{
  "id": "2202.09660",
  "version": "v2",
  "published": "2022-02-19T19:03:57.000Z",
  "updated": "2022-04-22T14:22:45.000Z",
  "title": "The heat flow conjecture for random matrices",
  "authors": [
    "Brian C. Hall",
    "Ching-Wei Ho"
  ],
  "comment": "47 pages; 17 figures. Improved and expanded version",
  "categories": [
    "math.PR",
    "hep-th",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Recent results by various authors have established a \"model deformation phenomenon\" in random matrix theory. Specifically, it is possible to construct pairs of random matrix models such that the limiting eigenvalue distributions are connected by push-forward under an explicitly constructible map of the plane to itself. In this paper, we argue that the analogous transformation at the finite-$N$ level can be accomplished by applying an appropriate heat flow to the characteristic polynomial of the first model. Let the \"second moment\" of a random polynomial $p$ denote the expectation value of the square of the absolute value of $p.$ We find certain pairs of random matrix models and we apply a certain heat-type operator to the characteristic polynomial $p_{1}$ of the first model, giving a new polynomial $q.$ We prove that the second moment of $q$ is equal to the second moment of the characteristic polynomial $p_{2}$ of the second model. This result leads to several conjectures of the following sort: when $N$ is large, the zeros of $q$ have the same bulk distribution as the zeros of $p_{2},$ namely the eigenvalues of the second random matrix model. At a more refined level, we conjecture that, as the characteristic polynomial of the first model evolves under the appropriate heat flow, its zeros will evolve close to the characteristic curves of a certain PDE. All conjectures are formulated in \"additive\" and \"multiplicative\" forms. As a special case, suppose we apply the standard heat operator for time $1/N$ to the characteristic polynomial $p$ of an $N\\times N$ GUE matrix, giving a new polynomial $q.$ We conjecture that the zeros of $q$ will be asymptotically uniformly distributed over the unit disk. That is, the heat operator converts the distribution of zeros from semicircular to circular.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-22T14:22:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "46L54"
    ],
    "keywords": [
      "heat flow conjecture",
      "characteristic polynomial",
      "appropriate heat flow",
      "second moment",
      "heat operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}