{
  "id": "2007.01869",
  "version": "v1",
  "published": "2020-07-03T18:00:00.000Z",
  "updated": "2020-07-03T18:00:00.000Z",
  "title": "New Recipes for Brownian Loop Soups",
  "authors": [
    "Valentino F. Foit",
    "Matthew Kleban"
  ],
  "comment": "23 pages, 3 figures",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "hep-th",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We define a large new class of conformal primary operators in the ensemble of Brownian loops in two dimensions known as the ``Brownian loop soup,'' and compute their correlation functions analytically and in closed form. The loop soup is a conformally invariant statistical ensemble with central charge $c = 2 \\lambda$, where $\\lambda > 0$ is the intensity of the soup. Previous work identified exponentials of the layering operator $e^{i \\beta N(z)}$ as primary operators. Each Brownian loop was assigned $\\pm 1$ randomly, and $N(z)$ was defined to be the sum of these numbers over all loops that encircle the point $z$. These exponential operators then have conformal dimension ${\\frac{\\lambda}{10}}(1 - \\cos \\beta)$. Here we generalize this procedure by assigning a more general random value to each loop. The operator $e^{i \\beta N(z)}$ remains primary with conformal dimension $\\frac {\\lambda}{10}(1 - \\phi(\\beta))$, where $\\phi(\\beta)$ is the characteristic function of the probability distribution used to assign random values to each loop. Using recent results we compute in closed form the exact two-point functions in the upper half-plane and four-point functions in the full plane of this very general class of operators. These correlation functions depend analytically on the parameters $\\lambda, \\beta_i, z_i$, and on the characteristic function $\\phi(\\beta)$. They satisfy the conformal Ward identities and are crossing symmetric. As in previous work, the conformal block expansion of the four-point function reveals the existence of additional and as-yet uncharacterized conformal primary operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-03T18:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian loop soup",
      "four-point function",
      "correlation functions",
      "characteristic function",
      "random value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}