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Large families of partition measures share the same moments, including two distinct sandpile-group laws for p=2.

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2026-07-10 05:04 UTC pith:7MUQG5HV

load-bearing objection Solid construction of multi-parameter moment-sharing families that cleanly unifies the Mészáros and bipartite p=2 sandpile cases; the bipartite conjecture is open but cleanly separated.

arxiv 2607.08607 v1 pith:7MUQG5HV submitted 2026-07-09 math.CO math.NTmath.PR

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

Jason Fulman , Nathan Kaplan , Deepesh Singhal , S. Ole Warnaar This is my paper
classification math.CO math.NTmath.PR MSC 05C8005E0515B5220K01
keywords sandpile groupsrandom bipartite graphsmethod of momentsHall–Littlewood functionsCohen–Lenstra heuristicsfinite abelian p-groupspartition measures
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 builds explicit families of probability measures on partitions (equivalently, on finite abelian p-groups) that all possess identical moments, even though the measures themselves are different. These families arise naturally when one studies the Sylow p-subgroups of sandpile groups of random graphs. In particular, the special distribution that appears for even-regular graphs at p=2 and the conjectured distribution for random bipartite graphs at p=2 are distinct, yet they belong to the same one-parameter family and therefore share every moment. The construction is obtained by weighting the classical symmetric-matrix cokernel measure by monomials in the conjugate partition and then taking affine combinations of sign-twisted versions; the resulting moments are evaluated with Hall–Littlewood polynomials and basic hypergeometric identities. The work therefore shows that the method of moments, while powerful when moments grow slowly, can leave large continuous moduli of distributions once the growth rate exceeds the Wood uniqueness threshold.

Core claim

For every strict partition κ of length r there exists a 2^{r−1}-parameter family of measures on partitions, all sharing the same collection of μ-moments as the weighted measure ˜P^{Sym,κ}_p. The two special p=2 sandpile distributions—one for even-regular graphs and one conjectured for bipartite graphs—both sit inside the r=1 subfamily and therefore have identical moments p^{n(μ)+ℓ(μ)}.

What carries the argument

The measures ˜P^{Sym,κ}_{p,α} obtained by multiplying the infinite symmetric cokernel measure by p^{∑κ_i λ′_i} and then taking affine combinations of the odd-sign twists indexed by subsets of [r]; their moments are computed by expressing the surjection counts via Hall–Littlewood skew polynomials and summing the resulting products of transition kernels K(a,b) with q-hypergeometric identities.

Load-bearing premise

The numerical evidence offered for the bipartite-sandpile conjecture consists of only 500 samples of graphs on 100 vertices and supplies no concentration bounds that would justify the infinite-n limit.

What would settle it

Compute the empirical distribution of the 2-Sylow of the sandpile group for random bipartite graphs with part sizes n and ⌈αn⌉, α>1/2, for n several thousand, and check whether the observed frequencies for partitions of length 1–4 converge to 2^{ℓ(λ)−1}P^{Sym}_{∞,2}(λ).

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper constructs large families of measures on partitions that share identical μ-moments, using Hall–Littlewood specializations and transition kernels K(a,b). For a strict partition κ of length r it produces a 2^{r-1}-parameter family ˜P^{Sym,κ}_{p,α} (α∈A_r) whose moments equal those of ˜P^{Sym,κ}_p (Theorems 1.8–1.9); the r=1 subfamily recovers both Mészáros’ even-d, p=2 distribution and the conjectural p=2 bipartite distribution, both having moments p^{n(μ)+l(μ)} (Theorem 1.10). The same machinery yields closed-form moments for Cohen–Lenstra, Malle-type and Garton-type measures (Theorems 1.11–1.13). Conjecture 1.3 on Sylow p-subgroups of random bipartite graphs is stated and supported by n=100 numerics.

Significance. The work supplies the first systematic source of non-uniqueness for the method of moments when moments exceed the Wood bound, and cleanly embeds two previously isolated special cases (Mészáros even-d p=2 and the bipartite p=2 conjecture) into a single geometric family. The proofs are self-contained, relying only on the skew Cauchy identity and classical q-hypergeometric summations; the polytope A_r is shown to be a cube, guaranteeing non-negativity of the affine combinations. The moment formulae for the Cohen–Lenstra, Malle and Garton families are new for general real parameters and finite d. The numerical experiments and accompanying code repository give concrete, falsifiable predictions for Conjecture 1.3.

minor comments (4)
  1. [Section 2] Section 2: the n=100, 500-sample experiments for Conjecture 1.3 lack error bars or concentration bounds; a short remark that the numerics are only heuristic would clarify the separation between proved theorems and open conjecture.
  2. [§1.5] After (1.7): the definition of the polytope A_r is clear, but a one-sentence reminder that the inequalities are exactly those needed for non-negativity of every coordinate would help the reader.
  3. [§4.1] Lemma 4.1 and Corollary 4.2: the appeal to dominated convergence for the a o∞ limit is correct but could be flagged more explicitly for readers less familiar with q-series.
  4. [References] References: the forthcoming preprint [20] of the third author is cited for the odd-p case of Conjecture 1.3; if a public arXiv link becomes available before publication it should be added.

Circularity Check

0 steps flagged

No significant circularity: moments are derived from Hall–Littlewood identities and q-series, not inserted by definition or self-citation.

full rationale

The paper’s central claims (Theorems 1.8–1.10) construct families of measures that share identical μ-moments and prove those moments equal p^{n(μ)+|κ|l(μ)} (or the normalized form p^{n(μ)+kl(μ)}). The derivation proceeds by rewriting P^Sym_∞,p via the transition kernels K(a,b) (Eqs. 4.1–4.3), expressing |Sur_p(λ,μ)| via Hall–Littlewood specializations (Prop. 1.5, Lemmas 4.3), evaluating the resulting multi-sums by induction with the q-hypergeometric identities of Lemmas 4.1, 4.4, 4.8 (proved from the 2φ0 evaluation in Prop. 6.1), and verifying that odd-cardinality sign flips leave the moments unchanged (Thm. 4.10). Normalization constants are fixed by the empty-partition moment so that the measures become probability distributions; the polytope A_r is defined by the linear inequalities that keep all probabilities non-negative (Lemma 6.5). None of these steps inserts the target moment formula by hand, fits a free parameter to data that is later “predicted,” or relies on a uniqueness theorem whose only support is a self-citation. Background formulas from Fulman–Kaplan 2019 are re-derived in Section 5 via the same Hall–Littlewood machinery. The open Conjecture 1.3 and its n=100 numerics are cleanly separated from the proved statements and do not underwrite them. Consequently the derivation chain is self-contained and free of circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The paper rests on standard combinatorial and q-series machinery plus previously published universality theorems for sandpile groups; no free parameters are fitted and no new physical entities are postulated. The only non-standard inputs are the definitions of the new measures themselves, which are pure constructions.

axioms (4)
  • domain assumption Wood’s universality theorem for cokernels of random symmetric p-adic matrices (Theorem 1.1 / [23])
    Used to identify the limiting sandpile distribution of Erdős–Rényi graphs with P^Sym_∞,p.
  • domain assumption Mészáros’s theorem for d-regular graphs (Theorem 1.2 / [15])
    Supplies the even-d, p=2 special case that is later shown to share moments with the bipartite law.
  • standard math Principal specializations and the skew Cauchy identity for Hall–Littlewood polynomials (Macdonald, Chapter III)
    Core computational engine for all moment evaluations (Sections 3–5).
  • domain assumption Koplewitz’s rank asymptotics for bipartite sandpile groups (Theorem 1.4 / [11])
    Justifies the necessity of the threshold α>1/p in Conjecture 1.3.
invented entities (1)
  • The signed measures ˜P^{Sym,κ}_{p,S} and the affine combinations ˜P^{Sym,κ}_{p,α} for α∈A_r no independent evidence
    purpose: To produce large families of distinct probability measures that share identical moments
    Purely combinatorial constructions; non-negativity is proved by showing A_r is a cube, but they have no independent existence outside the paper.

pith-pipeline@v1.1.0-grok45 · 35158 in / 2538 out tokens · 28465 ms · 2026-07-10T05:04:52.295069+00:00 · methodology

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Recently, there has been significant interest in applying the method of moments developed by Wood and others to study distributions of finite abelian groups that arise in number theory and combinatorics. When the moments do not grow too fast, they determine a unique distribution. We construct large families of distributions that have the same moments. These families include several distributions that arise naturally in the study of sandpile groups of families of random graphs. Wood determined the distribution of Sylow $p$-subgroups of sandpile groups of Erd\H{o}s--R\'enyi random graphs. This was extended by M\'esz\'aros to sandpile groups of random $d$-regular graphs, who observed an interesting special case when $d$ is even and $p = 2$. We study Sylow $p$-subgroups of sandpile groups of random bipartite graphs and similarly find a special case for $p =2$. Although this distribution differs from that of M\'esz\'aros, we show that they have the same moments and fit into our broader construction. To compute the moments of the distributions we study, we apply combinatorial tools from the theory of Hall--Littlewood functions.

Figures

Figures reproduced from arXiv: 2607.08607 by Deepesh Singhal, Jason Fulman, Nathan Kaplan, S. Ole Warnaar.

Figure 1
Figure 1. Figure 1: shows the results for seven partitions λ and compares the observed frequencies with the values predicted by the conjectural distribution P Sym ∞,3 (λ). λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 (1) (2) (3) (1, 1) (2, 1) (3, 1) conjecture (α, u) = (0.3, 0.5) (α, u) = (0.4, 0.5) (α, u) = (0.7, 0.2) (α, u) = (0.7, 0.5) (α, u) = (0.7, 0.8) (α, u) = (1, 0.25) (α, u) = (1, 0.5) (α, u) = (1, 0.75) [PITH_FULL_IMAGE:figures… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the conjectural distribution P Sym,1 2 and the empirical distribution of (SGα(100,u))2 for various choices of (α, u) and λ with sample size 500. The data corresponding to α = 0.45 is far from the conjectural distribution P Sym,1 2 , which is consistent with the fact that α < 1/2 = 1/p. In contrast, for the data with α > 1/2 the empirical frequencies for these seven partitions are all close to… view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of the 2-rank of the sandpile group of a random bipartite graph. Next, we compute the expected value of p l(λ) , for a random partition λ coming from P Sym,(k) p . This is a natural quantity to consider as p rank((SGα(n,u))p) = |SGα(n,u)/pSGα(n,u) |. Lemma 2.4. Let λ be a random partition chosen from the distribution P Sym,(k) p . The expected value of p l(λ) is 1 + p k . Proof of Lemma 2.4 as… view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of the 3-rank of the sandpile group of random bipartite graphs. However, based on our experiments this might not always be correct. Perhaps these limits hold for β < α ⩽ 1 for some β > 1/p [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Distribution of Sandpile groups of random bipartite graphs

    math.CO 2026-07 accept novelty 7.0

    For odd primes p and α>1/p the p-Sylow of the sandpile group of G_α(n,u) converges in distribution to P^Sym_∞,p after discarding a rare set of graphs with too many p-divisible degrees.

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