{
  "id": "2407.17608",
  "version": "v1",
  "published": "2024-07-24T19:35:58.000Z",
  "updated": "2024-07-24T19:35:58.000Z",
  "title": "Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices",
  "authors": [
    "James A. Mingo",
    "Daniel Munoz George"
  ],
  "comment": "62 pages, 13 Figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We compute the fluctuation moments $\\alpha_{m_1,\\dots,m_r}$ of a Complex Wigner Matrix $X_N$ given by the limit $\\lim_{N\\rightarrow\\infty}N^{r-2}k_r(Tr(X_N^{m_1}),\\dots,Tr(X_N^{m_r}))$. We prove the limit exists and characterize the leading order via planar graphs that result to be trees. We prove these graphs can be counted by the set of non-crossing partitioned permutations which permit us to express the moments $\\alpha_{m_1,\\dots,m_r}$ in terms of simpler quantities $\\overline{\\kappa}_{m_1,\\dots,m_r}$ which we call the pseudo-cumulants. We prove the pseudo-cumulants coincide with the higher order free cumulants up to $r=4$ which permit us to find the higher order free cumulants $\\kappa_{m_1,\\dots,m_r}$ associated to the moment sequence $\\alpha_{m_1,\\dots,m_r}$ up to order 4.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-24T19:35:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "46L54",
      "15B52"
    ],
    "keywords": [
      "higher order free cumulants",
      "complex wigner matrix",
      "asymptotic limit",
      "fluctuation moments",
      "planar graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}