A shape theorem for exploding sandpiles

TL;DR

This paper investigates the scaling limits of exploding Abelian sandpiles, establishing conditions for limit shape existence, providing criteria for explosiveness, and highlighting cases where convergence fails, thus advancing understanding of sandpile dynamics.

Contribution

It introduces new conditions for the existence of limit shapes in exploding sandpiles and offers criteria to identify explosive configurations, extending prior results.

Findings

Established sufficient conditions for limit shape existence.

Provided counterexamples where convergence does not occur.

Derived a simple criterion for sandpile explosiveness.

Abstract

We study scaling limits of exploding Abelian sandpiles using ideas from percolation and front propagation in random media. We establish sufficient conditions under which a limit shape exists and show via a family of counterexamples that convergence may not occur in general. A corollary of our proof is a simple criteria for determining if a sandpile is explosive; this strengthens a result of Fey, Levine, and Peres (2010).