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

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

2008-09-17

arxiv.org/abs/0809.2913v1

Electronic Journal of Probability 14 (2009), 895-911

Physics

Mathematical Physics

22 pages

Scientific paper

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 \mu. We show that for a stationary ergodic measure \mu with density \rho, for all \rho < 1/2, \mu is stabilizable; for all \rho >= 1, \mu is not stabilizable; for 1/2 <= \rho < 1, when \rho is near to 1/2 or 1, both possibilities can occur.

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