is near to $\frac{1}{2}$ or $1$, both possibilities can occur.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 32, 895-911.

Dates
Accepted: 27 April 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819493

Digital Object Identifier
doi:10.1214/EJP.v14-640

Mathematical Reviews number (MathSciNet)
MR2497456

Zentralblatt MATH identifier
1191.60087

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60F05: Central limit and other weak theorems 60B10: Convergence of probability measures 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Sandpile stationary distribution coupling critical density stabilizability

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Meester, Ronald; Fey-den Boer, Anne; Liu, Haiyan. Uniqueness of the stationary distribution and stabilizability in Zhang's sandpile model. Electron. J. Probab. 14 (2009), paper no. 32, 895--911. doi:10.1214/EJP.v14-640. https://projecteuclid.org/euclid.ejp/1464819493


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References

  • P. Bak, C. Tang, K.Wiesenfeld. Self-organized criticality. Phys. Rev. A (3) 38 (1988), no. 1, 364–374.
  • Benjamini, Itai, Lyons, Russell, Peres, Yuval, Schramm, Oded. Critical percolation on any nonamenable group has no infinite clusters. Ann. Prob. 27 (1999), no. 3, 1347–1356.
  • D. Dhar. Studying self-organized criticality with exactly solved models, Arxiv, 1999. cond-mat/9909009.
  • R. Dickman, M. Munoz, A. Vespagnani and S. Zapperi. Paths to self-organized criticality. Brazilian Journal of Physics 30 (2000), 27-41.
  • W. Feller. An introduction to probability theory and its applications. Vol. II. John Wiley & Sons, Inc., New York-London-Sydney 1966 xviii+636 pp.
  • A. Fey-den Boer, R. Meester, C. Quant, F. Redig. A probabilistic approach to Zhang's sandpile model, Comm. Math. Phys. 280 (2) (2008), 351-388.
  • A. Fey-den Boer, R. Meester, F. Redig. (2008) Stabilizability and percolation in the infinite volume sandpile model. To appear in Annals of Probability.
  • A. Fey-den Boer, F. Redig. Organized versus self-organized criticality in the abelian sandpile model. Markov Process. Related Fields 11 (2005), no. 3, 425–442.
  • I.M. Janosi. Effect of anisotropy on the self-organized critical state. Phys. Rev. A 42 (2) (1989), 769-774.
  • R. Meester, C. Quant. Connections between `self-organised' and `classical' criticality. Markov Process. Related Fields 11 (2005), no. 2, 355–370. (
  • R. Meester, F. Redig, D. Znamenski. The abelian sandpile: a mathematical introduction. Markov Process. Related Fields 7 (2001), no. 4, 509–523. (
  • J.R. Norris. Markov chains. Reprint of 1997 original. Cambridge Series in Statistical and Probabilistic Mathematics, 2. Cambridge University Press, Cambridge, 1998. xvi+237 pp. ISBN: 0-521-48181-3 (
  • H. Thorisson. Coupling, stationarity, and regeneration. Probability and its Applications (New York). Springer-Verlag, New York, 2000. xiv+517 pp. ISBN: 0-387-98779-7
  • Y.-C. Zhang. Scaling theory of Self-Organized Criticality. Phys. Rev. Lett. 63 (5) (1989), 470-473.