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

Abstract: 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 Erdős--Rényi random graphs. This was extended by Mészáros 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 Mészáros, 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.

• The paper constructs large families of distinct probability distributions on partitions with identical moments, including a 2^{r-1}-parameter family for every strict partition of length r.
• Hall–Littlewood symmetric functions and basic hypergeometric identities yield explicit moment formulas, prove moment bounds, and extend calculations for Cohen–Lenstra, Malle, and Garton-type distributions.
• Numerical evidence supports the conjectured bipartite-graph threshold α=1/p, while the p=2 limit and the behavior of expected p-ranks near this threshold remain open problems.

Overview

This paper by Fulman, Kaplan, Singhal, and Warnaar studies the distribution of Sylow $p$-subgroups of sandpile groups of random bipartite graphs and, more broadly, constructs large families of distinct distributions on partitions that share the same moments. The work sits at the intersection of the method-of-moments program for finite abelian $p$-groups initiated by Wood [2017], the combinatorics of Hall–Littlewood symmetric functions, and the theory of basic hypergeometric series.

The central observation is a moment-degeneracy phenomenon: while Wood's universality theorem guarantees that moments bounded by $p^{n(\mu)}$ determine a unique measure on partitions, several natural distributions arising from random graph sandpile groups have larger moments, and the paper shows these moments fail to be injective in a structured way. Specifically, two distributions—one conjectured for Sylow 2-subgroups of sandpile groups of random bipartite graphs, and one proved by Mészáros for even-degree random regular graphs—are different yet have identical moments.

Background: sandpile groups and moment methods

For a connected graph $\Gamma$, the sandpile group $S_\Gamma$ is the cokernel of the reduced Laplacian; its order counts spanning trees by Kirchhoff's theorem. Wood proved that for Erdős–Rényi random graphs $G(n,u)$, the Sylow $p$-subgroups $(S_{G(n,u)})_p$ converge in distribution to the measure $P^{\mathrm{Sym}}_{\infty,p}$ arising as the cokernel distribution of Haar-random symmetric matrices over $\mathbb{Z}_p$, whose $p$0-moment is exactly $p$1. Mészáros extended this to random $p$2-regular graphs, observing an anomaly when $p$3 is even and $p$4: the limiting 2-Sylow distribution has odd rank always, with probabilities $p$5 on partitions of odd length, and its $p$6-moments equal $p$7—exceeding Wood's uniqueness threshold.

The paper's bipartite analogue concerns $p$8 with fixed $p$9. Koplewitz's result on expected $p^{n(\mu)}$0-ranks establishes a sharp threshold: for $p^{n(\mu)}$1 all limiting probabilities vanish, while for $p^{n(\mu)}$2 the paper conjectures (Conjecture 1.1) convergence to $p^{n(\mu)}$3 for odd $p^{n(\mu)}$4 and to $p^{n(\mu)}$5 for $p^{n(\mu)}$6. A proof for odd primes appears in forthcoming work of the third author; the $p^{n(\mu)}$7 case remains open here.

Families of distributions with identical moments

The main structural contribution is a general construction. For any partition $p^{n(\mu)}$8 of length $p^{n(\mu)}$9, define $\Gamma$0, and for subsets $\Gamma$1 introduce signed variants multiplying by $\Gamma$2. The key theorem states:

• The $\Gamma$3-moments of $\Gamma$4 are bounded by $\Gamma$5, so they exceed Wood's threshold whenever $\Gamma$6.
• If $\Gamma$7 is strict, then for every odd-cardinality subset $\Gamma$8, the measures $\Gamma$9 and $S_\Gamma$0 have identical $S_\Gamma$1-moments for all $S_\Gamma$2.

Fixing a strict $S_\Gamma$3 of length $S_\Gamma$4 thus yields $S_\Gamma$5 distinct measures with common moments, and affine combinations parametrized by a polytope $S_\Gamma$6—shown to be a $S_\Gamma$7-dimensional cube via a Hadamard-type orthogonality argument—give a full $S_\Gamma$8-parameter family of probability distributions sharing those moments, with injectivity of the parametrization established.

The case $S_\Gamma$9 is highlighted: for each positive integer $G(n,u)$0 there is a one-parameter family $G(n,u)$1, $G(n,u)$2, all having $G(n,u)$3-moments $G(n,u)$4. The Mészáros even-$G(n,u)$5, $G(n,u)$6 distribution is the instance $G(n,u)$7, while the conjectural bipartite $G(n,u)$8 limit is $G(n,u)$9—two genuinely different distributions with the same moments. This connects to Poonen–Rains heuristics for $p$0-Selmer ranks of elliptic curves: the rank distribution under $p$1 matches their conjecture, and Wood's ICM results show any measure with the corresponding rank moments agrees with some member of the family on ranks.

Moment computations via Hall–Littlewood theory

The technical engine is a transition-probability factorization. Writing $p$2 for explicit Markov transition weights, the identity $p$3 reduces moment sums to iterated applications of $p$4-hypergeometric summations, notably a terminating $p$5 transformation and the Rogers–Szegő evaluation at $p$6. Moments of $p$7 are expressed through a Srivastava–Daoust type multiple series $p$8, whose evaluation yields both the moment bounds and the equality of moments across the family (the latter following from the vanishing of alternating sums $p$9).

The same machinery reproves Fulman–Kaplan's closed form for moments of the Cohen–Lenstra distributions $(S_{G(n,u)})_p$0 (moments $(S_{G(n,u)})_p$1), and computes moments for two generalizations tied to class group heuristics in the presence of roots of unity:

• A Malle/Lipnowski–Sawin–Tsimerman type family $(S_{G(n,u)})_p$2, with moments $(S_{G(n,u)})_p$3, valid for real $(S_{G(n,u)})_p$4 and non-prime $(S_{G(n,u)})_p$5.
• A Garton type family $(S_{G(n,u)})_p$6, with moments $(S_{G(n,u)})_p$7, extending results previously known only for integer parameters.

These computations hold for arbitrary real $(S_{G(n,u)})_p$8, where surjection counts are defined algebraically through Hall–Littlewood skew functions via the Nguyen–Van Peski formula.

Numerical evidence

Simulations in Sage ($(S_{G(n,u)})_p$9, 500 samples per parameter pair) support Conjecture 1.1. For both $P^{\mathrm{Sym}}_{\infty,p}$0 and $P^{\mathrm{Sym}}_{\infty,p}$1, empirical frequencies of $P^{\mathrm{Sym}}_{\infty,p}$2 closely match the predicted distributions once $P^{\mathrm{Sym}}_{\infty,p}$3, while data at $P^{\mathrm{Sym}}_{\infty,p}$4 deviates sharply, exhibiting heavier rank tails consistent with Koplewitz's linear growth of expected rank below the threshold. Rank-distribution experiments show the same threshold behavior.

A notable tension emerges in the expected values of $P^{\mathrm{Sym}}_{\infty,p}$5: the conjectures imply limits of $P^{\mathrm{Sym}}_{\infty,p}$6 (odd $P^{\mathrm{Sym}}_{\infty,p}$7) and $P^{\mathrm{Sym}}_{\infty,p}$8 ($P^{\mathrm{Sym}}_{\infty,p}$9). Empirically, expectations are near these values for $\mathbb{Z}_p$0 comfortably above $\mathbb{Z}_p$1, but at $\mathbb{Z}_p$2 for $\mathbb{Z}_p$3 the observed value was about $\mathbb{Z}_p$4, driven by two outlier graphs of ranks 12 and 9; excluding five outliers brings it to roughly $\mathbb{Z}_p$5. The authors explicitly concede that the conjectured limits may fail near the threshold and may only hold for $\mathbb{Z}_p$6 with some $\mathbb{Z}_p$7—an open question the data raises but does not resolve.

Limitations and open questions

Several caveats bound the results. The bipartite conjecture itself is unproven in the paper for all primes: odd $\mathbb{Z}_p$8 is deferred to separate forthcoming work, and $\mathbb{Z}_p$9 remains entirely conjectural, supported only by computation. The numerical experiments use $p$00 with 500 samples, so agreement near the $p$01 threshold—where expected ranks grow polynomially and outliers dominate empirical means—is weak evidence. The moment-equality construction requires $p$02 strict; whether analogous degeneracies exist for general $p$03 is not addressed. Finally, the paper deliberately excludes questions of pairings on sandpile groups and joint distributions across multiple primes, and the relationship between the moment families constructed here and the general uniqueness criteria of Sawin–Wood is left implicit rather than developed.

Conclusion

The paper identifies a concrete moment-degeneracy phenomenon for distributions on finite abelian $p$04-groups, embeds the $p$05 anomalies for random regular and bipartite graph sandpile groups into a unified $p$06-parameter family, and supplies a Hall–Littlewood/hypergeometric toolkit that computes moments for a range of Cohen–Lenstra-type distributions beyond the prime setting. Its main open problems—the $p$07 bipartite conjecture and the behavior of expected $p$08-ranks near the $p$09 threshold—define the immediate boundary of what is currently established.