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

TL;DR

This paper proves the uniqueness of the stationary distribution in Zhang's sandpile model on finite sites and investigates the conditions under which infinite volume configurations are stabilizable, revealing phase transitions based on density.

Contribution

It generalizes the uniqueness result of the stationary measure for Zhang's sandpile model and analyzes stabilizability in infinite volume, identifying critical density thresholds.

Findings

Unique stationary measure for all 0 <= a < b <= 1.

Stabilizability for densities below 1/2.

Non-stabilizability for densities at or above 1.

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 \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.