{
  "id": "1412.7108",
  "version": "v1",
  "published": "2014-12-22T19:33:48.000Z",
  "updated": "2014-12-22T19:33:48.000Z",
  "title": "The eigenvectors of Gaussian matrices with an external source",
  "authors": [
    "Romain Allez",
    "Joël Bun",
    "Jean-Philippe Bouchaud"
  ],
  "comment": "27 pages, 4 figures",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech"
  ],
  "abstract": "We consider a diffusive matrix process $(X_t)_{t\\ge 0}$ defined as $X_t:=A+H_t$ where $A$ is a given deterministic Hermitian matrix and $(H_t)_{t\\ge 0}$ is a Hermitian Brownian motion. The matrix $A$ is the \"external source\" that one would like to estimate from the noisy observation $X_t$ at some time $t>0$. We investigate the relationship between the non-perturbed eigenvectors of the matrix $A$ and the perturbed eigenstates at some time $t$ for the three relevant scaling relations between the time $t$ and the dimension $N$ of the matrix $X_t$. We determine the asymptotic (mean-squared) projections of any given non-perturbed eigenvector $|\\psi_j^0>$, associated to an eigenvalue $a_j$ of $A$ which may lie inside the bulk of the spectrum or be isolated (spike) from the other eigenvalues, on the orthonormal basis of the perturbed eigenvectors $|\\psi_i^t>,i\\neq j$. In the case of one isolated eigenvector $|\\psi_j^0>$, we prove a central limit Theorem for the overlap $< \\psi_j^0|\\psi_j^t>$. When properly centered and rescaled by a factor $\\sqrt{N}$, this overlap converges in law towards a centered Gaussian distribution with an explicit variance depending on $t$. Our method is based on analyzing the eigenvector flow under the Dyson Brownian motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-22T19:33:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "external source",
      "gaussian matrices",
      "dyson brownian motion",
      "deterministic hermitian matrix",
      "hermitian brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.7108A"
    }
  }
}