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

Abstract

In this paper, we study the joint distribution of the cokernels of random $p$-adic matrices. Let $p$ be a prime and $P_1(t), \cdots, P_l(t) \in \mathbb{Z}_p[t]$ be monic polynomials whose reductions modulo $p$ in $\mathbb{F}_p[t]$ are distinct and irreducible. We determine the limit of the joint distribution of the cokernels $\text{cok} (P_1(A)), \cdots, \text{cok}(P_l(A))$ for a random $n \times n$ matrix $A$ over $\mathbb{Z}_p$ with respect to Haar measure as $n \rightarrow \infty$. 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 $B_n$ is a fixed $n \times n$ matrix over $\mathbb{Z}_p$ for each $n$ and $A$ is a random $n \times n$ matrix over $\mathbb{Z}_p$.