Finiteness of Two-Dimensional Sandpile Avalanches

Definitions and mathematical statement

Use the nearest-neighbour graph on ${\mathbb{Z}}^2$. Adopt the height convention in which a stable configuration is $\eta\in\{0,1,2,3\}^{{\mathbb{Z}}^2}$, and a toppling at a vertex removes four grains there and sends one grain to each of its four neighbours. Let $\nu$ denote the stationary measure referred to in the supplied record; its precise construction is not supplied.

Fix an addition site $x\in{\mathbb{Z}}^2$. To describe the avalanche without assuming that infinite-volume stabilization terminates, let $V_n=\{-n,\ldots,n\}^2$, with $n$ sufficiently large that $x\in V_n$. Restrict $\eta$ to $V_n$, add one grain at $x$, and stabilize with all grains leaving $V_n$ absorbed by a sink. Let $u_{n,\eta}^{x}(y)$ be the number of topplings at $y$, extended by zero outside $V_n$, and define the extended toppling count \[ u_{\eta}^{x}(y):=\sup_{n:\,x\in V_n}u_{n,\eta}^{x}(y)\in{\mathbb{N}}\cup\{0,\infty\}. \] The auxiliary sink specifies the finite-volume approximation; there is no dissipative vertex at a finite site in the infinite lattice. Write \[ A_x(\eta):=\{y\in{\mathbb{Z}}^2:u_{\eta}^{x}(y)>0\},\qquad T_x(\eta):=\sum_{y\in{\mathbb{Z}}^2}u_{\eta}^{x}(y). \] Here an avalanche is called finite when $T_x(\eta)<\infty$, equivalently when its toppled set is finite and every vertex topples finitely many times.

Question. For the intended infinite-volume stationary sandpile measure $\nu$, is \[ \nu\left(\{\eta:T_x(\eta)<\infty\}\right)=1 \] for each fixed addition site $x\in{\mathbb{Z}}^2$?

The supplied statement presents finiteness of the number of topplings at the addition site, \[ \nu\left(\{\eta:u_{\eta}^{x}(x)<\infty\}\right)=1, \] as a sufficient route to an affirmative answer. This sufficiency is retained as a source assertion, not established here. In particular, finiteness of a single toppling count does not by definition give finiteness of $T_x$.

Unresolved qualifications. The source does not specify the construction of $\nu$, the infinite-volume stabilization convention, or whether "finite avalanche" explicitly means finitely many topplings or finitely many toppled vertices. The definitions above give a conventional precise formulation for review; their agreement with the intended source formulation, and the asserted sufficient route, require confirmation.

Short English statement

After adding one grain at a fixed site of a stationary sandpile on ${\mathbb{Z}}^2$, must the resulting avalanche contain only finitely many topplings with probability one? The source proposes proving that the addition site topples only finitely often as a sufficient route. The intended stationary measure and the justification of that route require review.

Sources

• Local supplied problem registry, permanent UnsolvedMath ID $30003723$, title Finiteness of Two-Dimensional Sandpile Avalanches, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The record gives proposed year 2018.
• Supplied report citation: Strongly Correlated Random Interacting Processes (2019), Oberwolfach Reports, pp. 187--253, DOI 10.4171/owr/2018/4. Supplied source link: https://doi.org/10.4171/owr/2018/4. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labelled "original OWR report": https://doi.org/10.4171/OWR/2018/4.
• Supplied literature source, labelled "Járai, sandpile models survey": https://arxiv.org/abs/1401.0354.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that almost-sure avalanche finiteness after one grain addition on ${\mathbb{Z}}^2$ remains open and that "the cited implication" has not been established. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial metadata, not independent verification of the mathematical formulation or literature status.