{
  "id": "1910.06382",
  "version": "v1",
  "published": "2019-10-14T18:54:35.000Z",
  "updated": "2019-10-14T18:54:35.000Z",
  "title": "Particle-number distribution in large fluctuations at the tip of branching random walks",
  "authors": [
    "A. H. Mueller",
    "S. Munier"
  ],
  "comment": "24 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech",
    "hep-ph"
  ],
  "abstract": "We investigate properties of the particle distribution near the tip of one-dimensional branching random walks at large times $t$, focusing on unusual realizations in which the rightmost lead particle is very far ahead of its expected position -- but still within a distance smaller than the diffusion radius $\\sim\\sqrt{t}$. Our approach consists in a study of the generating function $G_{\\Delta x}(\\lambda)=\\sum_n \\lambda^n p_n(\\Delta x)$ for the probabilities $p_n(\\Delta x)$ of observing $n$ particles in an interval of given size $\\Delta x$ from the lead particle to its left, fixing the position of the latter. This generating function can be expressed with the help of functions solving the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation with suitable initial conditions. In the infinite-time and large-$\\Delta x$ limits, we find that the mean number of particles in the interval grows exponentially with $\\Delta x$, and that the generating function obeys a nontrivial scaling law, depending on $\\Delta x$ and $\\lambda$ through the combined variable $[\\Delta x-f(\\lambda)]^{3}/\\Delta x^2$, where $f(\\lambda)\\equiv -\\ln(1-\\lambda)-\\ln[-\\ln(1-\\lambda)]$. From this property, one may conjecture that the growth of the typical particle number with the size of the interval is slower than exponential, but, surprisingly enough, only by a subleading factor at large $\\Delta x$. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-14T18:54:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large fluctuations",
      "particle-number distribution",
      "one-dimensional branching random walks",
      "large times",
      "typical particle number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}