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

Problem Statement
The paper investigates the distribution of finite abelian $p$-groups, specifically the Sylow $p$-subgroups of sandpile groups associated with random graphs. While the distributions for Erdős–Rényi random graphs and random $d$-regular graphs are largely understood (converging to the Cohen–Lenstra heuristics or specific variations thereof), the behavior of sandpile groups for random bipartite graphs remains less explored. A central problem addressed is the determination of the limiting distribution of the Sylow $p$-subgroups of the sandpile group of a random bipartite graph $G(n, \lceil \alpha n \rceil, u)$ as $n \to \infty$.

Furthermore, the paper addresses a broader theoretical question in the context of the "method of moments": whether moments uniquely determine a distribution. While Wood established that moments determine a unique distribution if they do not grow too fast, this paper explores the construction of large families of distinct distributions that share the exact same moments, particularly in cases where the moments grow rapidly (e.g., when $p=2$).

Methodology
The authors employ a combination of probabilistic combinatorics, algebraic number theory, and the theory of symmetric functions. Key methodological components include:

1. Sandpile Groups and Random Graphs: The study utilizes the definition of the sandpile group $S_\Gamma$ as the cokernel of the reduced Laplacian of a graph $\Gamma$. The authors analyze random bipartite graphs $G(n_1, n_2, u)$ where edges are included with probability $u$.
2. Moment Method: The primary analytical tool is the computation of moments. For a distribution $\nu$ on partitions and a fixed partition $\mu$, the $\mu$-moment is defined as the expected number of surjective homomorphisms from a random group (chosen according to $\nu$) to the fixed group $G_\mu$.
3. Hall–Littlewood Symmetric Functions: To compute these moments, the authors translate the problem into the language of Hall–Littlewood symmetric polynomials $P_\lambda$ and $Q_\lambda$. They utilize the relationship between the number of surjections $|\text{Sur}_p(\lambda, \mu)|$ and these polynomials, as established by Nguyen and Van Peski.
4. $q$-Hypergeometric Series: The evaluation of sums involving these polynomials relies heavily on identities from basic hypergeometric series (specifically $r\phi_s$ series) and transformation formulas (such as those by Rogers–Szegő and others).
5. Markov Chain Interpretations: The distributions are interpreted via transition probabilities $K(a, b)$, allowing the authors to express the probability of a partition $\lambda$ as a product of these transition weights. This facilitates the calculation of moments by summing over the conjugate parts of the partitions.

Key Contributions and Results

• Conjecture on Random Bipartite Graphs: The paper proposes Conjecture 1.3, which posits the limiting distribution of the Sylow $p$-subgroups of sandpile groups of random bipartite graphs.

  • For odd primes $p$, the distribution is predicted to be the standard symmetric matrix distribution $P^{\text{Sym}}_{\infty, p}$.
  • For $p=2$, the distribution is predicted to be a modified version: $2^{l(\lambda)-1} P^{\text{Sym}}_{\infty, 2}(\lambda)$, where $l(\lambda)$ is the length of the partition.
  • The authors provide numerical evidence (Section 2) supporting this conjecture for $p=2$ and $p=3$, observing a threshold phenomenon where the parameter $\alpha$ (the ratio of the smaller vertex set to the larger) must satisfy $\alpha > 1/p$ for the distribution to converge to the predicted form.

• Families of Distributions with Identical Moments: A major theoretical contribution is the construction of large families of distinct measures that share the same moments.

  • The authors define a family of measures $\tilde{P}^{\text{Sym}, \kappa}_{p, \alpha}$ parameterized by a strict partition $\kappa$ and a vector $\alpha$ in a polytope $A_r$.
  • Theorem 1.9 states that for a fixed strict partition $\kappa$ of length $r$, there exists a $2^{r-1}$-parameter family of measures that all possess the same moments.
  • This construction generalizes the specific case observed by Mészáros for random $d$-regular graphs (where $d$ is even and $p=2$) and the special case for random bipartite graphs ($p=2$). The paper demonstrates that these seemingly different distributions are actually members of the same moment-equivalence family.

• Moment Computations for Various Distributions: The paper provides closed-form formulas for the moments of several distributions, often reproving known results or extending them to real parameters $p > 1$:

  • Theorem 1.11: Moments of the Cohen–Lenstra distribution $P_{d, u}$.
  • Theorem 1.12: Moments of a generalization of Malle's distribution (motivated by class groups of number fields with roots of unity).
  • Theorem 1.13: Moments of a generalization of Garton's distribution.
  • Theorem 1.10: Moments of the normalized measures $P^{M, (k)}_{p, \alpha}$, showing they equal $p^{n(\mu) + kl(\mu)}$.

• Technical Lemmas: The paper includes technical proofs regarding the properties of the transition functions $K(a, b)$ and the geometry of the polytope $A_r$, proving that $A_r$ is a $2^{r-1}$-dimensional cube.

Significance and Claims
The paper claims significance in two primary areas:

1. Advancing the Understanding of Sandpile Groups: By formulating and providing evidence for the distribution of sandpile groups in random bipartite graphs, the work extends the universality results known for Erdős–Rényi and regular graphs to a new class of random graphs. The identification of a special case for $p=2$ highlights the nuanced behavior of sandpile groups depending on the graph structure and the prime $p$.

2. Challenging the Uniqueness of Moment Determination: The construction of large families of distributions with identical moments serves as a counterpoint to the method of moments when moments grow too rapidly. The authors explicitly state that while Wood's theorem guarantees uniqueness under certain growth conditions, their work demonstrates that "moments no longer determine the measure for larger values of the moments." This provides concrete examples of non-uniqueness in the context of finite abelian groups, fitting into broader recent developments in the method of moments (citing Wood, Sawin, and others).

The paper does not claim to prove Conjecture 1.3 for all cases (noting that a proof for odd primes is forthcoming in separate work) but asserts that the construction of moment-equivalent families is a complete and rigorous result that unifies several previously observed phenomena in the literature.