{
  "id": "2310.14132",
  "version": "v1",
  "published": "2023-10-21T23:04:18.000Z",
  "updated": "2023-10-21T23:04:18.000Z",
  "title": "Spectral measure for uniform $d$-regular digraphs",
  "authors": [
    "Arka Adhikari",
    "Amir Dembo"
  ],
  "comment": "65 pages",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Consider the matrix $A_{\\mathcal{G}}$ chosen uniformly at random from the finite set of all $N$-dimensional matrices of zero main-diagonal and binary entries, having each row and column of $A_{\\mathcal{G}}$ sum to $d$. That is, the adjacency matrix for the uniformly random $d$-regular simple digraph $\\mathcal{G}$. Fixing $d \\ge 3$, it has long been conjectured that as $N \\to \\infty$ the corresponding empirical eigenvalue distributions converge weakly, in probability, to an explicit non-random limit, %measure $\\mu_d$ on $\\mathbb{C}$, which is given by the Brown measure of the free sum of $d$ Haar unitary operators. We reduce this conjecture to bounding the decay in $N$ of the probability that the minimal singular value of the shifted matrix $A(w) = A_{\\mathcal{G}} - w I$ is very small. While the latter remains a challenging task, the required bound is comparable to the recently established control on the singularity of $A_{\\mathcal{G}}$. The reduction is achieved here by sharp estimates on the behavior at large $N$, near the real line, of the Green's function (aka resolvent) of the Hermitization of $A(w)$, which is of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-21T23:04:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L53",
      "60B10",
      "60B20",
      "05C50",
      "05C20"
    ],
    "keywords": [
      "spectral measure",
      "regular digraphs",
      "regular simple digraph",
      "explicit non-random limit",
      "minimal singular value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}