{
  "id": "1503.07955",
  "version": "v1",
  "published": "2015-03-27T04:02:20.000Z",
  "updated": "2015-03-27T04:02:20.000Z",
  "title": "Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition",
  "authors": [
    "Peter J. Forrester",
    "Dang-Zheng Liu"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CA",
    "math.MP"
  ],
  "abstract": "The singular values squared of the random matrix product $Y = G_r G_{r-1} \\cdots G_1 (G_0 + A)$, where each $G_j$ is a rectangular standard complex Gaussian matrix while $A$ is non-random, are shown to be a determinantal point process with correlation kernel given by a double contour integral. When all but finitely many eigenvalues of $A^*A$ are equal to $bN$, the corresponding correlation kernel is shown to admit a well-defined hard edge scaling, in which a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of $0<b<1$ is independent of $b$, and is in fact the same as that known for the case $b=0$ due to Kuijlaars and Zhang[arXiv:1308.1003]. The critical regime of $b=1$ allows for a double scaling limit by choosing $b =(1 - \\tau/\\sqrt{N})^{-1}$, and for this the critical kernel and outlier phenomenon are established, while a distribution corresponding to a finite product is proven to be the scaling limit in the supercritical regime of $b>1$ with two distinct scaling rates. In the simplest case $r=0$, which is closely related to non-intersecting squared Bessel paths, the latter gives rise to the finite LUE distribution. Similar results also hold true for the random matrix product $T_r T_{r-1} \\cdots T_1 (G_0 + A)$, with each $T_j$ being a truncated unitary matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-27T04:02:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "30E15"
    ],
    "keywords": [
      "complex ginibre matrices",
      "hard edge limit",
      "phase transition",
      "singular values",
      "random matrix product"
    ],
    "publication": {
      "doi": "10.1007/s00220-015-2507-5",
      "journal": "Communications in Mathematical Physics",
      "year": 2015,
      "month": "Nov",
      "pages": 241
    },
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015CMaPh.tmp..241F"
    }
  }
}