2024/07/28 by Roger Van Peski, Van Peski, Roger
Computer Science · Engineering · Mathematics · #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2407.19578
openalex publication_date 2024/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider uniformly random strictly upper-triangular matrices in Matn(\mathbbFq). 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.