Avalanche Tails and Height Correlations in Sandpiles

Definitions and mathematical statement

Let $d\in{\mathbb{Z}}$, $d\geq2$, and consider the Abelian sandpile model on ${\mathbb{Z}}^d$. Write $o=(0,\ldots,0)$, let $\nu$ denote the infinite-volume measure intended by the source, and let $\eta(x)$ denote the height at $x$ in a configuration sampled from $\nu$. The source does not specify the height convention or the construction and boundary conditions defining $\nu$; these must be fixed before a fully precise formulation is possible.

1. Let $\operatorname{Av}$ denote the avalanche intended by the source. Determine the asymptotic behavior of \[ \nu\left(|\operatorname{Av}|>t\right),\qquad t\to\infty, \] for every integer $d\geq2$. The addition site, stabilization procedure, and meaning of $|\operatorname{Av}|$ are unspecified. In particular, counting distinct toppled sites and counting all topplings are different observables; the supplied record does not select one. It also does not specify whether the requested tail description is an exponent, matching bounds, or an asymptotic equivalent.

2. The original statement additionally asks whether the centered height-zero field $h_\varepsilon$ converges, as $\varepsilon\downarrow0$, to the generalized field denoted by \[ :\!|\nabla h|^2\!: \] in a suitable space. This question is absent from the clean statement but is retained here because it is part of the supplied original statement. The record does not define $h_\varepsilon$, its scaling or normalization, the underlying field $h$, the meaning of the displayed renormalized expression, the dimension, the spatial domain, or the topology and mode of convergence. Consequently, this part remains a formal target requiring those specifications.

3. In dimension $d=2$, define the height-indicator covariance by \[ C_{i,j}(x) :=\operatorname{Cov}_{\nu}\!\left(\mathbf{1}_{\{\eta(o)=i\}},\mathbf{1}_{\{\eta(x)=j\}}\right) =\nu\left(\eta(o)=i,\eta(x)=j\right) -\nu\left(\eta(o)=i\right)\nu\left(\eta(x)=j\right). \] For every $i,j\in\{1,2,3\}$, prove or disprove the supplied conjectural asymptotic \[ C_{i,j}(x)\sim a_{i,j}(\log^2|x|)\,|x|^{-4}, \qquad x\in{\mathbb{Z}}^2,\quad |x|\to\infty. \] The source does not define $a_{i,j}$. If it denotes a nonzero constant multiplying $(\log|x|)^2$, the assertion means \[ \lim_{\substack{x\in{\mathbb{Z}}^2\\|x|\to\infty}} \frac{C_{i,j}(x)}{a_{i,j}(\log|x|)^2|x|^{-4}}=1. \] This conditional interpretation requires confirmation: the notation could instead indicate a function evaluated at $\log^2|x|$. A zero coefficient does not define an asymptotic equivalent in the usual ratio sense. The height labeling, meaning of $|x|$, and any directional restriction on the limit also require confirmation.

The supplied dated curation labels the problem partially solved and reports a mean-field avalanche-size tail exponent in dimensions at least five, with the requested remaining low-dimensional behavior unresolved. This is a supplied literature assessment, not an independently verified theorem or status determination.

Short English statement

Determine avalanche-size tails for the Abelian sandpile on ${\mathbb{Z}}^d$, $d\geq2$, and test the stated two-dimensional height-correlation asymptotic for every $i,j\in\{1,2,3\}$. Retain the original question about convergence of the centered height-zero field, which the clean statement omits. Avalanche conventions, field definitions, and the correlation coefficients require clarification. The supplied partially solved label is dated curation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003725$, title Avalanche Tails and Height Correlations in Sandpiles, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2018 and supplies no named proposer.
• Supplied source citation: Strongly Correlated Random Interacting Processes (2019), Oberwolfach Reports, pp. 187--253, DOI 10.4171/owr/2018/4. Supplied source URL: https://doi.org/10.4171/owr/2018/4. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's 2019 designation and the record's 2018 year are retained without reconciliation.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/OWR/2018/4.
• Supplied literature source, labeled "Hutchcroft, universality of high-dimensional sandpiles": https://doi.org/10.1007/s00440-019-00923-3.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem partially solved and reports that the avalanche-size tail has the mean-field exponent in dimensions at least five, while the requested remaining low-dimensional behavior is unresolved. This assessment is supplied dated curation derived from the OWR material, not independently verified here. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the clean statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial claims; they do not resolve the omitted field-convergence question or establish mathematical correctness. No cited source was independently consulted for this draft.