Scaling Limit of the Sandpile Height-Zero Field

Definitions and mathematical statement

Let $D\subset{\mathbb{C}}$ be a bounded smooth domain and, for $\varepsilon>0$, put $D_\varepsilon=D\cap\varepsilon{\mathbb{Z}}^2$. Let $\eta$ have the Abelian sandpile measure $\nu_\varepsilon$ on $D_\varepsilon$, as named in the supplied record. Define the centered height-zero observable by \[ h_\varepsilon(z)=\mathbf{1}_{\{\eta(z)=0\}}-\nu_\varepsilon\left(\eta(z)=0\right),\qquad z\in D_\varepsilon. \] The record does not specify the boundary or sink convention defining $\nu_\varepsilon$, or the convention for labeling sandpile heights.

The proposed scaling-limit question is whether a suitably rescaled realization of $h_\varepsilon$ converges, as $\varepsilon\downarrow0$, in a suitable function or distribution space, to the proposed generalized field \[ :\!|\nabla h|^2\!:\,. \] Here the continuum field $h$, its law, normalization and boundary conditions, and the meaning and existence of the indicated renormalized quadratic-gradient field are not defined in the supplied record.

For example, a distributional formulation could seek deterministic factors $a_\varepsilon$, a specified space $\mathcal X$ of distributions on $D$, and a construction of $Y=:\!|\nabla h|^2\!:$, such that the random distributions defined by \[ \langle X_\varepsilon,\varphi\rangle =a_\varepsilon\varepsilon^2\sum_{z\in D_\varepsilon}h_\varepsilon(z)\varphi(z), \qquad \varphi\in C_c^\infty(D), \] satisfy \[ X_\varepsilon\xrightarrow[\varepsilon\downarrow0]{\mathrm{law}}Y \quad\text{in }\mathcal X. \] This pairing and convergence in law are an explicitly conditional way to formulate the question; the source does not select this embedding or mode of convergence. The normalization, topology, target construction and sandpile boundary conventions remain unresolved specifications. Thus the supplied record determines an intended scaling-limit question, but not a fully specified convergence proposition.

Short English statement

The question asks whether the centered indicator of sandpile height zero on a fine square lattice in a bounded smooth planar domain has a rescaled continuum limit described by $:\!|\nabla h|^2\!:$. The scaling, boundary conventions, continuum field, renormalization and convergence space require clarification. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30003726$, title Scaling Limit of the Sandpile Height-Zero Field, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2018.
• Supplied report citation: Strongly Correlated Random Interacting Processes (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/4, pp. 187--253. Supplied source URL: https://doi.org/10.4171/owr/2018/4. The background attributes the problem collection to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/OWR/2018/4.
• Supplied literature source, labeled "Elvidge--Járai, height-zero scaling": https://arxiv.org/abs/2607.26169. No fuller bibliographic citation is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports rigorous correlation and one-point asymptotic results for the height-zero field, and states that convergence of the renormalized field to the proposed quadratic-gradient field in a specified function space remains unproved. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports checking it against the official Oberwolfach report. These are supplied provenance assertions; no source has been independently visited or checked for this draft.