{
  "id": "2206.03029",
  "version": "v1",
  "published": "2022-06-07T06:03:46.000Z",
  "updated": "2022-06-07T06:03:46.000Z",
  "title": "Liouville quantum gravity from random matrix dynamics",
  "authors": [
    "Paul Bourgade",
    "Hugo Falconet"
  ],
  "comment": "39 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We establish the first connection between $2d$ Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if $(U_t)$ is a Brownian motion on the unitary group at equilibrium, then the measures $$ |\\det(U_t - e^{i \\theta})|^{\\gamma} dt d\\theta $$ converge in the limit of large dimension to the $2d$ LQG measure, a properly normalized exponential of the $2d$ Gaussian free field. Gaussian free field type fluctuations associated with these dynamics were first established by Spohn (1998) and convergence to the LQG measure in $2d$ settings was conjectured since the work of Webb (2014), who proved the convergence of related one dimensional measures by using inputs from Riemann-Hilbert theory. The convergence follows from the first multi-time extension of the result by Widom (1973) on Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols. To prove these, we develop a general surgery argument and combine determinantal point processes estimates with stochastic analysis on Lie group, providing in passing a probabilistic proof of Webb's $1d$ result. We believe the techniques will be more broadly applicable to matrix dynamics out of equilibrium, joint moments of determinants for classes of correlated random matrices, and the characteristic polynomial of non-Hermitian random matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-07T06:03:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville quantum gravity",
      "random matrix dynamics",
      "gaussian free field type fluctuations",
      "lqg measure",
      "determinantal point processes estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}