Stability of patterns in the Abelian sandpile

TL;DR

This paper proves the stability of patterns in the Abelian sandpile model by combining structure theory with elliptic equation regularity, enabling improved convergence results for certain solutions.

Contribution

It introduces a novel stability proof for sandpile patterns using elliptic regularity techniques, enhancing understanding of pattern convergence.

Findings

Patterns in the Abelian sandpile are stable.

Stability results improve convergence from weak-* to pattern convergence.

The proof integrates structure theory with elliptic regularity machinery.

Abstract

We show that the patterns in the Abelian sandpile are stable. The proof combines the structure theory for the patterns with the regularity machinery for non-divergence form elliptic equations. The stability results allows one to improve weak-* convergence of the Abelian sandpile to pattern convergence for certain classes of solutions.