Distribution of Sandpile groups of random bipartite graphs

Deepesh Singhal

For odd primes, the p-part of the sandpile group of a random bipartite graph has the same limiting law as for ordinary random graphs.

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2026-07-14 00:42 UTC pith:BYPMA4D2

load-bearing objection Solid proof of the odd-p bipartite sandpile conjecture via a clean truncation that tames diverging moments; p=2 left open but honestly so.

arxiv 2607.10056 v1 pith:BYPMA4D2 submitted 2026-07-11 math.CO math.PR

Distribution of Sandpile groups of random bipartite graphs

Deepesh Singhal This is my paper

classification math.CO math.PR MSC 05C8015B5220K0160B20

keywords sandpile groupsrandom bipartite graphsCohen–Lenstra heuristicsp-Sylow subgroupsrandom matricescokernelsWood universalitysurjective moments

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 a conjecture on the limiting distribution of the p-Sylow subgroup of the sandpile group of a random Erdős–Rényi bipartite graph, when the two sides have sizes n and roughly αn with α > 1/p and p is an odd prime. The limit is the same universal distribution that appears for ordinary random graphs and for cokernels of random symmetric matrices. The technical obstacle is that the usual surjective moments diverge because a rare class of graphs (those with too many vertices of degree divisible by p) contributes an infinite amount; after discarding that class the moments become finite and match the expected values, so Wood’s universality theorem supplies the distribution. The same moment calculation works for p = 2 and also yields the limiting law of the p-rank for every prime.

Core claim

When p is odd, 1/p < α ≤ 1 and G is any finite abelian p-group, the probability that the p-Sylow of the sandpile group of the random bipartite graph G_α(n,u) is isomorphic to G tends to the symmetric Cohen–Lenstra probability P^Sym_∞,p(G). Equivalently, the cokernel of the associated random bipartite Laplacian matrix with ε-balanced entries obeys the same limit. For every prime the rank distribution also converges to the explicit q-series formula predicted by the conjecture.

What carries the argument

Conditional surjective (or Hom) moments after restricting to the high-probability set of matrices in which fewer than (1/p + ρ)n row-sums vanish mod p. On that set the moments converge to |Λ²G| (or |G/2G|·|Λ²G| when p=2); Wood’s theorem then upgrades matching moments to convergence in distribution for odd p.

Load-bearing premise

After one throws away the rare graphs that have too many vertices whose degree is divisible by p, the remaining graphs have the same surjective moments as the conjectured limiting distribution.

What would settle it

Compute the exact distribution of the p-Sylow of the sandpile group for a sequence of moderate-sized bipartite graphs with α slightly larger than 1/p and check whether the empirical frequencies approach P^Sym_∞,p; or verify that the untruncated moments really diverge while the truncated ones stay bounded.

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

If this is right

The p-Sylow of the sandpile group of a random bipartite graph is, for odd p, distributed exactly like that of an ordinary random graph or a random symmetric matrix.
The limiting probability that the p-rank equals r is given by the explicit infinite-product formula involving (p^{-1};p^{-1})_r for every prime p.
The same moment method, after the same degree-mod-p truncation, applies to other structured random Laplacian ensembles whose raw moments diverge.
For p=2 the moments and the probability of odd rank are consistent with the conjectured one-parameter family, giving strong evidence that the full distribution is the Mészáros measure with a=0.

Where Pith is reading between the lines

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

The same truncation technique should extend the result to the full sandpile group (not just its p-parts) once a global integral-matrix universality theorem becomes available.
The divergence of raw moments is likely generic for bipartite or rectangular Laplacian models whenever the aspect ratio α is close to 1/p; controlling the “too many zero degrees mod p” event may be the universal fix.
For p=2 the missing uniqueness of the moments suggests that an independent computation of the probability of each 2-group of even rank would finish the conjecture.

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

Solid proof of the odd-p bipartite sandpile conjecture via a clean truncation that tames diverging moments; p=2 left open but honestly so.

read the letter

This paper settles the odd-p case of the Bhargava–de Pascale–Koenig / Fulman–Kaplan–Singhal–Warnaar conjecture on the p-Sylow of the sandpile group of an Erdős–Rényi bipartite graph G_α(n,u) with α > 1/p. The limiting law is exactly Wood’s P^Sym_∞,p, the same one that appears for ordinary and regular graphs.

What is new is the obstruction and the fix. Raw surjective moments diverge because a rare set of matrices (too many row-sums ≡ 0 mod p) contribute huge spikes. Singhal discards that set (the complement of E_{n,ρ} with 0 < ρ < α − 1/p), shows its probability tends to 1 (Lemma 9.8), and proves that the conditional Hom-moments converge to the known moments of P^Sym_∞,p (Proposition 2.8 / 2.11). Wood’s universality theorem then finishes the job for odd p. The same truncated moments work for p=2 and give the correct rank distribution, but do not pin down the full group law; that is stated clearly.

The technical work is careful and standard: translated codes, δ-depth, robust characters, special-character enumeration, and two error regimes that decay exponentially once δ is small. The character-sum estimates and robustness counts are direct adaptations of Wood’s method; I see no gap that would invalidate the conditional-moment limit. Citations are appropriate and the circularity burden is essentially zero.

Soft spots are minor and acknowledged. The p=2 distribution remains open (moments alone are insufficient), and the argument is long and technical, so a referee will need to check the exponential-decay constants carefully. Neither issue undermines the main theorem.

This is for people already working on sandpile groups, Cohen–Lenstra heuristics for graphs, or random-matrix cokernels. It is a clean, self-contained advance inside that program and deserves a serious referee. I would accept it for peer review and expect it to appear after the usual polishing.

Referee Report

0 major / 4 minor

Summary. The paper proves that for odd primes p and 1/p < α ≤ 1, the p-Sylow subgroup of the sandpile group of the random bipartite Erdős–Rényi graph G_α(n,u) converges in distribution to the Clancy–Kaplan–Leake–Payne–Wood measure P^Sym_∞,p. The same limit is established for the cokernel of the associated random bipartite Laplacian with independent ε-balanced p-adic entries (Theorems 1.4 and 1.7). The argument proceeds by showing that, after conditioning on the high-probability event that fewer than (1/p + ρ)n row sums vanish mod p, the surjective (equivalently Hom) moments converge to those of P^Sym_∞,p; Wood’s universality theorem then yields the distributional limit. For p = 2 the same truncated moments are obtained and the probability of odd rank is shown to be 1/2, which determines the limiting rank distribution (Theorem 1.6) but not the full group distribution.

Significance. The result settles a natural bipartite analogue of Wood’s theorem on sandpile groups of Erdős–Rényi graphs and of Mészáros’s theorem for regular graphs, for all odd primes. The technical contribution is the systematic truncation that removes a rare set of matrices whose contribution makes the raw moments diverge, together with a complete adaptation of the translated-code / robust-character machinery to the bipartite Laplacian. The rank distribution is obtained for every prime, including p = 2. The work therefore both confirms the conjectures of Bhargava–de Pascale–Koenig and Fulman–Kaplan–Singhal–Warnaar for odd p and supplies a reusable truncation technique for other bipartite or rectangular matrix models.

minor comments (4)

[§2.3] The definition of δ-depth (Definition 2.10) is stated to conflict with Wood’s earlier usage; a one-sentence comparison would help readers who consult both papers.
[§6.2] In the proof of Proposition 6.2 the main-term contribution is written with denominator |T|^{n1+n2-1} p^{n1}; the subsequent Corollary 6.2.1 absorbs the missing |T| factor correctly, but a brief remark that the -1 arises from the free choice of the constant character would improve readability.
[§9.3] Lemma 9.8 is cited from the companion paper [16]; a short self-contained sketch (or an explicit reference to the corresponding concentration argument) would make the present manuscript more independent.
[§8] A few typographical inconsistencies appear (e.g., “Ra w moments” in the section title of §8, occasional missing spaces after punctuation). These are easily corrected in production.

Circularity Check

0 steps flagged

No significant circularity: truncated moments are derived by direct character-sum estimates and matched to an external known measure via Wood’s black-box universality theorem.

full rationale

The derivation chain is self-contained and non-circular. The target measure P^Sym_∞,p is imported from Clancy–Kaplan–Leake–Payne–Wood (external) with known surjective moments |Λ²G| (Theorem 2.1). The paper’s original work is the direct estimation, after discarding the rare event η(L_n^(α)) ∉ E_{n,ρ}, that the conditional Hom/Sur moments of the bipartite Laplacian equal exactly those values (Propositions 2.8/2.11, proved via special-character enumeration, robustness counts, and exponential error bounds in §§3–6). Wood’s external universality theorem (Theorem 2.4) then converts matching moments into distributional convergence for odd p. The truncation itself is justified by an independent probability estimate (Lemma 9.8) and by the explicit divergence of the untruncated moments (Proposition 8.1). Self-citations to the companion note [8] merely supply the original conjecture statement and the moments of a one-parameter family for p=2; they are not used to force uniqueness or to define the bipartite moments. No equation reduces the claimed limit to a quantity fitted from the same ensemble, and no uniqueness theorem is imported from the author’s own prior work as an external fact. The argument therefore stands as an independent verification that the bipartite ensemble realizes the already-known measure.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The argument rests on standard p-adic probability, Wood’s universality theorems for cokernels of ε-balanced matrices, and elementary character-sum estimates. No free parameters are fitted; the only modelling choices are the fixed edge probability u∈(0,1), the aspect ratio α>1/p, and the ε-balanced hypothesis on the entry measure, all of which are stated as hypotheses rather than fitted.

axioms (3)

standard math Wood’s universality theorem (Theorem 2.4 / [17, Thm 8.3]): matching surjective moments bounded by |Λ^{2}G| imply convergence in distribution of random finitely generated Z_p-modules.
Invoked after Proposition 2.3 to upgrade moment convergence to distributional convergence for odd p.
domain assumption Entries of the random matrix are independent and ε-balanced for a fixed ε>0.
Used throughout the character-sum estimates (Lemma 2.15) and the concentration of η(L) (Lemma 9.8).
domain assumption α > 1/p so that a positive-density set of degree vectors lies outside the bad set E_{n,ρ}^c.
Necessary for the probability of the good set to tend to 1 and for the Type-2 error term to vanish.

invented entities (2)

Translated codes and δ-depth of maps Hom(Z_p^n,G) no independent evidence
purpose: Partition the Hom space so that main-term and error-term contributions can be estimated separately.
Technical bookkeeping devices adapted from Wood’s codes; no independent physical or combinatorial meaning claimed outside the moment calculation.
The rare set of matrices with too many p-divisible row sums (E_{n,ρ}^c) no independent evidence
purpose: Identify the source of diverging moments so that conditioning removes the divergence.
Defined ad hoc for the bipartite Laplacian; its probability tends to zero by a standard concentration argument.

pith-pipeline@v1.1.0-grok45 · 50928 in / 2483 out tokens · 22901 ms · 2026-07-14T00:42:59.568064+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 Erd\H{o}s--R\'enyi bipartite graph $G_{\alpha}(n,u)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceil\alpha n\rceil$, and edge probability $0<u<1$. The authors of [1] and [8] conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $G_{\alpha}(n,u)$ as $n\to\infty$. We prove this conjecture for odd primes $p$. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group to $H$, for each finite abelian $p$-group $H$. However, in our setting, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this issue by discarding the graphs for which too many vertices have degrees divisible by $p$. Once we remove the contribution of this rare set of graphs, then the surjective moments converge to the expected values. When $p$ is odd, applying Wood's universality theorem yields the desired convergence in distribution. For $p=2$, our computed moments (after excluding the rare set of graphs) match those of the conjectured distribution. However, these moments do not uniquely determine a distribution.

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

19 extracted references · 3 linked inside Pith · cited by 1 Pith paper

[1]

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

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[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

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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.arXiv preprint arXiv:2303.09125, 2023

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

Pith/arXiv arXiv 2023
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JulienClancy, NathanKaplan, TimothyLeake, SamPayne, andMelanieMatchettWood.Onacohen–lenstra heuristic for jacobians of random graphs.Journal of Algebraic Combinatorics, 42(3):701–723, 2015

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Ole Warnaar

Jason Fulman, Nathan Kaplan, Deepesh Singhal, and S. Ole Warnaar. Sandpile groups of random bipartite graphs and families of distributions with the same moments.arXiv preprint arXiv:2607.08607, 2026

Pith/arXiv arXiv 2026
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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
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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
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Joint distribution of the cokernels of random p-adic matrices

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2023
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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
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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
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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

arXiv 2026
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Distribution of sandpile groups of random directed bipartite graphs.arXiv preprint arXiv:2606.10214, 2026

Deepesh Singhal. Distribution of sandpile groups of random directed bipartite graphs.arXiv preprint arXiv:2606.10214, 2026

Pith/arXiv arXiv 2026
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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
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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

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Probability theory for random groups arising in number theory

Melanie Matchett Wood. Probability theory for random groups arising in number theory. InInternational Congress of Mathematicians, pages 4476–4508. European Mathematical Society-EMS-Publishing House GmbH, 2023

2023

[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.arXiv preprint arXiv:2303.09125, 2023

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

Pith/arXiv arXiv 2023

[6] [6]

JulienClancy, NathanKaplan, TimothyLeake, SamPayne, andMelanieMatchettWood.Onacohen–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

1989

[8] [8]

Ole Warnaar

Jason Fulman, Nathan Kaplan, Deepesh Singhal, and S. Ole Warnaar. Sandpile groups of random bipartite graphs and families of distributions with the same moments.arXiv preprint arXiv:2607.08607, 2026

Pith/arXiv arXiv 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

arXiv 2026

[16] [16]

Distribution of sandpile groups of random directed bipartite graphs.arXiv preprint arXiv:2606.10214, 2026

Deepesh Singhal. Distribution of sandpile groups of random directed bipartite graphs.arXiv preprint arXiv:2606.10214, 2026

Pith/arXiv arXiv 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

[19] [19]

Probability theory for random groups arising in number theory

Melanie Matchett Wood. Probability theory for random groups arising in number theory. InInternational Congress of Mathematicians, pages 4476–4508. European Mathematical Society-EMS-Publishing House GmbH, 2023

2023