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

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 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.