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2009 Uniqueness of the stationary distribution and stabilizability in Zhang's sandpile model
Ronald Meester, Anne Fey-den Boer, Haiyan Liu
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Electron. J. Probab. 14: 895-911 (2009). DOI: 10.1214/EJP.v14-640

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.

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Ronald Meester. Anne Fey-den Boer. Haiyan Liu. "Uniqueness of the stationary distribution and stabilizability in Zhang's sandpile model." Electron. J. Probab. 14 895 - 911, 2009. https://doi.org/10.1214/EJP.v14-640

Information

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

zbMATH: 1191.60087
MathSciNet: MR2497456
Digital Object Identifier: 10.1214/EJP.v14-640

Subjects:
Primary: 60J27
Secondary: 60B10 , 60F05 , 82B20

Keywords: coupling , critical density , Sandpile , stabilizability , stationary distribution

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