{
  "id": "2401.14942",
  "version": "v1",
  "published": "2024-01-26T15:24:48.000Z",
  "updated": "2024-01-26T15:24:48.000Z",
  "title": "Noise-like analytic properties of imaginary chaos",
  "authors": [
    "Juhan Aru",
    "Guillaume Baverez",
    "Antoine Jego",
    "Janne Junnila"
  ],
  "comment": "40 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this note we continue the study of imaginary multiplicative chaos $\\mu_\\beta := \\exp(i \\beta \\Gamma)$, where $\\Gamma$ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of $|\\mu_\\beta(Q(x,r))|$ as $r \\to 0$, where $Q(x,r)$ is a square of side-length $2r$ centred at $x$. More precisely, we prove monofractality of this process, a law of the iterated logarithm as $r \\to 0$ and analyse its exceptional points, which have a close connection to fast points of Brownian motion. Some of the technical ideas developed to address these questions also help us pin down the exact Besov regularity of imaginary chaos, a question left open in [JSW20]. All the mentioned properties illustrate the noise-like behaviour of the imaginary chaos. We conclude by proving that the processes $x \\mapsto |\\mu_\\beta(Q(x,r))|^2$, when normalised additively and multiplicatively, converge as $r \\to 0$ in law, but not in probability, to white noise; this suggests that all the information of the multiplicative chaos is contained in the angular parts of $\\mu_\\beta(Q(x,r))$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-26T15:24:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G20",
      "60G60"
    ],
    "keywords": [
      "imaginary chaos",
      "noise-like analytic properties",
      "two-dimensional continuum gaussian free field",
      "multiplicative chaos",
      "question left open"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}