A note on the abelian sandpile in Z^d

TL;DR

This paper provides a new, concise proof that the radius of the toppled cluster in the abelian sandpile model on Z^d with a specific initial configuration grows proportionally to n^{1/d}.

Contribution

It offers a simplified proof of the known theorem regarding the growth of the toppled cluster radius in the abelian sandpile model on Z^d.

Findings

The toppled cluster radius is O(n^{1/d}) for the given initial configuration.

The proof simplifies previous arguments and techniques.

Confirms the growth rate of the toppled cluster radius in high-dimensional lattices.

Abstract

We analyse the abelian sandpile model on $\mathbbm{Z}^d$ for the starting configuration of $n$ particles in the origin and $2d-2$ particles otherwise. We give a new short proof of the theorem of Fey, Levine and Peres \cite{FLP} that the radius of the toppled cluster of this configuration is $O(n^{1/d})$.