{
  "id": "2007.15438",
  "version": "v1",
  "published": "2020-07-28T21:54:45.000Z",
  "updated": "2020-07-28T21:54:45.000Z",
  "title": "Non-Hermitian random matrices with a variance profile (II): properties and examples",
  "authors": [
    "Nicholas A. Cook",
    "Walid Hachem",
    "Jamal Najim",
    "David Renfrew"
  ],
  "comment": "35 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1612.04428",
  "categories": [
    "math.PR"
  ],
  "abstract": "For each $n$, let $A_n=(\\sigma_{ij})$ be an $n\\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\\times n$ random matrix with i.i.d. centered entries of unit variance. In the companion article Cook et al., we considered the empirical spectral distribution $\\mu_n^Y$ of the rescaled entry-wise product \\[ Y_n = \\frac 1{\\sqrt{n}} A_n\\odot X_n = \\left(\\frac1{\\sqrt{n}} \\sigma_{ij}X_{ij}\\right) \\] and provided a deterministic sequence of probability measures $\\mu_n$ such that the difference $\\mu^Y_n - \\mu_n$ converges weakly in probability to the zero measure. A key feature in Cook et al. was to allow some of the entries $\\sigma_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence $(\\mu_n)$, described by a family of Master Equations. We consider these equations in important special cases such as separable variance profiles $\\sigma^2_{ij}=d_i \\widetilde d_j$ and sampled variance profiles $\\sigma^2_{ij} = \\sigma^2\\left(\\frac in, \\frac jn \\right)$ where $(x,y)\\mapsto \\sigma^2(x,y)$ is a given function on $[0,1]^2$. Associate examples are provided where $\\mu_n^Y$ converges to a genuine limit. We study $\\mu_n$'s behavior at zero and provide examples where $\\mu_n$'s density is bounded, blows up, or vanishes while an atom appears. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al., we prove that except maybe in zero, $\\mu_n$ admits a positive density on the centered disc of radius $\\sqrt{\\rho(V_n)}$, where $V_n=(\\frac 1n \\sigma_{ij}^2)$ and $\\rho(V_n)$ is its spectral radius.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-28T21:54:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-hermitian random matrices",
      "variance profile",
      "random matrix",
      "important special cases",
      "standard deviation profiles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}