2025/04/16 by András Mészáros, Mészáros, András · 2 citations
Computer Science · Mathematics · #15B52 #60B20 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2504.12275
openalex publication_date 2025/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate uniform random block lower bidiagonal matrices over the finite field \mathbbFq, and prove that their rank undergoes a phase transition. First, we consider block lower bidiagonal matrices with (kn+1)× kn blocks where each block is of size n× n. We prove that if kn≪ qn/2, then these matrices have full rank with high probability, and if kn≫ qn/2, then the rank has Gaussian fluctuations. Second, we consider block lower bidiagonal matrices with kn× kn blocks where each block is of size n× n. We prove that if kn≪ qn/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 kn≫ qn/2, then the rank has Gaussian fluctuations. Finally, we also consider a truncated version of the first model, where we prove that at kn≈ qn/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.