The rank evolution of block bidiagonal matrices over finite fields

We investigate uniform random block lower bidiagonal matrices over the finite field $\mathbb{F}_q$, and prove that their rank undergoes a phase transition. First, we consider block lower bidiagonal matrices with $(k_n+1)\times k_n$ blocks where each block is of size $n\times n$. We prove that if $k_n\ll q^{n/2}$, then these matrices have full rank with high probability, and if $k_n\gg q^{n/2}$, then the rank has Gaussian fluctuations. Second, we consider block lower bidiagonal matrices with $k_n\times k_n$ blocks where each block is of size $n\times n$. We prove that if $k_n\ll q^{n/2}$, then the rank exhibits the same constant order fluctuations as the rank of the matrix products considered by Nguyen and Van Peski, and if $k_n\gg q^{n/2}$, then the rank has Gaussian fluctuations. Finally, we also consider a truncated version of the first model, where we prove that at $k_n\approx q^{n/2}$, 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.