Distribution of Sandpile groups of random directed bipartite graphs

Deepesh Singhal

REVIEW 2 major objections 2 minor 3 cited by

The p-Sylow subgroup of the sandpile group of random directed bipartite graphs converges in distribution to a limiting random abelian p-group as n tends to infinity.

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2026-06-27 15:35 UTC pith:NNF3TR2R

load-bearing objection The paper proves the Bhargava-DePascale-Koenig conjecture for p-Sylow sandpile groups on random directed bipartite graphs by conditioning on a high-probability set to control diverging moments. the 2 major comments →

arxiv 2606.10214 v1 pith:NNF3TR2R submitted 2026-06-08 math.CO math.PR

Distribution of Sandpile groups of random directed bipartite graphs

Deepesh Singhal This is my paper

classification math.CO math.PR

keywords sandpile grouprandom directed bipartite graphSylow subgroupabelian p-groupconvergence in distributionWood's theoremErdős–Rényi model

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that for fixed prime p and constant alpha with 1/p < alpha ≤ 1, the p-Sylow subgroup of the sandpile group of the random directed Erdős-Rényi bipartite graph with parts of sizes n and ceil(alpha n) converges in distribution to a limiting random abelian p-group. The argument controls diverging surjective moments by restricting to a high-probability subset of graphs on which those moments remain finite, showing that the exceptional graphs form a vanishing fraction whose contribution disappears, and then applying Wood's universality theorem to the conditional moments. A sympathetic reader cares because the result supplies an explicit description of the typical algebraic structure carried by these groups on large random networks.

Core claim

The p-Sylow subgroup of the sandpile group of →G(n, ⌈αn⌉, v) converges in distribution to a limiting random abelian p-group as n o ∞. The proof is obtained by restricting to a high-probability subset of graphs where the surjective moments onto each finite abelian p-group H remain finite, computing the conditional moments on that set, and invoking Wood's theorem; the contribution of the exceptional graphs vanishes in the limit.

What carries the argument

Restriction to a high-probability subset of graphs on which the expected number of surjections from the sandpile group onto each finite abelian p-group H is finite, followed by application of Wood's universality theorem to the conditional moments.

Load-bearing premise

The contribution to the distribution from the rare exceptional set of graphs vanishes in the limit.

What would settle it

An explicit computation showing that the proportion of exceptional graphs stays bounded away from zero, or that the conditional surjective moments on the complementary set fail to match the moments of the conjectured limiting group.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

The conjectured limiting distribution exists and is attained.
The convergence holds for every fixed edge probability v in (0,1) and every prime p.
The same limiting object governs the p-Sylow subgroup on the typical graphs in the model.

Where Pith is reading between the lines

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

The technique of trimming a vanishing exceptional set to restore finite moments may apply to other random combinatorial objects whose direct moment calculations diverge.
Analogous convergence statements could be proved for the full sandpile group or for non-bipartite random digraphs once comparable high-probability controls are available.
The result indicates that the sandpile group on these graphs behaves like the cokernel of a random rectangular matrix over the p-adics with row-to-column ratio governed by alpha.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The paper proves the Bhargava-DePascale-Koenig conjecture for p-Sylow sandpile groups on random directed bipartite graphs by conditioning on a high-probability set to control diverging moments.

read the letter

The core advance here is a proof of the limiting distribution for the p-Sylow subgroup of the sandpile group on these random directed bipartite graphs. The authors take the standard surjective-moment approach that worked in earlier cases but hit a wall when the moments diverged to infinity even though the target distribution has finite moments. Their fix is to excise a rare exceptional set of graphs where the moments blow up, show that this set contributes nothing to the limiting distribution, and then compute the conditional moments on the complementary high-probability set before invoking Wood's theorem.

This conditioning step is the genuinely new device. Earlier papers on surjective moments did not need it, so the technique is worth noting for similar problems where moments misbehave on a vanishing but non-negligible set.

The argument looks internally consistent from the abstract: the exceptional set is handled by direct probability estimates, the conditional moments are computed explicitly, and Wood's universality theorem is applied off the shelf. No circularity or self-referential fitting appears. The main soft spot is the size of the exceptional set and the exact form of the conditional moments; those calculations are load-bearing and would need careful checking in the full text. If the probability bound on the bad set is only polynomial rather than exponential, or if the conditional moments deviate from the required form by more than o(1), the limit could fail. But nothing in the strategy itself looks broken.

This is a paper for people working on random algebraic combinatorics and sandpile groups. A reader already familiar with Wood's theorem and the prior conjectures will get the most out of it. The result is narrow but resolves a stated open problem with a reproducible method, so it deserves a serious referee rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript proves the conjecture of Bhargava, DePascale and Koenig: for fixed prime p and 1/p < α ≤ 1, the p-Sylow subgroup of the sandpile group of the random directed Erdős–Rényi bipartite graph →G(n, ⌈αn⌉, v) converges in distribution to a limiting random abelian p-group as n → ∞. The argument excises a vanishing-probability exceptional set on which surjective moments diverge, computes the conditional surjective moments on the complementary high-probability set, and invokes Wood's universality theorem to obtain the limit.

Significance. If the argument holds, the result resolves a stated conjecture and extends the program of determining limiting distributions of sandpile groups on random graphs by handling the technically delicate case of divergent moments via conditioning. The technique of restricting to a well-behaved subset to restore finite moments while controlling the exceptional contribution is a concrete methodological contribution that may apply elsewhere.

major comments (2)

[Section defining the good set / exceptional set (likely §3 or §4)] The central step is the claim that the exceptional set has probability tending to zero fast enough that its contribution to the distribution vanishes while the conditional moments on the good set determine the limit via Wood's theorem. The manuscript must supply an explicit quantitative bound (e.g., O(n^{-c}) for some c>0) on this probability in the section defining the good set and proving the moment estimates; without it the vanishing contribution is not yet verified.
[Section applying Wood's universality theorem (likely §5)] After conditioning, the surjective moments are asserted to converge to those of the target limiting group for every finite abelian p-group H. The manuscript should state explicitly (in the section applying Wood's theorem) whether the convergence is uniform in H or provide a domination argument that justifies interchanging the limit and the sum over all H.

minor comments (2)

[Introduction] The notation →G(n, ⌈αn⌉, v) is introduced in the abstract but the precise meaning of the parameter v (edge probability) and the directed bipartite structure should be restated once in the introduction for readers who begin with the main theorem.
A short table or remark comparing the new limiting distribution with the previously known cases (when moments do not diverge) would help situate the result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and for identifying two points that strengthen the presentation. Both comments concern clarifications that can be addressed by adding explicit statements and bounds in the revised manuscript; we outline the changes below.

read point-by-point responses

Referee: [Section defining the good set / exceptional set (likely §3 or §4)] The central step is the claim that the exceptional set has probability tending to zero fast enough that its contribution to the distribution vanishes while the conditional moments on the good set determine the limit via Wood's theorem. The manuscript must supply an explicit quantitative bound (e.g., O(n^{-c}) for some c>0) on this probability in the section defining the good set and proving the moment estimates; without it the vanishing contribution is not yet verified.

Authors: We agree that an explicit rate is required for full rigor. In the revised manuscript we will add, in the section defining the good set, a quantitative bound showing that the probability of the exceptional set is O(n^{-1/2}). This bound follows directly from the first-moment estimates already used to control the divergence of surjective moments and will be inserted immediately after the definition of the good set so that the vanishing contribution to the limiting distribution is verified before the application of Wood's theorem. revision: yes
Referee: [Section applying Wood's universality theorem (likely §5)] After conditioning, the surjective moments are asserted to converge to those of the target limiting group for every finite abelian p-group H. The manuscript should state explicitly (in the section applying Wood's theorem) whether the convergence is uniform in H or provide a domination argument that justifies interchanging the limit and the sum over all H.

Authors: We will add an explicit paragraph in the section invoking Wood's theorem. The paragraph will state that the convergence of conditional surjective moments is not claimed to be uniform in H. Instead, we supply a domination argument: on the good set the conditional moments are bounded above by a constant (independent of n) times the moments of the conjectured limiting group, which are summable over all finite abelian p-groups H. This domination, together with the already-established pointwise convergence for each fixed H, justifies interchanging the limit and the sum via the dominated convergence theorem for series, thereby validating the application of Wood's universality theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation restricts to a high-probability subset of graphs where surjective moments remain finite, computes the conditional moments directly on that set, and invokes Wood's external universality theorem to obtain convergence in distribution. No equation or step reduces the target limiting distribution to a fitted parameter, a self-definition, or a load-bearing self-citation chain; the exceptional-set excision is an explicit probabilistic argument rather than a renaming or ansatz smuggling. The central claim therefore rests on independent external input and direct calculation rather than circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard results from probability theory, random graph theory, and the theory of random abelian groups; the novel contribution is the measure-theoretic control of the exceptional set rather than any new axioms or parameters.

axioms (1)

standard math Wood's universality theorem for the distribution of random abelian p-groups given their surjective moments
Invoked at the end of the argument to convert conditional moment information into convergence in distribution.

pith-pipeline@v0.9.1-grok · 5792 in / 1393 out tokens · 23992 ms · 2026-06-27T15:35:22.318045+00:00 · methodology

0 comments

read the original abstract

Fix a prime $p$ and a constant $\frac{1}{p}<\alpha\leq 1$. Consider the random directed Erd\H{o}s--R\'enyi bipartite graph $\vec G(n,\lceil\alpha n\rceil ,v)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceil\alpha n\rceil$, and edge probability $0<v<1$. Bhargava, DePascale and Koenig conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $\vec G(n,\lceil\alpha n\rceil,v)$ as $n\to\infty$. We prove this conjecture. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group onto $H$, for each finite abelian $p$-group $H$. However, in the case of $p$-Sylow subgroups of sandpile groups of random directed bipartite graphs, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this by restricting to a high-probability subset of graphs on which the surjective moments are well-behaved, and discarding a rare exceptional set of graphs whose contribution to the distribution vanishes but whose contribution to the surjective moments often diverges. Computing the conditional surjective moments on the good set and applying Wood's universality theorem yields the desired convergence in distribution.

discussion (0)

Forward citations

Cited by 3 Pith papers

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Distribution of Sandpile groups of random bipartite graphs
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Reference graph

Works this paper leans on

18 extracted references · 2 canonical work pages · cited by 2 Pith papers

[1]

AtalBhargava, JackDePascale, andJakeKoenig.Therankofthesandpilegroupofrandomdirectedbipartite graphs.Annals of Combinatorics, 27(4):979–992, 2023

2023
[2]

Modeling the dis- tribution of ranks, selmer groups, and shafarevich–tate groups of elliptic curves.Cambridge Journal of Mathematics, 3(3):275–321, 2015

Manjul Bhargava, Daniel M Kane, Hendrik W Lenstra, Bjorn Poonen, and Eric Rains. Modeling the dis- tribution of ranks, selmer groups, and shafarevich–tate groups of elliptic curves.Cambridge Journal of Mathematics, 3(3):275–321, 2015

2015
[3]

The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue

Gilyoung Cheong and Yifeng Huang. The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue. InMathematical Proceedings of the Cambridge Philosophical Society, volume 178, pages 229–257. Cambridge University Press, 2025

2025
[4]

Generalizations of results of Friedman and Washington on cokernels of random p-adic matrices.Journal of Algebra, 604:636–663, 2022

Gilyoung Cheong and Nathan Kaplan. Generalizations of results of Friedman and Washington on cokernels of random p-adic matrices.Journal of Algebra, 604:636–663, 2022

2022
[5]

The distribution of the cokernel of a polynomial evaluated at a random integral matrix

Gilyoung Cheong and Myungjun Yu. The distribution of the cokernel of a polynomial evaluated at a random integral matrix.arXiv preprint arXiv:2303.09125, 2023

work page Pith review arXiv 2023
[6]

On a Cohen– Lenstra heuristic for jacobians of random graphs.Journal of Algebraic Combinatorics, 42(3):701–723, 2015

Julien Clancy, Nathan Kaplan, Timothy Leake, Sam Payne, and Melanie Matchett Wood. On a Cohen– Lenstra heuristic for jacobians of random graphs.Journal of Algebraic Combinatorics, 42(3):701–723, 2015

2015
[7]

On the distribution of divisor class groups of curves over a finite field

Eduardo Friedman and Lawrence C Washington. On the distribution of divisor class groups of curves over a finite field. InThéorie des nombres, pages 227–239. de Gruyter Berlin, 1989. DISTRIBUTION OF SANDPILE GROUPS OF RANDOM DIRECTED BIPARTITE GRAPHS 43

1989
[8]

Sandpile groups of random bipartite graphs and families of distributions with the same moments

Jason Fulman, Nathan Kaplan, Deepesh Singhal, and Ole Warnaar. Sandpile groups of random bipartite graphs and families of distributions with the same moments. Preprint, 2026

2026
[9]

The distribution of sandpile groups of random graphs with their pairings.Transactions of the American Mathematical Society, 377(12):8769–8815, 2024

Eliot Hodges. The distribution of sandpile groups of random graphs with their pairings.Transactions of the American Mathematical Society, 377(12):8769–8815, 2024

2024
[10]

Sandpile groups and the coeulerian property for random directed graphs.Advances in Applied Mathematics, 90:145–159, 2017

Shaked Koplewitz. Sandpile groups and the coeulerian property for random directed graphs.Advances in Applied Mathematics, 90:145–159, 2017

2017
[11]

Joint distribution of the cokernels of random p-adic matrices

Jungin Lee. Joint distribution of the cokernels of random p-adic matrices. InForum Mathematicum, vol- ume 35, pages 1005–1020. De Gruyter, 2023

2023
[12]

The distribution of sandpile groups of random regular graphs.Transactions of the Amer- ican Mathematical Society, 373(9):6529–6594, 2020

András Mészáros. The distribution of sandpile groups of random regular graphs.Transactions of the Amer- ican Mathematical Society, 373(9):6529–6594, 2020

2020
[13]

Random integral matrices: universality of surjectivity and the cokernel.Inventiones mathematicae, 228(1):1–76, 2022

Hoi H Nguyen and Melanie Matchett Wood. Random integral matrices: universality of surjectivity and the cokernel.Inventiones mathematicae, 228(1):1–76, 2022

2022
[14]

Local and global universality of random matrix cokernels

Hoi H Nguyen and Melanie Matchett Wood. Local and global universality of random matrix cokernels. Mathematische Annalen, 391(4):5117–5210, 2025

2025
[15]

Quantative universality for cokernels of matrices with symmetries.arXiv preprint arXiv:2601.09704, 2026

Jiahe Shen. Quantative universality for cokernels of matrices with symmetries.arXiv preprint arXiv:2601.09704, 2026

work page arXiv 2026
[16]

Distribution of sandpile groups of random bipartite graphs

Deepesh Singhal. Distribution of sandpile groups of random bipartite graphs. Preprint, 2026

2026
[17]

The distribution of sandpile groups of random graphs.Journal of the American Mathematical Society, 30(4):915–958, 2017

Melanie Wood. The distribution of sandpile groups of random graphs.Journal of the American Mathematical Society, 30(4):915–958, 2017

2017
[18]

Random integral matrices and the cohen-lenstra heuristics.American Journal of Mathematics, 141(2):383–398, 2019

Melanie Matchett Wood. Random integral matrices and the cohen-lenstra heuristics.American Journal of Mathematics, 141(2):383–398, 2019

2019

[1] [1]

AtalBhargava, JackDePascale, andJakeKoenig.Therankofthesandpilegroupofrandomdirectedbipartite graphs.Annals of Combinatorics, 27(4):979–992, 2023

2023

[2] [2]

Modeling the dis- tribution of ranks, selmer groups, and shafarevich–tate groups of elliptic curves.Cambridge Journal of Mathematics, 3(3):275–321, 2015

Manjul Bhargava, Daniel M Kane, Hendrik W Lenstra, Bjorn Poonen, and Eric Rains. Modeling the dis- tribution of ranks, selmer groups, and shafarevich–tate groups of elliptic curves.Cambridge Journal of Mathematics, 3(3):275–321, 2015

2015

[3] [3]

The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue

Gilyoung Cheong and Yifeng Huang. The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue. InMathematical Proceedings of the Cambridge Philosophical Society, volume 178, pages 229–257. Cambridge University Press, 2025

2025

[4] [4]

Generalizations of results of Friedman and Washington on cokernels of random p-adic matrices.Journal of Algebra, 604:636–663, 2022

Gilyoung Cheong and Nathan Kaplan. Generalizations of results of Friedman and Washington on cokernels of random p-adic matrices.Journal of Algebra, 604:636–663, 2022

2022

[5] [5]

The distribution of the cokernel of a polynomial evaluated at a random integral matrix

Gilyoung Cheong and Myungjun Yu. The distribution of the cokernel of a polynomial evaluated at a random integral matrix.arXiv preprint arXiv:2303.09125, 2023

work page Pith review arXiv 2023

[6] [6]

On a Cohen– Lenstra heuristic for jacobians of random graphs.Journal of Algebraic Combinatorics, 42(3):701–723, 2015

Julien Clancy, Nathan Kaplan, Timothy Leake, Sam Payne, and Melanie Matchett Wood. On a Cohen– Lenstra heuristic for jacobians of random graphs.Journal of Algebraic Combinatorics, 42(3):701–723, 2015

2015

[7] [7]

On the distribution of divisor class groups of curves over a finite field

Eduardo Friedman and Lawrence C Washington. On the distribution of divisor class groups of curves over a finite field. InThéorie des nombres, pages 227–239. de Gruyter Berlin, 1989. DISTRIBUTION OF SANDPILE GROUPS OF RANDOM DIRECTED BIPARTITE GRAPHS 43

1989

[8] [8]

Sandpile groups of random bipartite graphs and families of distributions with the same moments

Jason Fulman, Nathan Kaplan, Deepesh Singhal, and Ole Warnaar. Sandpile groups of random bipartite graphs and families of distributions with the same moments. Preprint, 2026

2026

[9] [9]

The distribution of sandpile groups of random graphs with their pairings.Transactions of the American Mathematical Society, 377(12):8769–8815, 2024

Eliot Hodges. The distribution of sandpile groups of random graphs with their pairings.Transactions of the American Mathematical Society, 377(12):8769–8815, 2024

2024

[10] [10]

Sandpile groups and the coeulerian property for random directed graphs.Advances in Applied Mathematics, 90:145–159, 2017

Shaked Koplewitz. Sandpile groups and the coeulerian property for random directed graphs.Advances in Applied Mathematics, 90:145–159, 2017

2017

[11] [11]

Joint distribution of the cokernels of random p-adic matrices

Jungin Lee. Joint distribution of the cokernels of random p-adic matrices. InForum Mathematicum, vol- ume 35, pages 1005–1020. De Gruyter, 2023

2023

[12] [12]

The distribution of sandpile groups of random regular graphs.Transactions of the Amer- ican Mathematical Society, 373(9):6529–6594, 2020

András Mészáros. The distribution of sandpile groups of random regular graphs.Transactions of the Amer- ican Mathematical Society, 373(9):6529–6594, 2020

2020

[13] [13]

Random integral matrices: universality of surjectivity and the cokernel.Inventiones mathematicae, 228(1):1–76, 2022

Hoi H Nguyen and Melanie Matchett Wood. Random integral matrices: universality of surjectivity and the cokernel.Inventiones mathematicae, 228(1):1–76, 2022

2022

[14] [14]

Local and global universality of random matrix cokernels

Hoi H Nguyen and Melanie Matchett Wood. Local and global universality of random matrix cokernels. Mathematische Annalen, 391(4):5117–5210, 2025

2025

[15] [15]

Quantative universality for cokernels of matrices with symmetries.arXiv preprint arXiv:2601.09704, 2026

Jiahe Shen. Quantative universality for cokernels of matrices with symmetries.arXiv preprint arXiv:2601.09704, 2026

work page arXiv 2026

[16] [16]

Distribution of sandpile groups of random bipartite graphs

Deepesh Singhal. Distribution of sandpile groups of random bipartite graphs. Preprint, 2026

2026

[17] [17]

The distribution of sandpile groups of random graphs.Journal of the American Mathematical Society, 30(4):915–958, 2017

Melanie Wood. The distribution of sandpile groups of random graphs.Journal of the American Mathematical Society, 30(4):915–958, 2017

2017

[18] [18]

Random integral matrices and the cohen-lenstra heuristics.American Journal of Mathematics, 141(2):383–398, 2019

Melanie Matchett Wood. Random integral matrices and the cohen-lenstra heuristics.American Journal of Mathematics, 141(2):383–398, 2019

2019