The rank of a random triangular matrix over Fq

We consider uniformly random strictly upper-triangular matrices in Matₙ(Fq). For such a matrix Aₙ, we show that n-rank(Aₙ) ≈ logq n as n → ∞, and find that the fluctuations around this limit are finite-order and given by explicit Z-valued random variables. More generally, we consider the random partition whose parts are the sizes of the nilpotent Jordan blocks of Aₙ: its k largest parts (rows) were previously shown by Borodin to have jointly Gaussian fluctuations as N → ∞, and its columns correspond to differences rank(Aₙⁱ⁻¹) - rank(Aₙⁱ). We show the fluctuations of the columns converge jointly to a discrete random point configuration Lt,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.