Fey–Meester–Redig divisible-sandpile percolation question

Fix $d\ge 2$ . Let $\eta=(\eta(x))_{x\in\mathbb{Z}^d}$ be an i.i.d. initial mass configuration for the divisible sandpile, with mean density $\rho$ , and let $\mathcal{T}_\rho=\{x\in\mathbb{Z}^d:x\text{ topples at least once during stabilization}\}$ . Does there exist a density $\rho<1$ such that $\mathbb{P}(\mathcal{T}_\rho$ contains an infinite nearest-neighbor connected component $)>0$ ? Equivalently, is the critical percolation density $\rho_c:=\inf\{\rho:\mathbb{P}(\mathcal{T}_\rho\text{ contains an infinite component})>0\}$ strictly less than $1$ ?

References

Primary source

arXiv

Additional references

Quantitative explosion and percolation of the divisible sandpile — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new paper claims to settle a variant of the long-standing question, but it does not establish that the original question itself is solved.

The Fey–Meester–Redig question concerns percolation in the divisible-sandpile model. The newly reported result addresses a variant of that question rather than clearly settling the original formulation.

September 2, 2026 report

The paper Quantitative explosion and percolation of the divisible sandpile claims dimension- and tail-sensitive explosion rates and a nontrivial percolation phase transition below the mean-one threshold. This is substantial progress and is presented as resolving a longstanding variant, but the exact relationship to the original question remains unverified.

Current status (as of September 2026): A variant of the question is claimed solved by the new paper, while the original formulation remains unsettled.

Sources

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