Fixed point forms of the parallel symmetric sandpile model

TL;DR

This paper introduces a parallel symmetric sandpile model that generalizes the traditional model by allowing simultaneous collapses, and proves that its fixed points match those of the sequential model through explicit construction.

Contribution

It presents a new parallel update scheme for the symmetric sandpile model and proves the equivalence of fixed points with the sequential version.

Findings

Fixed points are the same in parallel and sequential models.

Explicit construction method for fixed points in parallel scheme.

Parallel model preserves the set of fixed points of the sequential model.

Abstract

This paper presents a generalization of the sandpile model, called the parallel symmetric sandpile model, which inherits the rules of the symmetric sandpile model and implements them in parallel. In this new model, at each step the collapsing of the collapsible columns happens at the same time and one collapsible column is able to collapse on the left or on the right but not both. We prove that the set of forms of fixed points of the symmetric sandpile model is the same as the one of that model using parallel update scheme by constructing explicitly the way (in the parallel update scheme) to reach the form of an arbitrary fixed point of the sequential model.