Universality of the cokernels of random $p$-adic matrices with inhomogeneously balanced columns

Jungin Lee; Sungjin Park

A vanishing exponential sum on column balance parameters forces cokernels of random p-adic matrices to match the Haar-random limiting law.

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2026-06-28 12:47 UTC pith:VIZ4N46S

load-bearing objection This extends Nguyen-Wood to the inhomogeneous balanced-columns case under an explicit exponential-sum decay condition on the alphas.

arxiv 2606.02180 v1 pith:VIZ4N46S submitted 2026-06-01 math.NT math.PR

Universality of the cokernels of random p-adic matrices with inhomogeneously balanced columns

Jungin Lee , Sungjin Park This is my paper

classification math.NT math.PR

keywords cokernelsrandom matricesp-adic integersuniversalitybalanced columnsHaar measureconvergence in distributionfinite abelian p-groups

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that if each column of a random n by (n+u) matrix over Z_p is α_n(i)-balanced and the sum over i of exp(-ε α_n(i) n) tends to zero for every ε>0, then the distribution of the cokernel converges to the same limit obtained from Haar-random matrices. This condition allows the balance levels to vary across columns while still guaranteeing the universality result. The extension matters because it identifies a quantitative threshold on inhomogeneity that preserves the limiting statistics without requiring every column to have the same balance parameter. The proof transfers the known convergence for the uniform case by controlling the deviation introduced by the varying α_n(i).

Core claim

Let u be a fixed nonnegative integer. Let A(n) be a random n by (n+u) matrix over Z_p whose i-th column is α_n(i)-balanced. If the sum from i=1 to n+u of exp(-ε α_n(i) n) tends to zero as n tends to infinity for every ε>0, then the cokernels of A(n) converge in distribution to the same limiting law as the cokernels of Haar-random n by (n+u) matrices over Z_p.

What carries the argument

The α_n(i)-balanced condition on individual columns together with the exponential-sum vanishing condition that averages out the inhomogeneity.

Load-bearing premise

The matrix entries are drawn so that each column exactly satisfies its prescribed α_n(i)-balanced property, and this modeling choice plus the sum condition suffices to equate the distributions.

What would settle it

Construct a sequence α_n(i) for which the exponential sum vanishes yet, for some fixed finite abelian p-group G and some n large, the probability that the cokernel equals G differs from the corresponding Haar probability by more than a fixed positive constant.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

The limiting cokernel distribution depends only on the vanishing of the sum and is otherwise insensitive to the particular choice of the sequence α_n.
The same convergence holds for any fixed excess number of columns u.
Universality statements that previously required uniform balance now apply to a strictly larger family of column distributions.
The result supplies an explicit quantitative criterion that can be checked for concrete random-matrix ensembles arising in p-adic arithmetic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sum condition may be sufficient to prove universality for Smith normal forms or for cokernels over other local rings.
One could numerically sample matrices with chosen α_n sequences satisfying the sum condition and compare empirical cokernel frequencies against the known Haar probabilities for small p.
The technique might extend to models of random p-adic modules with column-dependent generation probabilities that still satisfy an averaged balance requirement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

This extends Nguyen-Wood to the inhomogeneous balanced-columns case under an explicit exponential-sum decay condition on the alphas.

read the letter

The main new piece is the transfer of the cokernel universality limit from Haar-random matrices to the setting where each column i has its own alpha_n(i)-balance level, as long as sum exp(-eps alpha_n(i) n) goes to zero for every eps > 0. That condition is the device that keeps the inhomogeneity from affecting the limiting distribution.

The paper states the hypothesis cleanly and frames the result as a direct extension rather than a wholesale reworking. It re-uses the alpha-balanced notion from the literature without introducing new parameters or circular definitions. The claim itself is precise and matches the abstract exactly.

The soft spots are limited. The condition is shown to be sufficient but its necessity is not addressed, so it is not clear how sharp the threshold is. The full estimates that turn the sum condition into control on the cokernel probabilities are not visible in the provided text, which leaves the derivation steps unexamined. No internal contradictions appear in the statement, and the modeling choice is consistent with prior work.

This is for people already working on random p-adic matrices and cokernel statistics in arithmetic geometry. Readers who know the Nguyen-Wood theorem will see the incremental value right away.

It deserves a serious referee because the extension is targeted, the hypothesis is checkable, and the result sits in an active subfield.

Referee Report

0 major / 0 minor

Summary. The paper proves that for a random n × (n+u) matrix A(n) over Z_p with i-th column α_n(i)-balanced, the condition ∑_{i=1}^{n+u} exp(−ε α_n(i) n) → 0 as n → ∞ for every ε > 0 implies that the cokernels of A(n) converge in distribution to the same limiting law as the cokernels of Haar-random n × (n+u) matrices over Z_p. This is presented as a direct extension of the Nguyen-Wood universality theorem to the inhomogeneous balanced-column model.

Significance. If the result holds, it broadens the scope of cokernel universality results in p-adic random matrix theory by accommodating varying column balances under an explicit, checkable exponential-sum hypothesis. The extension preserves the limiting law without introducing new parameters or altering the target distribution, which is a strength for applications in arithmetic statistics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the main theorem and for recognizing the significance of extending the Nguyen–Wood universality result to the inhomogeneous balanced-column model under the stated exponential-sum hypothesis. No major comments were raised in the report, so we have no specific points to address point-by-point at this time.

Circularity Check

0 steps flagged

No significant circularity; extension of external theorem under explicit hypothesis

full rationale

The derivation extends the Nguyen-Wood universality theorem (external citation, non-overlapping authors) to the inhomogeneous α_n(i)-balanced model. The summability condition ∑ exp(−ϵ α_n(i) n) → 0 is an explicit hypothesis controlling deviation from homogeneity, not derived from or fitted to the target cokernel law. The limiting distribution is imported from the independent Haar-random case rather than defined in terms of the paper's own parameters or prior self-citations. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the claimed chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the prior universality result of Nguyen and Wood for the homogeneous case and on the standard definition of an α-balanced column; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption The universality theorem of Nguyen and Wood holds for Haar-random matrices over Z_p.
The current work extends this theorem to the inhomogeneous case.

pith-pipeline@v0.9.1-grok · 5707 in / 1406 out tokens · 27175 ms · 2026-06-28T12:47:32.833654+00:00 · methodology

0 comments

read the original abstract

In this paper, we prove universality of the distribution of the cokernels of a random $p$-adic matrix with inhomogeneously balanced columns. More precisely, let $u \ge 0$ be an integer and $A(n)$ be a random $n \times (n+u)$ matrix over $\mathbb{Z}_p$ whose $i$-th column is $\alpha_n(i)$-balanced. We prove that if $\sum_{i=1}^{n+u} \exp(-\epsilon \alpha_n(i)n) \to 0$ as $n \to \infty$ for every $\epsilon>0$, then the cokernels of $A(n)$ converge in distribution, as $n \to \infty$, to the same limiting law as the cokernels of Haar-random $n \times (n+u)$ matrices over $\mathbb{Z}_p$. This extends a universality theorem of Nguyen and Wood to random $p$-adic matrices with inhomogeneously balanced columns.

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Cohen and H

H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer, Berlin, 1984, 33–62

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[2]

Friedman and L

E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over a finite field, in Th´ eorie des Nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, 227–239

1987
[3]

J. Jung, J. Lee and M. Yu, Sharp threshold for universality of cokernels of classical random matrix models over thep-adic integers, arXiv:2603.12879

work page arXiv
[4]

D. Y. Kang, J. Lee and M. Yu, Randomp-adic matrices with fixed zero entries and the Cohen–Lenstra heuristics, arXiv:2409.01226, to appear in Selecta Math. (N.S.)

work page internal anchor Pith review arXiv
[5]

Lee, Universality of the cokernels of randomp-adic Hermitian matrices, Trans

J. Lee, Universality of the cokernels of randomp-adic Hermitian matrices, Trans. Amer. Math. Soc. 376 (2023), no. 12, 8699–8732

2023
[6]

Sharp threshold for universality of cokernels of random matrices over finite fields.arXiv preprint arXiv:2511.13070, 2025

J. Lee, Sharp threshold for universality of cokernels of random matrices over finite fields, arXiv:2511.13070

work page arXiv
[7]

H. H. Nguyen and M. M. Wood, Random integral matrices: universality of surjectivity and the cokernel, Invent. Math. 228 (2022), 1–76

2022
[8]

H. H. Nguyen and M. M. Wood, Local and global universality of random matrix cokernels, Math. Ann. 391 (2025), no. 4, 5117–5210

2025
[9]

M. M. Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc. 30 (2017), no. 4, 915–958

2017
[10]

M. M. Wood, Random integral matrices and the Cohen–Lenstra heuristics, Amer. J. Math. 141 (2019), no. 2, 383–398

2019
[11]

M. M. Wood, Probability theory for random groups arising in number theory, Proc. Int. Cong. Math. 2022, Vol. 6, 4476– 4508. J. Lee – Department of Mathematics, Ajou University, Suwon 16499, Republic of Korea S. Park – Department of Mathematics, Ajou University, Suwon 16499, Republic of Korea Email address:jileemath@ajou.ac.kr, qkrtjd5421@ajou.ac.kr

2022

[1] [1]

Cohen and H

H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer, Berlin, 1984, 33–62

1983

[2] [2]

Friedman and L

E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over a finite field, in Th´ eorie des Nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, 227–239

1987

[3] [3]

J. Jung, J. Lee and M. Yu, Sharp threshold for universality of cokernels of classical random matrix models over thep-adic integers, arXiv:2603.12879

work page arXiv

[4] [4]

D. Y. Kang, J. Lee and M. Yu, Randomp-adic matrices with fixed zero entries and the Cohen–Lenstra heuristics, arXiv:2409.01226, to appear in Selecta Math. (N.S.)

work page internal anchor Pith review arXiv

[5] [5]

Lee, Universality of the cokernels of randomp-adic Hermitian matrices, Trans

J. Lee, Universality of the cokernels of randomp-adic Hermitian matrices, Trans. Amer. Math. Soc. 376 (2023), no. 12, 8699–8732

2023

[6] [6]

Sharp threshold for universality of cokernels of random matrices over finite fields.arXiv preprint arXiv:2511.13070, 2025

J. Lee, Sharp threshold for universality of cokernels of random matrices over finite fields, arXiv:2511.13070

work page arXiv

[7] [7]

H. H. Nguyen and M. M. Wood, Random integral matrices: universality of surjectivity and the cokernel, Invent. Math. 228 (2022), 1–76

2022

[8] [8]

H. H. Nguyen and M. M. Wood, Local and global universality of random matrix cokernels, Math. Ann. 391 (2025), no. 4, 5117–5210

2025

[9] [9]

M. M. Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc. 30 (2017), no. 4, 915–958

2017

[10] [10]

M. M. Wood, Random integral matrices and the Cohen–Lenstra heuristics, Amer. J. Math. 141 (2019), no. 2, 383–398

2019

[11] [11]

M. M. Wood, Probability theory for random groups arising in number theory, Proc. Int. Cong. Math. 2022, Vol. 6, 4476– 4508. J. Lee – Department of Mathematics, Ajou University, Suwon 16499, Republic of Korea S. Park – Department of Mathematics, Ajou University, Suwon 16499, Republic of Korea Email address:jileemath@ajou.ac.kr, qkrtjd5421@ajou.ac.kr

2022