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

In this paper, we study the joint distribution of the cokernels of random $p$-adic matrices. Let $p$ be a prime and $P_1(t), \cdots, P_l(t) \in $\mathbb{Z}$_p[t]$bemonicpolynomialswhosereductionsmodulo$p$in$$\mathbb{F}$_p[t]$ are distinct and irreducible. We determine the limit of the joint distribution of the cokernels $$\text{cok}$ (P_1(A)), \cdots, $\text{cok}$(P_l(A))$forarandom$n \times n$matrix$A$over$$\mathbb{Z}$_p$withrespecttoHaarmeasureas$n \rightarrow \infty$. By applying the linearization of a random matrix model, we also provide a conjecture which generalizes this result. Finally, we provide a sufficient condition that the cokernels $\text{cok}(A)$ and $\text{cok}(A+B_n)$ become independent as $n \rightarrow \infty$,where$B_n$isafixed$n \times n$matrixover$$\mathbb{Z}$_p$foreach$n$and$A$isarandom$n \times n$matrixover$$\mathbb{Z}$_p$.