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arXiv · 0809.2913

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

Anne Fey·Haiyan Liu·Ronald Meester
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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 <= a < b <= 1. This generalizes earlier results where this was shown in some special cases. We define the infinite volume Zhang's sandpile model in dimension d >= 1, in which topplings occur according to a Markov toppling process, and we study the stabilizability of initial configurations chosen according to some measure μ. We show that for a stationary ergodic measure μwith density ρ, for all ρ< 1/2, μis stabilizable; for all ρ>= 1, μis not stabilizable; for 1/2 <= ρ< 1, when ρis near to 1/2 or 1, both possibilities can occur.

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Anne Fey, Haiyan Liu, Ronald Meester. 2008-09-17. Uniqueness of the stationary distribution and stabilizability in Zhang's sandpile model. https://arxiv.org/abs/0809.2913

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