Divisible sandpiles via random walks in random scenery

TL;DR

This paper studies the behavior of divisible sandpiles on various graphs, extending previous results by analyzing an optimal stopping problem for random walks in random scenery, and establishing conditions for stabilization and explosion.

Contribution

It extends the understanding of sandpile stabilization and explosion criteria to general graphs, including unbounded-degree and bounded-degree cases, using a novel optimal stopping analysis.

Findings

Sandpiles with mean mass less than 1 stabilize on certain graphs.

Sandpiles with mean mass greater than or equal to 1 explode on all infinite bounded-degree graphs.

Conditions for stabilization and explosion are nearly sharp, with specific counterexamples provided.

Abstract

We analyze an optimal stopping problem for random walk in random scenery on general graphs, and determine when it has a finite optimum. We use this to extend a theorem of Levine, Murugan, Peres, and Ugurcan [2016]. They proved that on a vertex-transitive graph, the divisible sandpile with i.i.d. initial masses of mean $\mu$ stabilizes almost surely if $\mu < 1$, explodes if $\mu > 1$, and explodes if $\mu = 1$ with positive finite variance. Their proofs rely on conservation of mean mass under toppling. This conservation extends to unimodular random graphs, but fails on general graphs. We prove explosion for all infinite bounded-degree graphs whenever $\mu \geq 1$, and stabilization for $\mu<1$ provided the initial masses have finite $p$-th moment for some $p>3$. Our conditions are nearly sharp: we exhibit unbounded-degree graphs on which sandpiles with $\mu > 1$ stabilize, and for every $p < 3$ we construct bounded-degree graphs on which sandpiles with~$\mu < 1$ and finite $p$-th moment explode.