Joint distribution of the cokernels of random \(p\)-adic matrices II

• Published in: arXiv.org
• Main Authors: Jung, Jiwan, Lee, Jungin
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 14.01.2024
• Subjects: Combinatorial analysis; Integers
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2304.03583

In this paper, we study the combinatorial relations between the cokernels \(\text{cok}(A_n+px_iI_n)\) (\(1 \le i \le m\)) 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 \((\text{cok}(A_n+px_1I_n), \cdots, \text{cok}(A_n+px_mI_n)) = (H_1, \cdots, H_m)\) 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 \(\text{cok}(A_n+px_iI_n)\) (\(1 \le i \le m\)) converges as \(n \rightarrow \infty\).