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The rank of a random triangular matrix over $\mathbb{F}_q$

Roger Van Peski

TL;DR

This work analyzes the rank fluctuations of uniformly random strictly upper-triangular matrices over a finite field, linking the corank profile of powers $A_n^i$ to a discrete limiting object $\mathcal{L}_{t,\chi}$ that arises from Hall-Littlewood processes, a specialization of Macdonald processes. By de-Poissonizing a finite-$N$ integral formula from Hall-Littlewood theory, the authors obtain explicit prelimit probabilities and, through careful residue analysis, prove that the joint fluctuations of the first $k$ column lengths converge to the corresponding discrete limit law. The results show that $n-\operatorname{rank}(A_n)$ grows as $\log_q n$ with bounded fluctuations, and they provide a concrete framework for understanding the full joint column-fluctuation process via contour-integral representations. The work connects finite-field random matrices to universal limit objects seen in $p$-adic matrix-product limits, suggesting deep links between unipotent matrix ensembles, Hall-Littlewood processes, and nonarchimedean algebra.

Abstract

We consider uniformly random strictly upper-triangular matrices in $\operatorname{Mat}_n(\mathbb{F}_q)$. For such a matrix $A_n$, we show that $n-\operatorname{rank}(A_n) \approx \log_q n$ as $n \to \infty$, and find that the fluctuations around this limit are finite-order and given by explicit $\mathbb{Z}$-valued random variables. More generally, we consider the random partition whose parts are the sizes of the nilpotent Jordan blocks of $A_n$: its $k$ largest parts (rows) were previously shown by Borodin to have jointly Gaussian fluctuations as $N \to \infty$, and its columns correspond to differences $\operatorname{rank}(A_n^{i-1}) - \operatorname{rank}(A_n^i)$. We show the fluctuations of the columns converge jointly to a discrete random point configuration $\mathcal{L}_{t,χ}$ introduced in arXiv:2310.12275. The proofs use an explicit integral formula for the probabilities at finite $N$, obtained by de-Poissonizing a corresponding one in arXiv:2310.12275, which is amenable to asymptotic analysis.

The rank of a random triangular matrix over $\mathbb{F}_q$

TL;DR

This work analyzes the rank fluctuations of uniformly random strictly upper-triangular matrices over a finite field, linking the corank profile of powers to a discrete limiting object that arises from Hall-Littlewood processes, a specialization of Macdonald processes. By de-Poissonizing a finite- integral formula from Hall-Littlewood theory, the authors obtain explicit prelimit probabilities and, through careful residue analysis, prove that the joint fluctuations of the first column lengths converge to the corresponding discrete limit law. The results show that grows as with bounded fluctuations, and they provide a concrete framework for understanding the full joint column-fluctuation process via contour-integral representations. The work connects finite-field random matrices to universal limit objects seen in -adic matrix-product limits, suggesting deep links between unipotent matrix ensembles, Hall-Littlewood processes, and nonarchimedean algebra.

Abstract

We consider uniformly random strictly upper-triangular matrices in . For such a matrix , we show that as , and find that the fluctuations around this limit are finite-order and given by explicit -valued random variables. More generally, we consider the random partition whose parts are the sizes of the nilpotent Jordan blocks of : its largest parts (rows) were previously shown by Borodin to have jointly Gaussian fluctuations as , and its columns correspond to differences . We show the fluctuations of the columns converge jointly to a discrete random point configuration introduced in arXiv:2310.12275. The proofs use an explicit integral formula for the probabilities at finite , obtained by de-Poissonizing a corresponding one in arXiv:2310.12275, which is amenable to asymptotic analysis.
Paper Structure (9 sections, 20 theorems, 141 equations, 5 figures)

This paper contains 9 sections, 20 theorems, 141 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Symmetric function background
  3. Symmetric Laurent polynomials.
  4. Symmetric functions.
  5. Macdonald processes.
  6. Explicit formulas for the limit and prelimit
  7. The limiting random variable.
  8. Exact formula for the prelimit probability.
  9. Asymptotic analysis

Key Result

Theorem 1.2

Fix a prime power $q$ and for each $n \geq 1$, let $A_n$ be a uniformly random element of the finite set Then for every $k \in \mathbb{Z}_{\geq 1}$, where $\mathcal{L}^{(i)}_{q^{-1},\chi}$ is as defined in def:cL_series, and $D_\infty$ is the metric of def:metric on the set $\mathop{\mathrm{Sig}}\nolimits_k$.

Figures (5)

  • Figure 1: The Jordan block sizes $J(A) = (108, 45, 23, 12, 7, 3, 1, 1)$ of a uniformly random $A \in \mathfrak{g}(200,2)$, generated on Sage.
  • Figure 2: The Young diagram of $\lambda = (5,2,2,1)$ (left), and that of its conjugate partition $\lambda' = (4,3,1,1,1)$ obtained by flipping the diagram across the diagonal.
  • Figure 3: The contour $\mathop{\mathrm{\tilde{\Gamma}}}\nolimits$ in $\mathbb{C}$.
  • Figure 4: The contour $\Gamma(r,n)$ decomposed as in \ref{['eq:contours']}, with $\Gamma_1(n)$ in blue and $\Gamma_2(r,n)$ in red, and the poles of the integrand at $w_i = -t^\mathbb{Z}$ shown.
  • Figure 5: The contour $\mathop{\mathrm{\tilde{\Gamma}}}\nolimits$ decomposed as in \ref{['eq:tG_decomp']}, with $\Gamma_1(n)$ in blue and $\Gamma_3(n)$ in green.

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1: mac
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6: mac
  • Lemma 2.7
  • ...and 41 more