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The rank evolution of block bidiagonal matrices over finite fields

András Mészáros

TL;DR

The paper analyzes the rank evolution of uniform random block lower bidiagonal matrices over the finite field $\mathbb{F}_q$, revealing a phase transition in the corank near the critical index $k_n\approx q^{n/2}$. By modeling rank increments as a Markov chain with explicit transition probabilities and coupling this discrete process to a limiting continuous-time random walk $Z_t$, the authors obtain precise limiting distributions for the kernel dimension in various regimes, including Cohen-Lenstra, Gaussian, and heavy-tail cases. Key contributions include a detailed Markovian structure for rank increments, a strong coupling framework with convergence in total variation, and a unified treatment of even/odd dimension cases as well as truncated models that exhibit Cohen-Lenstra/non-Cohen-Lenstra transitions and localization/delocalization phenomena. The results illuminate deep connections between random matrix cokernels, matrix products, and $p$-adic band-matrix analogues, providing exact critical behavior and Gaussian fluctuations in broad settings with potential applications to number theory and random matrix theory.

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.

The rank evolution of block bidiagonal matrices over finite fields

TL;DR

The paper analyzes the rank evolution of uniform random block lower bidiagonal matrices over the finite field , revealing a phase transition in the corank near the critical index . By modeling rank increments as a Markov chain with explicit transition probabilities and coupling this discrete process to a limiting continuous-time random walk , the authors obtain precise limiting distributions for the kernel dimension in various regimes, including Cohen-Lenstra, Gaussian, and heavy-tail cases. Key contributions include a detailed Markovian structure for rank increments, a strong coupling framework with convergence in total variation, and a unified treatment of even/odd dimension cases as well as truncated models that exhibit Cohen-Lenstra/non-Cohen-Lenstra transitions and localization/delocalization phenomena. The results illuminate deep connections between random matrix cokernels, matrix products, and -adic band-matrix analogues, providing exact critical behavior and Gaussian fluctuations in broad settings with potential applications to number theory and random matrix theory.

Abstract

We investigate uniform random block lower bidiagonal matrices over the finite field , and prove that their rank undergoes a phase transition. First, we consider block lower bidiagonal matrices with blocks where each block is of size . We prove that if , then these matrices have full rank with high probability, and if , then the rank has Gaussian fluctuations. Second, we consider block lower bidiagonal matrices with blocks where each block is of size . We prove that if , then the rank exhibits the same constant order fluctuations as the rank of the matrix products considered by Nguyen and Van Peski, and if , then the rank has Gaussian fluctuations. Finally, we also consider a truncated version of the first model, where we prove that at , 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.

Paper Structure

This paper contains 23 sections, 49 theorems, 269 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A motivation for Theorem \ref{['theoremtruncated']}: The breakdown of the Cohen-Lenstra universality and the critical behavior of Haar-uniform band matrices over the $p$-adic integers
  3. A localization/delocalization phase transition for the kernel of $\widehat{C}_{2k_n}$
  4. The case of odd $n$
  5. The Markovian property of the rank increments -- The proof of Lemma \ref{['lemmaMarkov']}
  6. Constructing a coupling of $Z_t$ and $Y_t$ -- The proof of Theorem \ref{['theoremcoupling']}
  7. Bounding the number of jumps of $Z_t$
  8. The limit of $(D_{a,t},Z_{a,t})$ as $a\to -\infty$ -- The proof of Lemma \ref{['Dinftyexists']}
  9. The proof of Corollary \ref{['marginalconv']}
  10. The corank at criticality
  11. The proof of part \ref{['thmdimkerC2kpart2']} of Theorem \ref{['thmdimkerC2k']}
  12. The proof of parts \ref{['theoremtruncatedpart1']} and \ref{['theoremtruncatedpart2']} of Theorem \ref{['theoremtruncated']}
  13. Gaussian limits
  14. The proof of part \ref{['theoremtruncatedpart3']} of Theorem \ref{['theoremtruncated']}
  15. The asymptotic value of $\mu_n$ -- The proof of \ref{['munas']}
  16. ...and 8 more sections

Key Result

Lemma 1

The sequence $(X_i)$ is a time homogeneous irreducible aperiodic reversible Markov chain on the state space $\{0,1,\dots,n\}$ with transition matrix where we used the $q$-Pochhammer symbol $(a;q)_r=\prod_{i=0}^{r-1} (1-aq^i)$. Moreover, the stationary measure $\pi_n$ of this Markov chain is given by

Figures (1)

  • Figure 1: A schematic image of the block matrix $\widehat{C}_4$. All the cells correspond to $(n/2)\times (n/2)$ 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 $\mathbb{F}_q$.

Theorems & Definitions (81)

  • Lemma 1
  • Theorem 2
  • Corollary 3
  • Lemma 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Conjecture 9
  • Theorem 10
  • ...and 71 more