Distribution of Sandpile groups of random bipartite graphs

Abstract: Fix a prime $p$ and a constant $\frac{1}{p}&lt;α\leq 1$. Consider the random Erdős--Rényi bipartite graph $G_α(n,u)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceilαn\rceil$, 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 $G_α(n,u)$ 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.

• The paper establishes that the limiting distribution of Sylow p-subgroups in sandpile groups aligns with symmetric matrix cokernel distributions for odd primes, resolving key conjectures.
• It determines explicit rank distributions and parity constraints, notably showing that the 2-Sylow case has a 1/2 odd rank probability.
• The analysis innovatively refines the surjective moment method by excluding rare pathological graphs, bridging combinatorial graph theory with random matrix principles.

Distribution of Sandpile Groups in Random Bipartite Graphs

Introduction and Context

The paper "Distribution of Sandpile groups of random bipartite graphs" (2607.10056) offers a detailed analysis of the limiting distributions of Sylow $p$-subgroups of sandpile groups associated with the Erdős–Rényi random bipartite graphs $G_\alpha(n, u)$, where $|V_1| = n$, $|V_2| = \alpha n$ for fixed $\frac{1}{p}<\alpha\leq 1$, and each edge is present independently with probability $0random matrix theory, group theory, and probability, and generalizes the emergence of the Cohen–Lenstra and related distributions for sandpile groups in classical random graph models.

Prior results established the distributional convergence of sandpile groups or their Sylow $p$-subgroups for both dense and regular random graphs [wood2017distribution, meszaros2020distribution], and for several structured random matrix models [wood2019random, clancy2015cohen]. However, the bipartite case presents unique technical obstacles, notably the divergence of surjective moments—an issue not encountered in the symmetric or regular settings. The current work rigorously resolves the conjectures posed in [bhargava2023rank, FulmanKaplanSinghalWarnaar_SylowSandpileBipartite], establishing limiting distributions and clarifying the scope and limitations of the surjective moment method in the bipartite regime.

Main Theoretical Results

Limiting Distribution of Sylow $p$-subgroups

Let $S_{G_\alpha(n,u)}$ denote the sandpile group of the random bipartite graph $G_\alpha(n,u)$. The main theorem provides:

• For odd primes $G_\alpha(n, u)$0 and $G_\alpha(n, u)$1,

$G_\alpha(n, u)$2

for any finite abelian $G_\alpha(n, u)$3-group $G_\alpha(n, u)$4, where $G_\alpha(n, u)$5 matches the distribution arising for cokernels of large symmetric matrices with entries in $G_\alpha(n, u)$6-adic integers [clancy2015cohen, wood2017distribution]. This result is obtained by controlling the surjective moments after excising an exponentially small set of "bad degree" graphs, then invoking Wood's universality principle.

• For $G_\alpha(n, u)$7, while the surjective moments match those of the conjectured distribution after restriction, they do not uniquely specify the measure; however, the rank distribution is determined, and parity constraints (the odd rank probability is shown to be $G_\alpha(n, u)$8) further restrict the possibilities in line with the conjecture in [FulmanKaplanSinghalWarnaar_SylowSandpileBipartite].

Limiting Distribution of Ranks

The paper also establishes the limiting distribution for the $G_\alpha(n, u)$9-rank of these sandpile groups for all primes $|V_1| = n$0:

$|V_1| = n$1

where $|V_1| = n$2 has an explicit formula derived from the known symmetric pairing distributions, and for $|V_1| = n$3 is shown to satisfy $|V_1| = n$4.

Moment Calculation and Matrix Model

A crucial technical step is the formulation of the Laplacian as a random matrix with independent (but not always Haar-uniform) entries and the introduction of the so-called $|V_1| = n$5-balanced random matrix model. The authors show that the rare event of "too many degree zero mod $|V_1| = n$6" vertices can cause divergence of moments, but that removing these pathological instances (which are exponentially unlikely as $|V_1| = n$7) corrects the issue and recovers the conjectural distribution.

Divergence Phenomenon for Surjective Moments

The paper provides an example where, for certain parameters, the expected number of surjections from the sandpile group to a chosen $|V_1| = n$8-group diverges, revealing that the raw moment method cannot always suffice; the restriction to the high-degree subset and refined probabilistic analysis are then essential.

Implications and Methodological Innovations

Relationship to Random Matrix Theory & Number Theory

This work strengthens the random matrix-sandpile group analogy, illustrating that for a broader class of random graph Laplacians with structural constraints (bipartiteness; variable group sizes), the limiting group law is governed by universality principles akin to those in classical random matrix theory for symmetric or permutation-invariant models [wood2017distribution, wood2019random, nguyen2025local]. The use of the symmetric pairing law, rather than the Cohen–Lenstra law, is reinforced as the correct heuristic for sandpile groups—and further, the analysis clarifies which graph ensembles fall under each type.

Role and Limitation of Surjective Moments

The traditional method—matching expected surjective counts to all possible target abelian groups—proves insufficient in the bipartite case, as surjective moments generically diverge due to rare degree configurations. The novel contribution here is the systematic excision of these exceptional cases, validation of the method on the high-probability subset, and reduction to known universality theorems. The analysis of the $|V_1| = n$9 case highlights the limitations of moment methods for distinguishing certain measures, showing the necessity of additional symmetry or parity constraints.

Practical and Theoretical Significance

These results have substantial implications in arithmetic statistics, random graph theory, and mathematical physics, notably for problems where sandpile groups (chip-firing, rotor-routing) arise in network theory, tropical geometry, and abelian sandpile model studies. More broadly, this work informs the ongoing extension of random matrix heuristics and universality phenomena to new combinatorial and algebraic contexts.

Future Directions

Several avenues are open for further research:

• Uniqueness for $|V_2| = \alpha n$0: The precise characterization of the $|V_2| = \alpha n$1-Sylow sandpile group law for bipartite graphs remains open, pending an argument that parity plus surjective moments determine the distribution.
• Sparse/Other Ensembles: Analysis of sparser bipartite graphs, models with more complicated degree distributions, or random regular bipartite graphs could reveal new phenomena or require further refinement of the moment method.
• Directed and Hypergraphs: The companion results for directed bipartite graphs indicate broader applicability; extension to multi-edge, non-bipartite, or hypergraph Laplacians may bring new challenges.
• Effective Rates: Recent progress in effectivizing such universality theorems (providing error bounds or non-asymptotic statements) [shen2026quantative] invites refinement of current results toward practical sampling or computation.

Conclusion

This paper rigorously establishes the limiting law for Sylow $|V_2| = \alpha n$2-subgroups of sandpile groups of large random bipartite graphs for all odd primes and provides a matching rank law for all primes, resolving conjectures and clarifying subtleties of the surjective moment method in non-symmetric random matrix ensembles. The interplay between combinatorial phenomena (vertex degrees, rare events), group-theoretic distributional heuristics, and random matrix theory is elucidated with full technical rigor and points the way to further universality results in arithmetic and combinatorial probability (2607.10056).