{
  "id": "2208.01848",
  "version": "v1",
  "published": "2022-08-03T05:25:44.000Z",
  "updated": "2022-08-03T05:25:44.000Z",
  "title": "Bose-Einstein-Like condensation of deformed random matrix: A replica approach",
  "authors": [
    "Harukuni Ikeda"
  ],
  "comment": "17 pages, 11 figures",
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech"
  ],
  "abstract": "In this work, we investigate a symmetric deformed random matrix, which is obtained by perturbing the diagonal elements of the Wigner matrix by a constant $h_i$. The eigenvector $\\vec{x}_{\\rm min}$ of the minimal eigenvalue $\\lambda_{\\rm min}$ of the deformed random matrix tends to condensate at a site with the smallest $h_i$. In certain types of distribution of $h_i$ and in the limit of the large components, this condensation becomes a sharp phase transition, where the mechanism to cause the condensation can be identified with the Bose-Einstein condensation in a mathematical level. We study this Bose-Einstein like condensation phenomenon by means of the replica method. We first derive a formula to calculate the minimal eigenvalue and the overlap of $\\vec{x}_{\\rm min}$. Then, we apply the formula for two solvable cases: when the distribution $h_i$ has a double peak, and when it has a continuous peak. For the double peak, we find that at the transition point, the participation ratio changes discontinuously from a finite value to zero. On the contrary, in the case of the continuous distribution, the participation ratio continuously goes to zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-03T05:25:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "replica approach",
      "bose-einstein-like condensation",
      "minimal eigenvalue",
      "distribution",
      "symmetric deformed random matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}