{
  "id": "1403.1185",
  "version": "v1",
  "published": "2014-03-05T16:42:29.000Z",
  "updated": "2014-03-05T16:42:29.000Z",
  "title": "Phase transitions in the condition number distribution of Gaussian random matrices",
  "authors": [
    "Isaac Pérez Castillo",
    "Eytan Katzav",
    "Pierpaolo Vivo"
  ],
  "comment": "5 pag. + 7 pag. Suppl. Material. 3 Figures",
  "categories": [
    "cond-mat.stat-mech",
    "cs.CC",
    "cs.IT",
    "math-ph",
    "math.IT",
    "math.MP",
    "stat.OT"
  ],
  "abstract": "We study the statistics of the condition number $\\kappa=\\lambda_{\\mathrm{max}}/\\lambda_{\\mathrm{min}}$ (the ratio between largest and smallest squared singular values) of $N\\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\\mathcal{P}[\\kappa<x]$ and tail-cumulative $\\mathcal{P}[\\kappa>x]$ distributions of $\\kappa$. We find that these distributions decay as $\\mathcal{P}[\\kappa<x]\\approx\\exp\\left(-\\beta N^2 \\Phi_{-}(x)\\right)$ and $\\mathcal{P}[\\kappa>x]\\approx\\exp\\left(-\\beta N \\Phi_{+}(x)\\right)$, where $\\beta$ is the Dyson index of the ensemble. The left and right rate functions $\\Phi_{\\pm}(x)$ are independent of $\\beta$ and calculated exactly for any choice of the rectangularity parameter $\\alpha=M/N-1>0$. Interestingly, they show a weak non-analytic behavior at their minimum $\\langle\\kappa\\rangle$ (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around $\\langle\\kappa\\rangle$, we determine exactly the scale of typical fluctuations $\\sim\\mathcal{O}(N^{-2/3})$ and the tails of the limiting distribution of $\\kappa$. The analytical results are in excellent agreement with numerical simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-05T16:42:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.40.-a",
      "02.10.Yn",
      "02.50.Sk",
      "24.60.-k"
    ],
    "keywords": [
      "gaussian random matrices",
      "condition number distribution",
      "phase transition",
      "smallest squared singular values",
      "coulomb fluid technique"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1103/PhysRevE.90.050103",
      "journal": "Physical Review E",
      "year": 2014,
      "month": "Nov",
      "volume": 90,
      "number": 5,
      "pages": "050103"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014PhRvE..90e0103P"
    }
  }
}