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


 
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1. Title Title of document Uniqueness of the stationary distribution and stabilizability in Zhang's sandpile model
 
2. Creator Author's name, affiliation, country Ronald Meester; VU University Amsterdam
 
2. Creator Author's name, affiliation, country Anne Fey-den Boer; TU Delft
 
2. Creator Author's name, affiliation, country Haiyan Liu; VU University Amsterdam
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Sandpile, stationary distribution, coupling, critical density, stabilizability
 
3. Subject Subject classification 60J27, 60F05, 60B10, 82B20
 
4. Description 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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7. Date (YYYY-MM-DD) 2009-04-27
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/640
 
10. Identifier Digital Object Identifier 10.1214/EJP.v14-640
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 14
 
12. Language English=en
 
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