{
  "id": "2212.06499",
  "version": "v1",
  "published": "2022-12-13T11:26:56.000Z",
  "updated": "2022-12-13T11:26:56.000Z",
  "title": "Random matrices with row constraints and eigenvalue distributions of graph Laplacians",
  "authors": [
    "Pawat Akara-pipattana",
    "Oleg Evnin"
  ],
  "categories": [
    "cond-mat.dis-nn",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve $p_{\\mathrm{zrs}}(\\lambda)$ that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques. We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with $N$ vertices of mean degree $c$. In the regime $1\\ll c\\ll N$, the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of $p_{\\mathrm{zrs}}(\\lambda)$, centered at $c$ with width $\\sim\\sqrt{c}$. At smaller $c$, this curve receives corrections in powers of $1/\\sqrt{c}$ accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large $c$ limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-13T11:26:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random matrices",
      "row constraints",
      "fixed transition rate",
      "wigner semicircle",
      "zero row sums occur"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}