Parallel Sandpiles or Spurious Bidirectional Icepiles?

TL;DR

This paper critically analyzes a proposed global rule for sandpile dynamics, revealing it actually models icepile behavior with bidirectional rules, not standard sandpile dynamics, in a one-dimensional lattice.

Contribution

It clarifies the misinterpretation of the global rule and demonstrates it describes icepile rather than sandpile dynamics, highlighting the importance of correct semantic interpretation.

Findings

The FP global rule does not model standard sandpiles.

It describes height differences between piles, not the sandpile dynamics.

The rule corresponds to bidirectional icepile behavior with specific local rules.

Abstract

In a recent paper E. Formenti and K. Perrot (FP) introduce a global rule assumed to describe the discrete time dynamics associated with a sandpile model under the parallel application of a suitable local rule acting on d dimensional lattices of cells equipped with uniform neighborhood. In this paper we submit this approach to a critical analysis, in the simplest elementary particular case of a one-dimensional lattice, which can be divided in two parts. In the first part we prove that the FP global rule does not describe the dynamics of standard sandpiles, but rather furnishes a description of the quite different situation of height difference between consecutive piles. This is a semantic uncorrect difference of interpretation. In the second part we investigate the consequences of the uncorrect FP assumption proving that their global rule describes a bidirectional spurious dynamics of icepiles (rather than sandpiles), in the sense that this latter is the consequence of application of three local rules: bidirectional vertical rule, bidirectional horizontal rule (typical of icepiles), and a granule jump from the bottom to the top (spurious rule of the dynamics).