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 $$\underset{n\to \mathrm{\infty }}{\lim }(w_n-{\log }_p(n))=+\mathrm{\infty }\mathrm{.}$$