The divisible sandpile at critical density

TL;DR

This paper investigates the behavior of the divisible sandpile model at the critical density, showing that with finite variance initial masses, the process almost surely does not stabilize at the critical point on infinite graphs.

Contribution

It establishes that the divisible sandpile at critical density with finite variance initial masses almost surely does not stabilize, extending understanding of critical behavior in such models.

Findings

At critical density, the process does not stabilize with finite variance.

The number of topplings relates to a discrete biLaplacian Gaussian field.

Provides quantitative estimates on finite graphs.

Abstract

The divisible sandpile starts with i.i.d. random variables ("masses") at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses at most 1. The process stabilizes almost surely if m<1 and it almost surely does not stabilize if m>1, where $m$ is the mean mass per vertex. The main result of this paper is that in the critical case m=1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete biLaplacian Gaussian field.