{
  "id": "1711.07382",
  "version": "v1",
  "published": "2017-11-20T15:40:01.000Z",
  "updated": "2017-11-20T15:40:01.000Z",
  "title": "Spectral distribution of the free Jacobi process, revisited",
  "authors": [
    "Tarek Hamdi"
  ],
  "comment": "All comments are welcome",
  "categories": [
    "math.PR",
    "math.OA",
    "math.SP"
  ],
  "abstract": "We obtain a description for the spectral distribution of the free Jacobi process for any initial pair of projections. This result relies on a study of the unitary operator $RU_tSU_t^*$ where $R,S$ are two symmetries and $U_t$ a free unitary Brownian motion, freely independent from $\\{R,S\\}$. In particular, for non-null traces of $R$ and $S$, we prove that the spectral measure of $RU_tSU_t^*$ possesses two atoms at $\\pm1$ and an $L^\\infty$-density on the unit circle $\\mathbb{T}$, for every $t>0$. Next, via a Szeg\\H{o} type transform of this law, we obtain a full description of the spectral distribution of $PU_tQU_t^*$ beyond the $\\tau(P)=\\tau(Q)=1/2$ case. Finally, we give some specializations for which these measures are explicitly computed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-20T15:40:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free jacobi process",
      "spectral distribution",
      "free unitary brownian motion",
      "type transform",
      "unit circle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}