On infinite bounded-degree graphs, divisible sandpiles with i.i.d. initial masses of mean μ stabilize almost surely if μ < 1 and masses have finite p-moment for p > 3, but explode if μ ≥ 1; the conditions are nearly sharp via counterexamples on other graphs.
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Divisible sandpiles via random walks in random scenery
On infinite bounded-degree graphs, divisible sandpiles with i.i.d. initial masses of mean μ stabilize almost surely if μ < 1 and masses have finite p-moment for p > 3, but explode if μ ≥ 1; the conditions are nearly sharp via counterexamples on other graphs.