{
  "id": "2007.09421",
  "version": "v1",
  "published": "2020-07-18T13:02:04.000Z",
  "updated": "2020-07-18T13:02:04.000Z",
  "title": "Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform",
  "authors": [
    "Pierre Mergny",
    "Marc Potters"
  ],
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "math.MP"
  ],
  "abstract": "In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for $\\beta =2$ , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called $R$-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all $\\beta >0$ in the multiplicative case, where the asymptotic is governed by the logarithm of the $S$-transform. As a consequence of this result one can calculate the asymptotic behavior of complete homogeneous symmetric polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-18T13:02:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behavior",
      "multiplicative counterpart",
      "harish-chandra integral",
      "complete homogeneous symmetric polynomials",
      "similar result holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}