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

Ronald Meester, Anne Fey-den Boer, Haiyan Liu
2009 Electronic Journal of Probability  
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 of [6] 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
more » ... t for a stationary ergodic measure µ with density ρ, for all ρ < 1 2 , µ is stabilizable; for all ρ ≥ 1, µ is not stabilizable; for 1 2 ≤ ρ < 1, when ρ is near to 1 2 or 1, both possibilities can occur.
doi:10.1214/ejp.v14-640 fatcat:jvug7bzxjfaupkcy565jtjb7cy