Any shape can ultimately cross information on two-dimensional abelian sandpile models

TL;DR

This paper proves that in two-dimensional abelian sandpile models, any scaled neighborhood derived from a continuous non-flat shape can eventually facilitate crossing, expanding understanding of information flow in such systems.

Contribution

It establishes that all scaled neighborhoods from continuous non-flat shapes enable crossing in 2D abelian sandpile models, a novel generalization in the field.

Findings

Any scaled neighborhood from a continuous non-flat shape can perform crossing.

The result applies to all such neighborhoods, regardless of shape specifics.

This broadens the class of neighborhoods known to support crossing in sandpile models.

Abstract

In this paper we study the abelian sandpile model on the two-dimensional grid with uniform neighborhood, and prove that any family of neighborhoods defined as scalings of a continuous non-flat shape can ultimately perform crossing.