Dynamic dimensional reduction in the Abelian sandpile

TL;DR

This paper proves the dimensional reduction conjecture for the Abelian sandpile on the hypercube, demonstrating that key properties persist during toppling and confirming earlier empirical observations.

Contribution

It provides a rigorous proof of the dimensional reduction conjecture and shows the persistence of symmetry and regularity during the sandpile's evolution.

Findings

Dimensional reduction holds during the toppling process

Symmetry and regularity persist throughout the dynamics

Empirical observations are theoretically validated

Abstract

We prove the dimensional reduction conjecture of Fey, Levine, and Peres (2010) on the hypercube. The proof shows that dimensional reduction, symmetry, and regularity of the Abelian sandpile persist during the parallel toppling process. This stronger result verifies empirical observations first documented by Liu, Kaplan, and Gray (1990).