Joint distribution of the cokernels of random $p$-adic matrices II

In this paper, we study the combinatorial relations between the cokernels $\text{cok}(A_n+p{x}_i{I}_n)$ ($1\le i\le m$) where $A_n$ is an $n\times n$ matrix over the ring of $p$-adic integers ${\mathbb{Z}}_p$, $I_n$ is the $n\times n$ identity matrix and $x_1,\cdots ,x_m$ are elements of ${\mathbb{Z}}_p$ whose reductions modulo $p$ are distinct. For a positive integer $m\le 4$ and given $x_1,\cdots ,x_m\in {\mathbb{Z}}_p$, we determine the set of $m$-tuples of finitely generated ${\mathbb{Z}}_p$-modules $(H_1,\cdots ,H_m)$ for which $(\text{cok}(A_n+p{x}_1{I}_n),\cdots ,\text{cok}(A_n+p{x}_m{I}_n))=(H_1,\cdots ,H_m)$ for some matrix $A_n$. We also prove that if $A_n$ is an $n\times n$ Haar random matrix over ${\mathbb{Z}}_p$ for each positive integer $n$, then the joint distribution of $\text{cok}(A_n+p{x}_i{I}_n)$ ($1\le i\le m$) converges as $n\to \mathrm{\infty }$.