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Criticality of the abelian sandpile in dimension 1

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Pith's one-line read A single recursive ratio decides whether a one-dimensional sandpile with traps is critical, splitting all trap configurations into three regimes.

desk verdict Genuinely new classification for 1D dissipative sandpile with arbitrary traps, but the proof of the practical non-criticality criterion (Thm 2.10) is internally inconsistent as printed and needs repair. read the letter →

arxiv 2509.03697 v1 pith:DO5WUDAL submitted 2025-09-03 math.PR

Adrien Rezzouk This is my paper · ORCID
classification math.PR MSC 60K3582C2760G50
keywords abeliansandpiledissipativetrappedrandomwalkexpectedstoppingtimecriticalityphasetransitiongoldenratio
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks when a walker on the integer line, with traps that absorb with probability 1/3, dies quickly enough for the equivalent dissipative abelian sandpile to be non-critical. It proves that this phase transition is governed by an explicit recursive sequence attached to the trap positions, and that the whole line can be analyzed through two half-line walks. The resulting classification has three regimes separated by the square of the golden ratio, $(3+\sqrt5)/2$: trap sequences growing slower than that geometric rate are always non-critical, faster-growing sequences can be either critical or non-critical, and sequences growing faster than every double exponential $a^{2^k}$ are always critical. A striking consequence is that spacing traps farther apart can shorten the expected lifetime, reversing the naive monotonicity intuition.

What carries the argument

The machine that carries the argument is the recursion of Theorem 2.6. Writing the finite-volume expected absorption time at trap $k$ as $E_k=p_k A-q_k$ and imposing the boundary condition $E_N=0$ gives $A(N)=q_N/p_N$, so the full expected absorption time is finite exactly when the increasing sequence $q_n/p_n$ has a finite limit. The proof then rewrites $q_n/p_n$ as $\varepsilon_n+\sum_k \mu_k x_k$, a barycenter of trap positions with weights $\mu_k$, and translates the recursion into the identity $R_{k-1}-R_k=\sum_{j>k} x_k(1-x_k/x_j)(R_{j-1}-R_j)$ for the tail sums $R_k$. Bounding this renewal-type identity from below produces the characteristic polynomial $X^2-3X+1$, whose larger root is $(3+\sqrt5)/2$, the geometric threshold; bounding it from above yields the ratio test $x_{n+1}/x_n^2$ that separates the intermediate regimes.

What would settle it

Compute the recursion and the expected absorption time for the explicit trap sequence $x_k=\lfloor \rho^k\rfloor$ with a fixed $\rho$ just below $(3+\sqrt5)/2$; the paper predicts finite $E_0[T]$ for every such sequence, so a single $\rho$ in that range with diverging expectation would disprove Theorem 2.7. A cheaper check is to evaluate displayed formula (9) for the half-line with one trap: as printed it omits the $+1$ for the first step and will not reproduce the true expectation.

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Extended reading notes

Core claim

The paper establishes that for a dissipative abelian sandpile on $\mathbb{Z}$ with traps at arbitrary positions, the model is non-critical exactly when the expected absorption time of the associated trapped random walk is finite, and that this is decided by a single explicit recursion. For the ordered trap sequence $x_1<x_2<\cdots$ on the positive half-line, define $p_1=x_1$, $q_1=x_1^2$, and for $n\ge 2$, $p_n=x_n+\sum_{i<n}(x_n-x_i)p_i$ and $q_n=x_n^2+\sum_{i<n}(x_n-x_i)(q_i+1)$. Then $q_n/p_n$ is increasing, and the model is critical if and only if $q_n/p_n\to\infty$. Around this criterion the paper finds a sharp geometric threshold at the square of the golden ratio, $\phi^2=(3+\sqrt5)/2$: slower-than-$\phi^2$ growth is always non-critical; faster growth admits both critical and non-critical configurations, with $x_{n+1}/x_n^2$ lying below or above $1$ as a practical discriminator; and any trap sequence that eventually exceeds $a^{2^k}$ for every $a>1$ is necessarily critical.

Load-bearing premise

The entire classification rests on assuming that the walk from 0 breaks into independent excursions, each ending at 0 or at the sink, so that the total absorption time is a geometric sum of excursion lengths; if that decomposition failed for some trap arrangement, the half-line reduction (and hence the three-regime picture) would not follow.

Editorial extensions

If this is right

  • If trap positions grow at most like $\rho^k$ with $\rho<(3+\sqrt5)/2$, the model is non-critical: the expected absorption time stays finite.
  • If $\limsup_{n\to\infty} x_{n+1}/x_n^2<1$, the model is non-critical; if $\liminf_{n\to\infty} x_{n+1}/x_n^2>1$, the model is critical.
  • Any trap sequence that eventually has $x_k>a^{2^k}$ for every $a>1$ is critical.
  • Adding or removing finitely many traps, and in particular whether $0$ is a trap, does not change criticality.
  • In the symmetric case, the full system is critical exactly when its positive half-line process is critical.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The boundary case $x_k\sim \phi^{2k}$ is left partly open; the ratio and series criteria together suggest that criticality there depends on subleading constants and logarithmic factors, which a direct computation of $q_n/p_n$ could settle.
  • The non-monotonicity under dilating trap spacing implies that criticality is controlled by trap geometry, not merely trap density, so any higher-dimensional analogue would need to track the arrangement of dissipative sites.
  • Remark 2.19 indicates the threshold should shift with the trap absorption probability $p$ through $\alpha=2p/(1-p)$; extending the three-regime analysis to general $p$ is a concrete continuation of the same method.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the dissipative Abelian sandpile model on Z with trap sites at arbitrary locations (x_k), equivalently a nearest-neighbor random walk on Z that is absorbed at rate 1/3 when at a trap and at an additional sink. The main result is a three-regime classification of criticality (infinite expected avalanche size / infinite expected absorption time). The paper proves that criticality of the full system is equivalent to criticality of the two half-line walks, and for a half-line with traps x_1<x_2<... it introduces explicit sequences p_n, q_n defined by (2), showing the model is critical iff q_n/p_n -> infinity. It then derives: non-criticality for x_k = O(rho^k) with rho < phi = (3+sqrt(5))/2; existence of critical sequences below every rho^k for rho > phi; and practical ratio criteria limsup x_{n+1}/x_n^2 < 1 implies non-critical, liminf > 1 implies critical, with a double-exponential boundary. The proofs are based on solving the linear system for expected hitting times on finite intervals, representing q_n/p_n as a barycenter, and bounding the barycentric weights.

Significance. If the results are correct, this gives a complete classification of criticality for the one-dimensional dissipative sandpile with arbitrary trap positions, going beyond the symmetric recursive case studied in [6] and identifying sharp thresholds. The paper's method is transparent and self-contained: the recursive sequences are derived from first principles, no parameters are fitted, and the examples recover [6] as a special case. The counterintuitive non-monotonicity (Theorem 2.8 and Section 2.2) is a genuine contribution. However, two central proof steps in the non-critical criterion (Theorem 2.10) contain algebraic errors that must be repaired before the classification is fully established; the half-line reduction also has a small error in the displayed excursion-length formula. The overall framework is convincing and the statements are likely true, but the printed proofs do not currently support every claim.

major comments (3)
  1. [Section 5.4, Theorem 2.10] The d'Alembert ratio test is applied to the series with terms a_k = x_{k+1} * product_{r=1}^k [1+x_r(1-x_r/x_{r+1})]^{-1}; the actual ratio is a_{k+1}/a_k = (x_{k+2}/x_{k+1}) / [1+x_{k+1}(1-x_{k+1}/x_{k+2})]. Condition (23) states limsup (x_{k+1}/x_k)(1+x_k(1-x_k/x_{k+1})) < 1, which is neither the correct ratio nor its reciprocal under the stated indexing. The convergence of the bounding series is therefore not established by the argument given. Since this is the only proof of the non-critical direction of the ratio criterion, this is a load-bearing gap.
  2. [Section 5.4, Eq. (24)] The function f_k(x) = (x/x_k)(1+x_k(1-x_k/x)) is linear with positive slope (1+x_k)/x_k, so it cannot be decreasing on (x_k+1, 2x_k^2/(x_k+1)) as claimed, and its value at x_k+1 is 2+1/x_k, not (x_k+1)^2/(x_k(2x_k+1)) <= 9/10. The stated bound and monotonicity are correct for the reciprocal function F_k(x) = x / [x_k(1+x_k(1-x_k/x))]. The definition of x_+(k,alpha) as the larger root of f_k(x)=alpha is therefore attached to the wrong function, and the derivation of limsup f_k(x_{k+1}) < 1 does not imply the desired ratio-test bound.
  3. [Section 4.2, Theorem 4.4, Eq. (9)] The formula for E[L_1] omits the +1 contributed by the first step from 0 to +/-1 in each non-absorbed excursion; it should read E[L_1] = 1 + (1-p)/2 E_1[tau] + (1-p)/2 E_{-1}[tau] (or equivalently p*1 + (1-p)/2 (1+E_1[tau]) + (1-p)/2 (1+E_{-1}[tau])). The displayed equality (9) is therefore not exact, although the finiteness equivalence survives because the omitted constant is finite. Please correct the displayed formula.
minor comments (5)
  1. [Section 5.1] The sentence 'the system is critical if and only if lim_{N->infty} A(N) < infinity' should read '= infinity', since A(N) is the expected stopping time of the truncated walk and increases with N.
  2. [Remark 2.18] The display 'the recurrence relation on |I_k| |I_{k+1}| = c|I_k|^2' is missing a colon; also the c=1 boundary case, which is covered by Example 2.17 with epsilon_k=0, should be made explicit.
  3. [Notation] The symbol 'J-n,nK' is nonstandard; consider replacing with { -n, ..., n } or [-n,n] intersect Z.
  4. [Theorem 2.8 proof] The sequence y_k is initially described as any increasing sequence with k phi^k = o(y_k); it is then written as lambda_k k phi^k with lambda_k -> infinity. This equivalence should be stated explicitly.
  5. [Section 2.2 and Remark 2.18] There are a few grammatical errors (e.g., 'by a homothety' in Section 2.2 and 'the recursion' in Remark 2.18) that should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the criticality criterion is derived from an explicit solution of the trapping equations, and prior work is used only as comparison, not as input.

full rationale

The paper's derivation chain is self-contained. The central object q_n/p_n is not fitted or assumed; it is obtained by solving the linear system (10) for the expected stopping time of the truncated half-line walk: writing E_k = p_k A - q_k and imposing the boundary condition E_N = 0 gives A(N) = q_n/p_n, so criticality is characterized by lim A(N) = infinity. The equivalence between model criticality and divergence of E_x(T) is proven in Section 3 via the Green-function identity (5)-(6), with only the definition of criticality taken from the external reference [5]. Subsequent results, including Theorems 2.7, 2.8, 2.10, 2.11 and the corollaries, are proven from the recurrence and from bounds on R_k derived from equation (15); no parameter is calibrated to a subset of data and no conclusion is assumed in its own proof. The cited work [6] is invoked only as motivation and as a comparison case in Remark 2.18, with the main theorems recovering those results rather than depending on them, so the self-citations are not load-bearing. The questionable displayed equation (9) and the alleged algebraic slip in the proof of Theorem 2.10 are correctness or rigor concerns, not circularity: even if those proofs needed repair, the statements would not reduce to their inputs by construction. No step in the derivation is equivalent by definition to the conclusion it is used to establish.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. The transition probabilities are fixed by the model (1/2 for normal moves, 1/3 into the sink at traps). The central derivation rests on standard stochastic tools (strong Markov property, Wald's equation), standard linear-recurrence asymptotics (Lemma 5.3), and the proven bridge between sandpile criticality and the trapped random walk (Section 3, Prop 3.6). The finite-trap case (Prop 4.3) and the half-line reduction (Theorem 4.4) carry the main structural load.

assumptions (3)
  • domain assumption The criticality of the dissipative abelian sandpile is equivalent to the finiteness of the expected stopping time of the associated trapped random walk (Theorem 2.1 / Prop 3.6).
    This is proven in the paper via the Green function and the bounded-degree observation (1/3 ≤ 1/deg(y) ≤ 1/2), but it is the bridge that transfers the sandpile problem to the random walk problem; if the factor 1/deg(y) broke boundedness, the equivalence would fail.
  • standard math Linear recurrence sequences with positive coefficients have a unique dominant positive root and exponential growth (Lemma 5.3).
    Used in the proof of Theorem 2.7 to control the bound R_p ≤ C'/ρ'^p via the characteristic polynomial X^{κ+1} - X^κ - λ(X^κ + ... + 1).
  • standard math Excursions of the random walk from 0 are i.i.d. and Wald's equation applies (Theorem 4.4).
    Used to reduce full-line criticality to half-line criticality; the proof relies on the strong Markov property and on the geometric number of excursions before absorption.

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Cite this review

Pith. "Pith review of Criticality of the abelian sandpile in dimension 1." pith.science (2026). https://pith.science/paper/DO5WUDAL

@misc{pith2026250903697,
  author       = {Pith},
  title        = {Pith review of: Criticality of the abelian sandpile in dimension 1},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DO5WUDAL}},
  note         = {Machine review of arXiv:2509.03697}
}
read the original abstract

We study the critical behavior of the dissipative abelian sandpile model on Z with dissipative sites at arbitrary positions (x_k). This is equivalent to studying whether the expected stopping time of a trapped random walk on Z is finite. Our main contribution is a precise description of this phase transition via three distinct regimes. Our analysis captures criticality through the asymptotic behavior of an explicit recursive sequence, revealing counterintuitive phenomena in which traps may be farther apart yet the stopping time becomes shorter.

Figures

Figures reproduced from arXiv: 2509.03697 by the authors.

Figure 1
Figure 1. Example with traps in x−1 = −2, x0 = 0 et x1 = 2 By adding grains of sand at uniformly chosen sites, one defines a Markov chain on the set of height configurations η over the graph. In finite volume, this chain admits a unique invariant measure, denoted µn for the volume J−n, nK, which is the uniform distribution over the set of recurrent configurations. For a concise yet precise introduction to the Abelian sandpile… view at source ↗

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Works this paper leans on

6 extracted references · 4 canonical work pages

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