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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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

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

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

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Zentralblatt MATH identifier

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

Sandpile stationary distribution coupling critical density stabilizability

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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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