{
  "id": "2307.10740",
  "version": "v1",
  "published": "2023-07-20T10:07:33.000Z",
  "updated": "2023-07-20T10:07:33.000Z",
  "title": "Conformally invariant fields out of Brownian loop soups",
  "authors": [
    "Antoine Jego",
    "Titus Lupu",
    "Wei Qian"
  ],
  "comment": "84 pages, 4 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a Brownian loop soup $\\mathcal{L}_D^\\theta$ with subcritical intensity $\\theta \\in (0,1/2]$ in some 2D bounded simply connected domain. We define and study the properties of a conformally invariant field $h_\\theta$ naturally associated to $\\mathcal{L}_D^\\theta$. Informally, this field is a signed version of the local time of $\\mathcal{L}_D^\\theta$ to the power $1-\\theta$. When $\\theta=1/2$, $h_\\theta$ is a Gaussian free field (GFF) in $D$. Our construction of $h_\\theta$ relies on the multiplicative chaos $\\mathcal{M}_\\gamma$ associated with $\\mathcal{L}_D^\\theta$, as introduced in [ABJL23]. Assigning independent symmetric signs to each cluster, we restrict $\\mathcal{M}_\\gamma$ to positive clusters. We prove that, when $\\theta=1/2$, the resulting measure $\\mathcal{M}_\\gamma^+$ corresponds to the exponential of $\\gamma$ times a GFF. At this intensity, the GFF can be recovered by differentiating at $\\gamma=0$ the measure $\\mathcal{M}_\\gamma^+$. When $\\theta<1/2$, we show that $\\mathcal{M}_\\gamma^+$ has a nondegenerate fractional derivative at $\\gamma=0$ defining a random generalised function $h_\\theta$. We establish a result which is analoguous to the recent work [ALS23] in the GFF case ($\\theta=1/2$), but for $h_\\theta$ with $\\theta \\in (0,1/2]$. Relying on the companion article [JLQ23], we prove that each cluster of $\\mathcal{L}_D^\\theta$ possesses a nondegenerate Minkowski content in some non-explicit gauge function $r \\mapsto r^2 |\\log r|^{1-\\theta+o(1)}$. We then prove that $h_\\theta$ agrees a.s.\\ with the sum of the Minkowski content of each cluster multiplied by its sign. We further extend the couplings between CLE$_4$, SLE$_4$ and the GFF to $h_\\theta$ for $\\theta\\in(0,1/2]$. We show that the (non-nested) CLE$_\\kappa$ loops form level lines for $h_\\theta$ and that there exists a constant height gap between the values of the field on either side of the CLE loops.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-20T10:07:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian loop soup",
      "conformally invariant field",
      "bounded simply connected domain",
      "loops form level lines",
      "non-explicit gauge function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 84,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}