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

TL;DR

This paper analyzes the rank distribution of random strictly upper-triangular matrices over finite fields, revealing a logarithmic relationship and explicit fluctuation behavior, with connections to nilpotent Jordan block structures and point configurations.

Contribution

It provides the asymptotic distribution of the rank and Jordan block sizes of random triangular matrices over finite fields, including explicit fluctuation results and new integral formulas.

Findings

The rank deficiency scales as log_q n with finite fluctuations.

Fluctuations of Jordan block sizes converge to a discrete point process.

Explicit integral formulas enable asymptotic analysis of matrix properties.

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 $\mathcal{L}_{t,\chi}$ 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.