{
  "id": "2309.12196",
  "version": "v1",
  "published": "2023-09-21T16:03:55.000Z",
  "updated": "2023-09-21T16:03:55.000Z",
  "title": "A variational approach to free probability",
  "authors": [
    "Octavio Arizmendi",
    "Samuel G. G. Johnston"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR",
    "math.FA",
    "math.OA"
  ],
  "abstract": "Let $\\mu$ and $\\nu$ be compactly supported probability measures on the real line with densities with respect to Lebesgue measure. We show that for all large real $z$, if $\\mu \\boxplus \\nu$ is their additive free convolution, we have \\begin{equation*} \\int_{-\\infty}^\\infty \\log(z - x) \\mu \\boxplus \\nu (\\mathrm{d}x) = \\sup_{\\Pi} \\left\\{ \\mathbb{E}_\\Pi[\\log(z - (X+Y)] - \\mathcal{E}[\\Pi]+\\mathcal{E}[\\mu]+\\mathcal{E}[\\nu] \\right\\}, \\end{equation*} where the supremum is taken over all probability laws $\\Pi$ on $\\mathbb{R}^2$ for a pair of real-valued random variables $(X,Y)$ with respective marginal laws $\\mu$ and $\\nu$, and given a probability law $P$ with density function $f$ on $\\mathbb{R}^k$, $\\mathcal{E}[P] := \\int_{\\mathbb{R}^k} f \\log f$ is its classical entropy. We prove similar formulas for the multiplicative free convolution $\\mu \\boxtimes \\nu$ and the free compression $[\\mu]_\\tau$ of probability laws. The maximisers in our variational descriptions of these free operations on measures can be computed explicitly, and from these we can then deduce the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution. We use our formulation to derive several new inequalities relating free and classical convolutions of random variables, such as \\begin{equation*} \\int_{-\\infty}^\\infty \\log(z - x) \\mu \\boxplus \\nu (\\mathrm{d}x) \\geq \\mathbb{E}[\\log(z - (X+Y)], \\end{equation*} valid for all large $z$, where on the right-hand side $X,Y$ are independent classical random variables with respective laws $\\mu,\\nu$. Our approach is based on applying a large deviation principle on the symmetric group to the celebrated quadrature formulas of Marcus, Spielman and Srivastava.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-21T16:03:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54",
      "60F10",
      "22C05",
      "28C10"
    ],
    "keywords": [
      "free probability",
      "variational approach",
      "probability law",
      "multiplicative free convolution",
      "independent classical random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}