The rank evolution of block bidiagonal matrices over finite fields

Andr\'as M\'esz\'aros

REVIEW 2 major objections 5 minor 58 references

The rank evolution of block bidiagonal matrices over finite fields

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Pith's one-line read For random block bidiagonal matrices over F_q, the corank undergoes a phase transition at k approximately q^{n/2}: near-zero, Cohen-Lenstra type, then Gaussian.

desk verdict A genuinely new phase transition for block bidiagonal rank over finite fields, rigorously proved in the even case; the odd-n results are stated without proof and need real checking before publication. read the letter →

arxiv 2504.12275 v1 pith:KEGG5PWO submitted 2025-04-16 math.PR

András Mészáros This is my paper · ORCID

classification math.PR

keywords rankmatricesblockbidiagonalprovethentimesbehavior

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Consider a matrix built from square blocks over a finite field, where the only nonzero blocks sit on the main diagonal and on the diagonal just below it. The paper asks how many vectors are sent to zero, that is, the size of the kernel. The main result is a sharp transition when the number of columns of blocks, k, is about q^{n/2}. When k is much smaller than q^{n/2}, the matrix has full rank with probability close to 1. When k is comparable to q^{n/2}, the kernel size converges to a new distribution built from a birth-death random walk. When k is much larger, the kernel size, after centering and scaling, becomes Gaussian.

The method is unusual. The increase in rank from one step to the next forms a Markov chain whose transition probabilities can be written explicitly. After rescaling, this chain converges to a continuous-time random walk on the integers with rates q^{-a} and q^a. The paper couples the two processes so that their embedded jumps agree for an exponentially long time. This coupling turns rank questions into questions about a random walk: how many down jumps it makes and where it sits at time t.

A truncated version of the model shows the same transition at the level of the Cohen-Lenstra distribution, the standard universal limit for cokernels of random matrices. The paper also proves that kernel vectors are spread out before the transition and can be supported on small pieces after it. This is a finite-field analog of the localization-delocalization transition known for random band matrices.

Extended reading notes

Core claim

Theorem 5 states the core phase transition: if t_n = q^{-n/2}/(q-1) k_n, then as n goes to infinity, (1) if t_n to 0, dim ker C^{(n)}_{2k_n} converges to 0 in probability; (2) if t_n to t in (0, infinity), dim ker C^{(n)}_{2k_n} converges to D_{-infinity,t} in total variation; (3) if t_n to infinity, (dim ker C^{(n)}_{2k_n} - mu_n t_n)/(sigma sqrt(t_n)) converges in distribution to a standard normal variable, with mu_n = 2 sum_{i>=1} q^{-i(i-1)} / (1 + 2 sum_{i>=1} q^{-i^2}) + O(q^{-n/2}).

Load-bearing premise

The coupling theorem (Theorem 18) only applies to initial states within 2 alpha n of the center m. For C_{2k}, the true initial state is X_0 = 0, i.e., Y_0 = -m, far outside this window. The proof bridges the gap with Lemma 33: starting from 0, the chain reaches the window m - 2 alpha n within time q^{m - alpha n} with high probability, and during those first steps the rank drops satisfy X_{2i-1} + X_{2i} = n. If Lemma 33 failed, the limiting object would not be the excursion-based variable D_{-infinity,t}.

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Desk Editor's Note

read the letter

This is a good paper and deserves a serious referee. What is actually new: a sharp phase transition in the rank of uniform random block lower bidiagonal matrices over F_q at k ~ q^{n/2}, with exact critical distributions (D_{-\infty,t}, L_t) and a localization/delocalization transition. The method—tracking rank increments as a Markov chain and strong-coupling that chain to a continuous-time random walk—is elegant and technically substantial.

The even-n core is proved in detail and reads honestly. The transition probabilities in Lemma 1, the coupling in Theorem 18, the bridge from X_0=0 via Lemma 33, the critical limit in Lemma 37, and the excursion-based CLT in Section 6 are all carefully done. Constants are derived, not fitted. The citation pattern is fine: self-citations supply background, not load-bearing steps.

The soft spot is the odd-n case. Section 1.3 states Theorem 14 and the odd-n analogue of Theorem 6, then says the proofs are 'almost the same' and only provides even-n proofs. That is an explicit gap, and it is not purely cosmetic: the odd-n limiting walk lives on Z+1/2, and the asymptotic for µ_n gains a sqrt(q) factor relative to the even-n formula. One has to re-run the coupling, the bridging lemma, and the excursion decomposition with half-integer centering, and check that the same σ_odd and the claimed µ_n emerge. The paper does not do this. The results may well be true, but as written they are unproved claims in the central part of the paper. A referee should ask for these proofs, or for the odd-n theorems to be clearly marked as conjectural or postponed.

There is also a minor quantifier issue in Theorem 10(2): 'for each L ≥ ε' should presumably mean integer L with L ≥ ⌈ε⌉. That is small.

The reader's conditional verdict is about right. I would send this to peer review: the even-n theorem alone is a solid contribution, and the odd-n gap is fixable or excisable. I would bring it to a reading group if the focus is random matrices over finite fields or p-adic cokernels.

Referee Report

2 major / 5 minor

Summary. The paper studies uniform random block lower bidiagonal matrices over the finite field F_q, tracking the rank increments X_i = rank(C_i) - rank(C_{i-1}). It proves that X_i is an explicit reversible Markov chain and uses a strong coupling of its rescaled even-indexed subsequence to a continuous-time random walk Z_t to establish a phase transition in the corank at block count k_n ≈ q^{n/2}. Below the critical scale the corank is either zero or follows Cohen–Lenstra-type fluctuations; at the critical scale it converges to explicitly described random variables built from Z_t; above the critical scale it has Gaussian fluctuations with derived constants μ_n and σ. The paper also proves a localization/delocalization transition for kernel vectors of a truncated version of the matrices. The even-n case is proved in detail; the odd-n analogues are stated in Section 1.3 with the comment that their proofs are almost the same and are omitted.

Significance. If the results are fully established, this is a strong contribution to the theory of random matrices over finite fields. The even-n core is genuinely substantive: the transition matrix in Lemma 1 is explicit and parameter-free, the coupling in Theorem 18 is strong (it controls embedded chains, jump times, and an additional structural event (d)), and the Gaussian limits in Section 6 are obtained from a regenerative-cycle CLT rather than by fitting constants. The paper also connects Cohen–Lenstra universality, heavy-tailed critical corank distributions, and kernel localization, which is of independent interest. The main caveat is that several central statements, namely the odd-n analogues, are not proved and in one case not even stated precisely. The even-n arguments appear sound, so the missing odd-n treatment is likely repairable, but it is currently a real completeness gap.

major comments (2)

[Section 1.3] The odd-n results (Theorem 11, Lemma 13, Theorem 14, and the asserted odd-n analogue of Theorem 6) are stated but their proofs are omitted. The sentence 'As the proofs in the case of odd n are almost the same as in the case of even n, we only provide proofs for the even case' is not a sufficient justification, because the odd-n case is not a literal repetition: the limiting process lives on Z + 1/2, the stationary measure is not symmetric about n/2, and the asymptotic value of μ_n given in Theorem 14(3) differs from the even-n value in (5). In particular, the Gaussian-limit proof in Section 6 is written for the even-indexed chain X_{2i} with center m = n/2 and uses the excursion decomposition and equation (30); for odd n the center is m + 1/2 and this decomposition has to be reworked. These are load-bearing claims of the paper, not corollaries that can be checked by inspection. Please provide the full odd-n proofs, or at minimum a rigorous reduction to the even-n arguments that addresses the parity-specific stationary and excursion issues. In addition, the announced odd-n analogue of Theorem 6 is never stated explicitly; the reader is only told that a similar analogue exists.
[Lemma 13] Lemma 13 (the existence of the total-variation limit (D^odd_{-∞,t}, Z^odd_{-∞,t}) as the initial state tends to -∞) is stated without proof. The proof of the even-n Lemma 4 relies on coupling independent copies of Z and on the jump-count estimates of Section 4; those ingredients are not transferred to Z^odd. Because Lemma 13 is used in Theorem 14(2) and in the odd-n analogue of Theorem 6, its proof cannot be deferred as 'almost the same.'

minor comments (5)

[Theorem 10(2)] The quantifier 'for each L ≥ ε' should read 'for each positive integer L ≥ ε (with L ≤ k_n+1 asymptotically)', since an L-localized kernel is defined only for positive integers L.
[Lemma 33, proof] The displayed summation has inconsistent upper limits: the first sum runs to m−αn while the following equality uses m−2αn. Please correct the indexing so that the displayed estimate is unambiguous.
[Section 7.1] Section 7.1 is headed 'The proof of part (1) of Theorem 5' but sits inside the localization section; renumber or cross-reference to avoid confusion.
[Lemma 27] The text says 'Cauchy-Schwartz'; the standard spelling is 'Cauchy–Schwarz'.
[Theorems 5(3) and 8(3)] These theorems assert the existence of a constant σ>0 without displaying its value. Consider explicitly defining σ^2 = Var(V∞)/EU∞, as is done in Section 6.1, so that the statement is more informative.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central argument, especially Lemma 1 and Theorem 18, is self-contained. No phenomenological constants are fitted to data; alpha = 0.08 is a technical exponent and the derived mu_n and sigma are not free parameters. The odd-n theorem and Theorem 7 rely on cited earlier results, which are treated as external assumptions here.

assumptions (4)

standard math Rank distribution of a uniform random (n-d) x n matrix over F_q is q^{-(n-d-r)(n-r)} (q^{-(n-d)};q)_r (q^{-n};q)_r / (q^{-r};q)_r.
Used in Lemma 1; proof is elementary Gaussian binomial counting in Lemma 15.
standard math Identity (56) from [36]: dim ker C'_{2k-1} = dim ker A'_1 A'_2 ... A'_k for bidiagonal matrices with identity off-diagonal blocks.
Used in Section 8.1 to connect block bidiagonal rank to matrix products; imported from the author's earlier paper.
domain assumption Nguyen-Van Peski local limit theorems for cokernels of random matrix products (Theorems from [39] and [51]).
Imported for Theorem 7 and the constant-order regime of Theorem 6(1); not reproved here.
domain assumption Wood's Cohen-Lenstra universality theorem for kernels of random matrices over F_p (Theorem from [54]).
Used in Section 1.1 to motivate J_0 and the Cohen-Lenstra distribution; not central to the new proofs.

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Pith. "Pith review of The rank evolution of block bidiagonal matrices over finite fields." pith.science (2026). https://pith.science/paper/KEGG5PWO

@misc{pith2026250412275,
  author       = {Pith},
  title        = {Pith review of: The rank evolution of block bidiagonal matrices over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KEGG5PWO}},
  note         = {Machine review of arXiv:2504.12275}
}

abstract

We investigate uniform random block lower bidiagonal matrices over the finite field $\mathbb{F}_q$, and prove that their rank undergoes a phase transition. First, we consider block lower bidiagonal matrices with $(k_n+1)\times k_n$ blocks where each block is of size $n\times n$. We prove that if $k_n\ll q^{n/2}$, then these matrices have full rank with high probability, and if $k_n\gg q^{n/2}$, then the rank has Gaussian fluctuations. Second, we consider block lower bidiagonal matrices with $k_n\times k_n$ blocks where each block is of size $n\times n$. We prove that if $k_n\ll q^{n/2}$, then the rank exhibits the same constant order fluctuations as the rank of the matrix products considered by Nguyen and Van Peski, and if $k_n\gg q^{n/2}$, then the rank has Gaussian fluctuations. Finally, we also consider a truncated version of the first model, where we prove that at $k_n\approx q^{n/2}$, we have a phase transition between a Cohen-Lenstra and a Gaussian limiting behavior of the rank. We also show that there is a localization/delocalization phase transition for the vectors in the kernels of these matrices at the same critical point. In all three cases, we also provide a precise description of the behavior of the rank at criticality. These results are proved by analyzing the limiting behavior of a Markov chain obtained from the increments of the ranks of these matrices.

Figures

Figures reproduced from arXiv: 2504.12275 by the authors.

**Figure 1.** A schematic image of the block matrix Cb4. All the cells correspond to (n/2) × (n/2) blocks. The entries of white cells are set to be 0. The entries of the gray cells are chosen as i.i.d. uniform elements of Fq. Let Cb2k be a truncated version of C2(k+1), which obtained from C2(k+1) by deleting the first n/2 rows and the last n/2 rows. Note that Cb2k is a (k + 1)n × (k + 1)n matrix. For u ≥ 0, let Ju be a Z≥0-valued… view at source ↗

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Theodoros Assiotis. Inﬁnite p-adic random matrices and ergodic decomposition of p-adic Hua measures. Transactions of the American Mathematical Society , 375(3):1745–1766, 2022

work page 2022

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