{
  "id": "2201.09572",
  "version": "v1",
  "published": "2022-01-24T10:30:29.000Z",
  "updated": "2022-01-24T10:30:29.000Z",
  "title": "Joint distribution of the cokernels of random $p$-adic matrices",
  "authors": [
    "Jungin Lee"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "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 $\\mathbb{Z}_p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-24T10:30:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "joint distribution",
      "adic matrices",
      "random matrix model",
      "sufficient condition",
      "haar measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}