Characterizing immutable sandpiles: A first look

TL;DR

This paper explores the concept of immutable sandpiles, where two notions of stability coincide, providing linear-algebraic characterizations for broad classes of such sandpiles on graphs.

Contribution

It introduces the notion of immutable sandpiles and offers linear-algebraic criteria to identify them, advancing understanding of sandpile stability.

Findings

Linear-algebraic characterizations for classes of immutable sandpiles

Equivalence of stability notions in certain sandpiles

Framework applicable to sandpiles over integers and reals

Abstract

By working with coefficients in $\mathbb{Z}$ or $\mathbb{R}$, one can define two different notions of stability for a sandpile on a graph. We call a sandpile immutable when these notions agree. Our main results give linear-algebraic characterizations for large classes of immutable sandpiles.