A probabilistic approach to Zhang's sandpile model

TL;DR

This paper analyzes Zhang's sandpile model, a less-studied variant of the abelian sandpile, demonstrating that its stationary distribution converges to that of the ASM in the infinite volume limit under certain conditions.

Contribution

It provides the first analytical investigation of Zhang's sandpile model's stationary distribution, confirming Zhang's conjecture in one dimension for specific parameters.

Findings

Stationary distribution converges to ASM distribution for large systems when a ≥ Ec/2.

Exact computations possible for single-site case.

Stationary expectation tends to √1/2 for a=0, b=1 case.

Abstract

The current literature on sandpile models mainly deals with the abelian sandpile model (ASM) and its variants. We treat a less known - but equally interesting - model, namely Zhang's sandpile. This model differs in two aspects from the ASM. First, additions are not discrete, but random amounts with a uniform distribution on an interval $[a,b]$. Second, if a site topples - which happens if the amount at that site is larger than a threshold value $E_c$ (which is a model parameter), then it divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the infinite volume limit, this model tends to behave like the ASM in the sense that the stationary measure for the system in large volumes tends to be peaked narrowly around a finite set. This belief is supported by simulations, but so far not by analytical investigations. We study the stationary distribution of this model in one dimension, for several values of $a$ and $b$. When there is only one site, exact computations are possible. Our main result concerns the limit as the number of sites tends to infinity, in the one-dimensional case. We find that the stationary distribution, in the case $a \geq E_c/2$, indeed tends to that of the ASM (up to a scaling factor), in agreement with Zhang's conjecture. For the case $a=0$, $b=1$ we provide strong evidence that the stationary expectation tends to $\sqrt{1/2}$.