{
  "id": "2009.03869",
  "version": "v1",
  "published": "2020-09-08T17:15:40.000Z",
  "updated": "2020-09-08T17:15:40.000Z",
  "title": "Free Convolution of Measures via Roots of Polynomials",
  "authors": [
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $\\mu^{\\boxplus k}$ for any real $k \\geq 1$. The purpose of this short note is to give an elementary description of $\\mu^{\\boxplus k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-08T17:15:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomials",
      "free convolution power",
      "asymptotic behavior",
      "compactly supported probability measure",
      "real line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}