{
  "id": "1008.3099",
  "version": "v3",
  "published": "2010-08-18T14:54:26.000Z",
  "updated": "2012-02-27T09:23:54.000Z",
  "title": "Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product",
  "authors": [
    "S. Belinschi",
    "B. Collins",
    "I. Nechita"
  ],
  "comment": "v3 changes: minor typographic improvements; accepted version",
  "journal": "Inventiones mathematicae, vol. 190, no. 3, 2012, pp. 647-697",
  "categories": [
    "math.PR",
    "math.OA",
    "quant-ph"
  ],
  "abstract": "Given two positive integers $n$ and $k$ and a parameter $t\\in (0,1)$, we choose at random a vector subspace $V_{n}\\subset \\mathbb{C}^{k}\\otimes\\mathbb{C}^{n}$ of dimension $N\\sim tnk$. We show that the set of $k$-tuples of singular values of all unit vectors in $V_n$ fills asymptotically (as $n$ tends to infinity) a deterministic convex set $K_{k,t}$ that we describe using a new norm in $\\R^k$. Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high eigenvalue degeneracy.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-02-27T09:23:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "52A22",
      "46L54"
    ],
    "keywords": [
      "large numbers",
      "random subspaces",
      "tensor product",
      "eigenvalues",
      "eigenvectors"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00222-012-0386-3",
      "journal": "Inventiones Mathematicae",
      "year": 2012,
      "month": "Dec",
      "volume": 190,
      "number": 3,
      "pages": 647
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012InMat.190..647B"
    }
  }
}