A phase transition for the cokernels of random band matrices over the p-adic integers

Let $\mathbf{B}_n$ be an $n\times n$ Haar-uniform band matrix over $\mathbb{Z}_p$ with band width $w_n$. We prove that $\text{cok}(\mathbf{B}_n)$ has Cohen-Lenstra limiting distribution if and only if $$\lim_{n\to\infty} \left(w_n-\log_p(n)\right)=+\infty.$$