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

In this paper, we study the combinatorial relations between the cokernels cok ⁡ ( A n + p ⁢ x i ⁢ I n ) {\operatorname{cok}(A_{n}+px_{i}I_{n})} ( 1 ≤ i ≤ m {1\leq i\leq m} ), where A n {A_{n}} is an n × n {n\times n} matrix over the ring of p-adic integers ℤ p {\mathbb{Z}_{p}} , I n {I_{n}} is the n × n {n\times n} identity matrix and x 1 , … , x m {x_{1},\dots,x_{m}} are elements of ℤ p {\mathbb{Z}_{p}} whose reductions modulo p are distinct. For a positive integer m ≤ 4 {m\leq 4} and given x 1 , … , x m ∈ ℤ p {x_{1},\dots,x_{m}\in\mathbb{Z}_{p}} , we determine the set of m-tuples of finitely generated ℤ p {\mathbb{Z}_{p}} -modules ( H 1 , … , H m ) {(H_{1},\dots,H_{m})} for which ( cok ⁡ ( A n + p ⁢ x 1 ⁢ I n ) , … , cok ⁡ ( A n + p ⁢ x m ⁢ I n ) ) = ( H 1 , … , H m ) (\operatorname{cok}(A_{n}+px_{1}I_{n}),\dots,\operatorname{cok}(A_{n}+px_{m}I_% {n}))=(H_{1},\dots,H_{m}) for some matrix A n {A_{n}} . We also prove that if A n {A_{n}} is an n × n {n\times n} Haar random matrix over ℤ p {\mathbb{Z}_{p}} for each positive integer n, then the joint distribution of cok ⁡ ( A n + p ⁢ x i ⁢ I n ) {\operatorname{cok}(A_{n}+px_{i}I_{n})} ( 1 ≤ i ≤ m {1\leq i\leq m} ) converges as n → ∞ {n\rightarrow\infty} .