{
  "id": "2307.04741",
  "version": "v1",
  "published": "2023-07-10T17:52:32.000Z",
  "updated": "2023-07-10T17:52:32.000Z",
  "title": "Cohen-Lenstra distribution for sparse matrices with determinantal biasing",
  "authors": [
    "András Mészáros"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let us consider the following matrix $B_n$. The columns of $B_n$ are indexed with $[n]=\\{1,2,\\dots,n\\}$ and the rows are indexed with $[n]^3$. The row corresponding to $(x_1,x_2,x_3)\\in [n]^3$ is given by $\\sum_{i=1}^3 e_{x_i}$, where $e_1,e_2,\\dots,e_n$ is the standard basis of $\\mathbb{R}^{[n]}$. Let $A_n$ be random $n\\times n$ submatrix of $B_n$, where the probability that we choose a submatrix $C$ is proportional to $|\\det(C)|^2$. Let $p\\ge 5$ be a prime. We prove that the asymptotic distribution of the $p$-Sylow subgroup of the cokernel of $A_n$ is given by the Cohen-Lenstra heuristics. Our result is motivated by the conjecture that the first homology group of a random two dimensional hypertree is also Cohen-Lenstra distributed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-10T17:52:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sparse matrices",
      "cohen-lenstra distribution",
      "determinantal biasing",
      "first homology group",
      "dimensional hypertree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}