{
  "id": "1505.00610",
  "version": "v1",
  "published": "2015-05-04T12:37:32.000Z",
  "updated": "2015-05-04T12:37:32.000Z",
  "title": "Correlation kernels for sums and products of random matrices",
  "authors": [
    "Tom Claeys",
    "Arno B. J. Kuijlaars",
    "Dong Wang"
  ],
  "comment": "33 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CA",
    "math.MP"
  ],
  "abstract": "Let $X$ be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of $GX$ and $TX$, where $G$ is a complex Ginibre matrix and $T$ is a truncated unitary matrix. We also consider the product of $X$ and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum $H + M$ where $H$ is a GUE matrix and $M$ is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of $H + M$ follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-04T12:37:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "60B20",
      "42C35",
      "42C05"
    ],
    "keywords": [
      "random matrix",
      "correlation kernel",
      "squared singular value",
      "polynomial ensemble",
      "truncated unitary matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150500610C"
    }
  }
}