{
  "id": "2209.10822",
  "version": "v1",
  "published": "2022-09-22T07:09:04.000Z",
  "updated": "2022-09-22T07:09:04.000Z",
  "title": "Clusters in the critical branching Brownian motion",
  "authors": [
    "Benoît Ferté",
    "Pierre Le Doussal",
    "Alberto Rosso",
    "Xiangyu Cao"
  ],
  "comment": "20 pages, 9 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Brownian particles that are replicated and annihilated at equal rate have strongly correlated positions, forming a few compact clusters separated by large gaps. We characterize the distribution of the particles at a given time, using a definition of clusters in terms a coarse-graining length recently introduced by some of us. We show that, in a non-extinct realization, the average number of clusters grows as $\\sim t^{D_{\\mathrm{f}}/2}$ where $D_{\\mathrm{f}} \\approx 0.22$ is the Haussdoff dimension of the boundary of the super-Brownian motion, found by Mueller, Mytnik, and Perkins. We also compute the distribution of gaps between consecutive particles. We find two regimes separated by the characteristic length scale $\\ell = \\sqrt{D/\\beta}$ where $D$ is the diffusion constant and $\\beta$ the branching rate. The average number of gaps greater than $g$ decays as $\\sim g^{D_{\\mathrm{f}}-2}$ for $g\\ll \\ell$ and $\\sim g^{-D_{\\mathrm{f}}}$ for $g \\gg \\ell$. Finally, conditioned on the number of particles $n$, the above distributions are valid for $g \\ll \\sqrt{n}$; the average number of gaps greater than $g \\gg \\sqrt{n}$ is much less than one, and decays as $\\simeq 4 (g/\\sqrt{n})^{-2}$, in agreement with the universal gap distribution predicted by Ramola, Majumdar, and Schehr. Our results interpolate between a dense super-Brownian motion regime and a large-gap regime, unifying two previously independent approaches.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-22T07:09:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical branching brownian motion",
      "average number",
      "gaps greater",
      "dense super-brownian motion regime",
      "characteristic length scale"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}