Cokernels of Random Integral Matrices

• Cokernels of random integral matrices are finite abelian groups defined as ℤⁿ modulo the image of a matrix, encoding key combinatorial and arithmetic properties.
• They exhibit universal behavior, with Sylow p-subgroups converging to Cohen–Lenstra distributions under mild sparsity and growth conditions.
• Advanced techniques including surjection moments, entropic large deviations, and Gaussian saddle-point methods reveal the robust interplay between randomness and algebraic structure.

Cokernels of Random Integral Matrices

A cokernel of an $n \times n$ integral matrix $A$ is the finite abelian group $\mathbb Z^n / A(\mathbb Z^n)$, which encodes the structure of the module generated by $\mathbb Z^n$ modulo the image of $A$. The study of the distribution of cokernels of random integral matrices, and their Sylow $p$-subgroups, has revealed deep universality phenomena connecting random matrix models, number theory, and group theory. A central discovery is that for various large classes of random integral ensembles, the distribution of these cokernels converges to universal laws, typically the Cohen–Lenstra measures, which also describe the distribution of class groups of quadratic fields.

1. Structured Row-Sparse Determinantal Ensembles

A recent advance considers ensembles of random $n \times n$ integral matrices $A_n$ formed as random submatrices of a highly structured, extremely row-sparse matrix $B_n$ of the form

$$B_n\;=\;\bigl(e_{b_1}+e_{b_2}+\cdots+e_{b_{k_n}}\bigr)_{(b_1,\dots,b_{k_n})\in[n]^{k_n}}$$

where $e_i$ are the standard basis vectors of $\mathbb Z^n$, and each row of $B_n$ sums exactly $k_n$ basis elements. For each $A_n$, its rows are sampled as a random $n$-element subset of the rows of $B_n$ using a determinantal measure: each set $X$ of $n$ rows is selected with probability proportional to $\det(B_n[X])^2$. The parameter $k_n \ge 3$ controls row sparsity and is allowed to grow slowly with $n$ (for example, $k_n = o(n^{1/30})$; for the 2-part one needs $k_n \gg \log\log n$).

This class of ensembles interpolates between dense i.i.d. matrix models and extremely sparse combinatorial objects (such as hypertrees), but with a specific combinatorial structure imposed by row-sum constraints.

2. Main Universality Theorem: Cohen–Lenstra Laws for Cokernels

The central result for determinantal row-sparse matrices is the asymptotic universality of their cokernel distributions. Let $G$ be a finite abelian group, and let $P$ be any finite set of primes containing all primes dividing $|G|$. Under the technical assumptions:

• For each $p \in P$, eventually $p\nmid k_n$
• $k_n = o(n^\delta)$ for some $\delta > 0$
• If $2 \in P$, then eventually $\log\log n < k_n$

the joint law of the Sylow-$p$ parts of $\mathrm{coker}(A_n)$ converges, as $n\to\infty$, to the product Cohen–Lenstra measure: $$\lim_{n\to\infty} \mathbb P\!\Bigl(\bigoplus_{p\in P}\mathrm{coker}(A_n)_p \cong G\Bigr) = \frac1{|\operatorname{Aut}(G)|} \prod_{p\in P} \prod_{i=1}^{\infty}(1 - p^{-i}) = \prod_{p\in P}\nu_{CL,p}(G_p)$$ In particular, for any fixed prime $p$ not dividing $k_n$, the $p$-Sylow subgroup $\mathrm{coker}(A_n)_p$ is distributed according to the Cohen–Lenstra measure $\nu_{CL,p}$ as $n\to \infty$.

This result generalizes and extends earlier work that treated the case of fixed $k_n = 3$ and larger primes $p \ge 5$ by A. Mészáros; the new theorem removes both the restriction on $k_n$ and the restriction on $p$ (Lee et al., 16 May 2025).

3. Proof Strategy: Surjection Moments, Entropic Large Deviations, and Gaussian Asymptotics

The proof employs the moment method, leveraging a key theorem of M. M. Wood that establishes that distributional convergence to the Cohen–Lenstra law is equivalent to convergence of expected surjection counts: $$\lim_{n\to\infty} \mathbb E\left|\mathrm{Sur}\bigl(\mathrm{coker}(A_n),G\bigr)\right| = 1$$ for all finite abelian groups $G$. The expected number of surjections is expanded as a sum over surjective $n$-tuples $q = (q_1,\ldots, q_n) \in G^n$: $$\mathbb E\left|\mathrm{Sur}(\mathrm{coker}(A_n), G)\right| = \sum_{\substack{q \in G^n\ \langle q_1,\ldots,q_n\rangle = G}} \mathbb P(A_n q = 0 \in G^n)$$ Leading asymptotics are found by grouping by empirical distribution vectors $\mathbf{n} = (n_a)$ encoding the frequency of each $a \in G$, and analyzing contributions via large-deviation theory. The dominant contribution arises from vectors with empirical distributions nearly uniform on $G$, penalized via a Kullback–Leibler divergence $\exp[-n D_{KL}(\nu\|\mu)]$; deviation from uniformity incurs entropic suppression. Fourier-analytic, Pinsker-type large-deviation bounds and Gaussian saddle-point analysis in $|G|-1$ dimensions yield the exact limiting mass $\frac1{|\operatorname{Aut}(G)|}\prod(1-p^{-i})$ (Lee et al., 16 May 2025).

4. Connections to Sparse Random Matrices and Sandpile Groups

Row-sparse determinantal models are part of a broader family of sparse random matrix ensembles with combinatorial structures. For such ensembles, a pivotal feature is the existence of a threshold phenomenon: below a critical number of random (or nonzero) entries, the limiting cokernel distribution fails to be Cohen–Lenstra, while above threshold (e.g., $\sim n + C \log n$ active entries), universality is recovered (Kang et al., 2024).

Implications extend to the topology of random simplicial complexes; the result provides evidence that the torsion in the first homology of combinatorial objects such as random simplicial hyper-forests follows the Cohen–Lenstra law, generalizing earlier counterexamples in the 2-torsion case.

5. Universality and Interpolation Between Dense and Sparse Regimes

The determinantal row-sparse ensembles $\{A_n\}$ interpolate between fully dense (i.i.d.) integer matrix models, where Cohen–Lenstra universality has been established (Wood, 2015, Maples, 2013), and extremely sparse, combinatorial models (hypertrees, sandpile groups in random graphs (Wood, 2014)). All these models, provided the randomness is sufficiently non-degenerate (typically some anti-concentration/balancedness condition for the entries and mild growth requirements), yield the same universal $p$-group laws for fixed $p$, even though the matrix structure varies dramatically.

Key threshold, moment, and large-deviation arguments apply across this spectrum, and the moment method combined with entropy-based suppression of unlikely code structures proves robust to significant sparsity and structural deviations in the random ensemble.

6. Technical Ingredients and Extensions

The principal proof components are:

• Diagonalization of random code probability contributions in terms of positive semidefinite matrices $M_{\mathbf n}$, controlled analytically in the uniform regime
• Relative entropy (KL divergence) exponential penalties for nonuniform codewords
• Pinsker and Fourier-analytic large deviation bounds to control non-generic code contributions
• Gaussian (saddle-point) analysis for the main term
• Interplay with moment determinacy results for surjection-moments, following Wood, Sawin, and collaborators

The universality mechanisms extend to sparse $p$-adic models, random matrices over Dedekind domains, and to joint distributions of successive matrix products (random flags) (Kang et al., 2024, Yan, 2023, Huang et al., 13 Aug 2025).

7. Significance and Open Directions

The discovery that determinantal row-sparse integral ensemble cokernels converge to the Cohen–Lenstra law as $n\to\infty$, with only mild constraints on sparsity and no requirement for dense randomness, demonstrates a remarkable universality in the arithmetic of random discrete structures. This unifies a variety of models: dense i.i.d. matrices, sparse combinatorial objects, determinantal probability measures, and random Laplacians. The techniques and results highlight the essential role of entropy, moment methods, and combinatorics in producing explicit distributional laws for random abelian group-valued invariants in algebraic, geometric, and probabilistic settings (Lee et al., 16 May 2025, Kang et al., 2024, Wood, 2015, Maples, 2013).