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

Abstract

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 \[\lim_{n\to\infty} \left(w_n-\log_p(n)\right)=+\infty.\]