{
  "id": "2306.00300",
  "version": "v1",
  "published": "2023-06-01T02:39:19.000Z",
  "updated": "2023-06-01T02:39:19.000Z",
  "title": "Eigenvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion",
  "authors": [
    "Syota Esaki",
    "Makoto Katori",
    "Satoshi Yabuoku"
  ],
  "comment": "LaTeX, 36 pages, no figure",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The non-Hermitian matrix-valued Brownian motion is the stochastic process of a random matrix whose entries are given by independent complex Brownian motions. The bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for their scale transformations with an invariant eigenvalue process. The eigenvector-overlap process is a Hermitian matrix-valued process, each element of which is given by a product of an overlap of right eigenvectors and that of left eigenvectors. We derive a set of stochastic differential equations for the coupled system of the eigenvalue process and the eigenvector-overlap process and prove the scale-transformation invariance of the system. The Fuglede-Kadison (FK) determinant associated with the present matrix-valued process is regularized by introducing an auxiliary complex variable. This variable is necessary to give the stochastic partial differential equations (SPDEs) for the time-dependent random field determined by the regularized FK determinant and for its logarithmic variation. Time-dependent point process of eigenvalues and its variation weighted by the diagonal elements of the eigenvector-overlap process are related to the derivatives of the logarithmic random-field of the regularized FK determinant. From the SPDEs a system of PDEs for the density functions of these two types of time-dependent point processes are obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-01T02:39:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "60H10",
      "60H15",
      "60G55"
    ],
    "keywords": [
      "non-hermitian matrix-valued brownian motion",
      "regularized fuglede-kadison determinant",
      "eigenvalue",
      "time-dependent point process",
      "eigenvector-overlap process"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}