Infinite-cluster percolation for the abelian sandpile

Determine whether there exists a stabilizable i.i.d. initial-height law for the infinite-volume abelian sandpile on \(\mathbb{Z}^d\) such that the set of sites that topple contains an infinite cluster.

Background

The paper studies percolation of toppled sites for the divisible sandpile and proves that, under suitable hypotheses, the toppled set percolates at densities strictly below one. It contrasts this result with the corresponding problem for the integer-valued abelian sandpile, where stabilization is possible below the critical density.

For the abelian sandpile started from i.i.d. heights, the unresolved issue is whether a stabilizable law can nevertheless produce an infinite connected component of toppled sites. The divisible-sandpile theorem establishes an analogue of such behavior but does not resolve the abelian case.