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arxiv: 2604.13968 · v1 · submitted 2026-04-15 · 🧮 math.PR

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Divisible sandpiles via random walks in random scenery

Ahmed Bou-Rabee, Ecaterina Sava-Huss, Yuval Peres

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Pith reviewed 2026-05-10 12:29 UTC · model grok-4.3

classification 🧮 math.PR
keywords divisible sandpilerandom walk in random sceneryoptimal stoppingstabilizationexplosionbounded-degree graphsinfinite graphs
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The pith

Divisible sandpiles on bounded-degree infinite graphs explode almost surely when mean initial mass is at least 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reduces the stabilization question for divisible sandpiles on arbitrary infinite graphs to an optimal stopping problem for random walk in random scenery. It proves that on any infinite graph with bounded vertex degrees, the sandpile explodes with probability one if the i.i.d. initial masses have mean μ at least 1. When μ is below 1, the process stabilizes almost surely provided the masses possess a finite moment of order p greater than 3. The proofs no longer rely on global mass conservation, which fails outside vertex-transitive graphs, and the authors supply matching examples showing that the degree bound and moment threshold cannot be relaxed much further.

Core claim

By analyzing an optimal stopping problem for random walk in random scenery, we prove that the divisible sandpile with i.i.d. initial masses of mean μ on an infinite bounded-degree graph explodes almost surely if μ ≥ 1 and stabilizes almost surely if μ < 1 provided the masses have finite p-th moment for some p > 3. We also show that these conditions are nearly sharp by exhibiting unbounded-degree graphs where sandpiles with μ > 1 stabilize and bounded-degree graphs where sandpiles with μ < 1 and finite p-moment for p < 3 explode.

What carries the argument

The optimal stopping problem for a random walk in random scenery, which decides whether the expected supremum of the cumulative scenery sums along stopped paths is finite.

If this is right

  • On every infinite bounded-degree graph the sandpile process fails to stabilize when the mean mass is at least 1.
  • When the mean mass is below 1 and moments exceed order 3, the toppling procedure terminates almost surely on bounded-degree graphs.
  • Unbounded degrees allow stabilization even for mean mass strictly above 1.
  • On some bounded-degree graphs, moments of order less than 3 permit explosion even when the mean mass is below 1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same stopping criterion could decide stabilization for other local redistribution rules that lack global conservation.
  • Degree growth rate appears to control whether large excursions of the walk can accumulate enough mass to prevent termination.
  • The moment threshold near 3 may be linked to integrability requirements for the maximum of a random walk's range on the given graph.
  • Explicit computation of the stopping threshold on regular trees would give a concrete test case for the general theory.

Load-bearing premise

The initial masses are i.i.d. and the graph is infinite.

What would settle it

Construct a bounded-degree infinite graph together with i.i.d. initial masses of mean greater than 1 on which the divisible sandpile stabilizes with positive probability.

Figures

Figures reproduced from arXiv: 2604.13968 by Ahmed Bou-Rabee, Ecaterina Sava-Huss, Yuval Peres.

Figure 1
Figure 1. Figure 1: Approximation of the odometer on a tree-of-pipes graph (Section 8.5), embedded in the Poincar´e disk with the root at the center and deeper pipes near the boundary. Edge color and width are proportional to the odometer. The i.i.d. initial masses have mean µ = 0.05 ≪ 1, yet a single heavy-tailed fluctuation near the boundary cascades inward through the tree hierarchy and drives the odometer at the root far … view at source ↗
Figure 2
Figure 2. Figure 2: Odometer u∞(v) on the Sierpi´nski gasket (level 6) with i.i.d. exponential initial masses of mean µ; vertex color and size are proportional to u∞(v). Left: subcritical (µ = 0.6); right: critical (µ = 1.0). The infinite Sierpi´nski gasket admits a stationary random rooting, so Theorem 2.4 gives stabilization for µ < 1 and explosion for µ > 1. A rooted graph is a pair (G, ρ) where G = (V, E) is a locally fin… view at source ↗
Figure 3
Figure 3. Figure 3: Trap selection in the proof of Proposition 6.6. The filled set Fk = S j≤k Cm(Yj ) consists of previously used trap sets (blue); the new center Yk+1 ∈ Ak lies far from Fk. Signs indicate ξ(v) at each vertex. In the red trap set every vertex has ξ(v) ≤ −ε, giving contribution ≤ −8m to h(Yk+1). Proof. The proof has two steps. First, we show E[u∞(o)] = ∞ by constructing a.s.-finite stopping times τm with E[Sτm… view at source ↗
Figure 4
Figure 4. Figure 4: Odometer u∞(v) on the comb lattice (Example 8.7) with i.i.d. Exp(µ) initial masses; vertex color and size are proportional to u∞(v). Left: subcritical (µ = 0.6); right: critical (µ = 1.0). The odometer concentrates along the spine (degree 4), which absorbs mass from the teeth (degree 2), illustrating the failure of conservation of mass. The comb is not stationary, but Theorems 1.2 and 1.3 still give explos… view at source ↗
Figure 5
Figure 5. Figure 5: The tree-of-pipes graph G with B = 3 (schematic; the proof requires B large enough to satisfy (43)). Each edge of the rooted B-ary tree is replaced by a pipe of length Ln at level n; the pipes grow geometrically longer at deeper levels. Filled circles are branching vertices; gray circles are pipe interior vertices (degree 2); open circles are leaves. Set K := 2 + 2b (Ccomb + 1) as in Proposition 8.8(d). Ca… view at source ↗
Figure 6
Figure 6. Figure 6: The graph G: a ray (thick horizontal line) with tree-of-pipes gadgets H(mk) attached at widely spaced positions s1 < s2 < s3 < · · · . Each gadget is a B-ary tree whose edges are replaced by pipes of increasing lengths. Later gadgets are larger; the radial intervals [sk, sk + Rmk ] are disjoint. at most Cball r df . In total, |B(o, r)| ≤ C rdf . If r ≥ sk + Rmk , then no later gadget is reached (by radial … view at source ↗
read the original abstract

We analyze an optimal stopping problem for random walk in random scenery on general graphs, and determine when it has a finite optimum. We use this to extend a theorem of Levine, Murugan, Peres, and Ugurcan [2016]. They proved that on a vertex-transitive graph, the divisible sandpile with i.i.d. initial masses of mean $\mu$ stabilizes almost surely if $\mu < 1$, explodes if $\mu > 1$, and explodes if $\mu = 1$ with positive finite variance. Their proofs rely on conservation of mean mass under toppling. This conservation extends to unimodular random graphs, but fails on general graphs. We prove explosion for all infinite bounded-degree graphs whenever $\mu \geq 1$, and stabilization for $\mu<1$ provided the initial masses have finite $p$-th moment for some $p>3$. Our conditions are nearly sharp: we exhibit unbounded-degree graphs on which sandpiles with $\mu > 1$ stabilize, and for every $p < 3$ we construct bounded-degree graphs on which sandpiles with~$\mu < 1$ and finite $p$-th moment explode.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes an optimal stopping problem for random walk in random scenery on general graphs and determines conditions for the stopping time to be finite. This tool is applied to extend the 2016 Levine-Murugan-Peres-Ugurcan theorem on divisible sandpiles: on any infinite bounded-degree graph, the sandpile with i.i.d. initial masses of mean μ explodes almost surely if μ ≥ 1, and stabilizes almost surely if μ < 1 provided the masses have finite p-th moment for some p > 3. The authors supply explicit counterexamples showing that the bounded-degree hypothesis cannot be dropped for the explosion claim when μ > 1, and that the moment threshold cannot be relaxed below p = 3 for stabilization on bounded-degree graphs.

Significance. If the results hold, the work provides a nearly sharp characterization of stabilization versus explosion for divisible sandpiles on arbitrary infinite graphs, replacing the mean-mass conservation argument (valid only on transitive or unimodular graphs) with a new optimal-stopping criterion. The explicit counterexamples for both the degree bound and the moment threshold p > 3 constitute a particular strength, as they demonstrate that the stated conditions are essentially optimal within the i.i.d. setting. The optimal-stopping analysis may have independent interest for other models involving random walks in random environments on non-transitive graphs.

major comments (2)
  1. [§3] §3 (optimal stopping criterion): The proof that the stopping time is finite a.s. when μ ≥ 1 on bounded-degree graphs uses the i.i.d. scenery and bounded degree to control the walk's range; it is not immediately clear from the argument whether the same conclusion holds if the degree bound is replaced by a uniform integrability condition on the degree distribution, which would be a natural extension.
  2. [§5] §5 (stabilization for μ < 1): The p > 3 moment assumption enters the optimal-stopping analysis via a specific embedding or maximal inequality; while the paper correctly shows sufficiency and supplies counterexamples for every p < 3, the derivation does not indicate whether the exponent 3 is an artifact of the current proof technique or intrinsic to the problem on bounded-degree graphs.
minor comments (2)
  1. [Introduction] The abstract states the results cleanly, but the introduction could include a short paragraph contrasting the new optimal-stopping approach with the mean-mass conservation used in the 2016 transitive case.
  2. [§2] Notation for the random scenery and the stopping time could be made uniform across sections to avoid minor redefinitions.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading, positive assessment, and recommendation of minor revision. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [§3] §3 (optimal stopping criterion): The proof that the stopping time is finite a.s. when μ ≥ 1 on bounded-degree graphs uses the i.i.d. scenery and bounded degree to control the walk's range; it is not immediately clear from the argument whether the same conclusion holds if the degree bound is replaced by a uniform integrability condition on the degree distribution, which would be a natural extension.

    Authors: Our proof in §3 for finiteness of the stopping time when μ ≥ 1 relies on the uniform bound on degrees to obtain uniform control over the range of the random walk, which is then combined with the i.i.d. scenery assumption. We do not know whether the same conclusion holds under the weaker uniform integrability condition on the degree distribution, as this would require a substantially different argument. However, the counterexamples in §4 already show that some restriction on degree growth is necessary for the explosion claim, since there exist unbounded-degree graphs on which the sandpile stabilizes for μ > 1. revision: no

  2. Referee: [§5] §5 (stabilization for μ < 1): The p > 3 moment assumption enters the optimal-stopping analysis via a specific embedding or maximal inequality; while the paper correctly shows sufficiency and supplies counterexamples for every p < 3, the derivation does not indicate whether the exponent 3 is an artifact of the current proof technique or intrinsic to the problem on bounded-degree graphs.

    Authors: The counterexamples constructed in §5 show that for every p < 3 there exists a bounded-degree graph on which the sandpile with i.i.d. masses of mean μ < 1 and finite p-moment explodes almost surely. This establishes that the moment threshold cannot be relaxed below 3 while preserving the stabilization result for all infinite bounded-degree graphs, so the exponent 3 is intrinsic to the problem on this class rather than an artifact of the proof technique. Our argument requires p > 3 because of the maximal inequality used in the optimal-stopping analysis, and whether the result holds at exactly p = 3 is left open. revision: no

standing simulated objections not resolved
  • Whether the finiteness of the stopping time (and thus explosion for μ ≥ 1) holds if the bounded-degree assumption is weakened to uniform integrability of the degree distribution.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper develops a new optimal-stopping criterion for random walk in random scenery on arbitrary graphs and uses it to prove explosion/stabilization thresholds for divisible sandpiles. The 2016 Levine-Murugan-Peres-Ugurcan result is invoked only to recover the already-known transitive case; the general-graph statements, the p>3 moment condition, and the matching counter-examples on unbounded-degree and low-moment graphs are all derived from the fresh analysis. No equation reduces to a prior fitted quantity, no uniqueness theorem is smuggled in via self-citation, and the central claims remain logically independent of the cited predecessor.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of probability on graphs (i.i.d. initial masses, infinite connected graphs) plus the new optimal-stopping analysis; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • domain assumption The graph is infinite, connected, and locally finite (bounded degree in the main theorems).
    Invoked to guarantee the random walk is transient or recurrent in controlled ways and to apply the optimal-stopping criterion.
  • domain assumption Initial masses are i.i.d. with finite mean μ and, for stabilization, finite p-moment p>3.
    Used to control the tail behavior that determines whether the stopping time is finite.

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Reference graph

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