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Random matrices with independent ε-balanced entries in a log(n)^{1+δ} band—and arbitrary entries outside—have cokernels approaching the Cohen–Lenstra distribution; up to αn per column and βn per row bad entries are also tolerated.

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arxiv 2607.06952 v2 pith:OHC52E5I submitted 2026-07-08 math.PR math.COmath.NT

Universality for cokernels of partially random integral matrices

Isaac Rajagopal This is my paper
classification math.PR math.COmath.NT
keywords entriesdistributionvarepsilonalphabalancedbetarandomallowing
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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The Cohen–Lenstra heuristics predict the distribution of the prime-p part of a number field's class group. Wood proved in 2019 that the same distribution appears as the cokernel of a random p-adic matrix whose entries are independent and 'ε-balanced'—no residue class is too likely. This paper asks how much randomness can be removed before the universality breaks. If at most αn entries per column and βn per row are allowed to be arbitrary or fixed, with α+β<1, the cokernel still converges to the Cohen–Lenstra distribution. The examples in Figures 1 and 2 show the condition is close to sharp: with α+β≥1 one can build matrices that never have full rank or that split into independent halves. The paper also permits some dependence inside each column: a column may be produced from independent random bits by an invertible linear change of basis, as long as enough 'good' coordinate directions are present across columns. The second main result is about band matrices: if the entries in a band of width log(n)^{1+δ} around the diagonal are ε-balanced and every other entry is completely arbitrary, the cokernel still converges to the Cohen–Lenstra distribution, answering a question of Kang, Lee, and Yu. The proof uses the moment method: instead of computing the cokernel directly, it counts expected surjections to every finite abelian p-group and shows these moments match the Cohen–Lenstra values. The hard part is bounding the number of surjections that fail to be 'codes' using inverse Littlewood–Offord estimates and a covering lemma for the path graph.

Core claim

Theorem 1.5: Let p be a prime, δ, ε>0, and B a finite abelian p-group. Let M(n) be a random n×n matrix over Z_p with independent entries, ε-balanced if |i−j|≤log(n)^{1+δ} and arbitrary otherwise. Then lim_{n→∞} P(cok(M(n))≃B)=∏_{k≥1}(1−p^{−k})/|Aut(B)|. Theorem 1.6 gives the analogous conclusion when at most αn entries per column of M(n) are ε-degenerate and a basis u_1,...,u_n exists with #{j: u_i∉W_j}≤βn for each i, with α+β<1, after allowing each column to become independent under an invertible linear change of basis. If the paper is correct, the Cohen–Lenstra universality class is far larger than Wood's original independent ε-balanced setting.

Load-bearing premise

The proof factorizes the moment as a product over columns, E(#Sur(cok(M),G))=∏_j P(F M_j=0) (Eq. 2.1), so the columns of M(n) must be independent; the paper explicitly says it relies on this and leaves cross-column dependence open. The dependent-column theorem additionally assumes condition (⋆): each column becomes independent after multiplying by some A_j∈GL_n(Z_p), and that a fixed basis u_1,...,u_n satisfies (1.1), i.e. each u_i is outside the ε-balanced locus W_j for at most βn columns. If all columns share a single index-p submodule, condition (1.1) fails and the conclusion can indeed fail (the paper's own example II). Thus the principal fragility is not the algebra but the geometric hypothesis that enough balanced directions are spread across many columns.

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Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No numbers are fitted to data. The theorem parameters α, β, δ, ε are hypotheses; internal proof constants such as λ, θ, and 40 are chosen by hand but do not enter the theorem conclusions. No new physical or probabilistic entities are postulated; the ε-balanced locus, robust image, and depth are proof devices defined from existing objects.

axioms (4)
  • domain assumption Wood's moment-method transfer: if E(#Sur(cok(M(n)),G))→|G|^{−u} for every finite abelian p-group G, then the cokernel distribution approaches the Cohen–Lenstra distribution.
    Invoked in Section 2 to derive Theorems 1.6 and 1.5 from the moment estimates in Theorems 2.1 and 2.2; this is a cited theorem from [18], not re-proved.
  • domain assumption Columns M_1,...,M_{n+u} of M(n) are independent; the moment factorizes as in Eq (2.1).
    Explicit in conditions (⋆) and Theorem 1.5; Eq (2.1) factors the G-moment into a product over columns. The paper notes in Section 1.3 that its proofs rely on this and do not address cross-column dependence.
  • domain assumption In Theorem 1.6, each column satisfies (⋆): for some A_j∈GL_n, A_j M_j has independent entries, and the ε-balanced loci satisfy condition (1.1) with a basis u_i and α+β<1.
    This is the definition of the allowed within-column dependence and the geometric spread of good directions; the proof uses it to count surjections in Section 5.
  • standard math Lemma 3.5 (Wood): for an ε-balanced variable y and nontrivial character χ, |E χ(y)|≤exp(−ε/a^2).
    Imported from [17, Lemma 4.2] and used in Lemma 3.4 to bound Fourier coefficients of the column probabilities.

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Given any $\varepsilon > 0$, let $M(n)$ be a random $n \times (n+u)$ matrix over $\mathbb{Z}_p$, with all entries independent and $\varepsilon$-balanced (lying in each residue class mod $p$ with probability at most $1-\varepsilon$). Wood proved that as $n \to \infty$ the distribution of $\mathrm{cok}(M(n))$ approaches Cohen and Lenstra's conjectured distribution of class groups. Given $\alpha,\beta >0$ such that $\alpha + \beta <1$, we prove that the distribution of $\mathrm{cok}(M(n))$ still approaches the Cohen--Lenstra distribution even if we weaken the hypothesis by allowing up to $\alpha n$ entries per column and up to $\beta n$ entries per row of $M(n)$ to not be $\varepsilon$-balanced. We also weaken the independence condition by allowing certain types of dependence between the entries of each column. In addition, we prove that, for any $\delta > 0$, the cokernels of random band matrices of width $\log(n)^{1+\delta}$ with $\varepsilon$-balanced entries in the band and arbitrary entries outside of it will also approach the Cohen--Lenstra distribution, which answers a question of Kang--Lee--Yu.

Figures

Figures reproduced from arXiv: 2607.06952 by Isaac Rajagopal.

Figure 1
Figure 1. Figure 1: These are three examples of random matrices M(n) valued in Mn×n(Zp) whose cokernels approach the Cohen–Lenstra distribution by Theorem 1.4. Blue regions represent ε-balanced entries and white regions represent entries with unrestricted distributions. For example, the entries in the white regions may all be fixed to zero. 0.7n 0.4n (d) 0 0.5n 0.5n 0.5n 0.5n (e) 0 0 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: These are two examples of random matrices M(n) valued in Mn×n(Zp) whose cokernels do not approach the Cohen–Lenstra distribution. Blue regions represent ε￾balanced entries and white regions represent entries which are all fixed to zero. of width log(n) 1+δ with ε-balanced entries in the band and arbitrary entries outside it, and is a universal version of [Mé24]. Section 1.2 contains Theorem 1.6, which gene… view at source ↗
Figure 3
Figure 3. Figure 3: The cokernels of M(n) in (f) and B(n) in (g) approach the Cohen–Lenstra distribution by Theorem 1.5 and [Mé24], respectively. Blue regions represent ε-balanced entries, gray regions represent Haar-uniform entries, and white regions represent entries with unrestricted distributions in (f) and entries fixed to zero in (g). matrices satisfy the assumptions of Theorem 1.4 except that in (d) we have α = 0.4 and… view at source ↗

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