For random block bidiagonal matrices over F_q, the corank undergoes a phase transition at k approximately q^{n/2}: near-zero, Cohen-Lenstra type, then Gaussian.
Random band ma trices in the delocalized phase I: Quantum unique ergodicity and universality
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The rank evolution of block bidiagonal matrices over finite fields
For random block bidiagonal matrices over F_q, the corank undergoes a phase transition at k approximately q^{n/2}: near-zero, Cohen-Lenstra type, then Gaussian.