{
  "id": "2003.04362",
  "version": "v1",
  "published": "2020-03-09T19:01:52.000Z",
  "updated": "2020-03-09T19:01:52.000Z",
  "title": "Monte Carlo study of the tip region of branching random walks evolved to large times",
  "authors": [
    "D. Le Anh",
    "A. H. Mueller",
    "S. Munier"
  ],
  "comment": "19 pages, 9 figures",
  "categories": [
    "cond-mat.stat-mech",
    "hep-ph"
  ],
  "abstract": "We implement a discretization of the one-dimensional branching Brownian motion in the form of a Monte Carlo event generator, designed to efficiently produce ensembles of realizations in which the rightmost lead particle at the final time $T$ is constrained to have a position $X$ larger than some predefined value $X_{\\text{min}}$. The latter may be chosen arbitrarily far from the expectation value of $X$, and the evolution time after which observables on the particle density near the lead particle are measured may be as large as $T\\sim 10^4$. We then calculate numerically the probability distribution $p_n(\\Delta x)$ of the number $n$ of particles in the interval $[X-\\Delta x,X]$ as a function of $\\Delta x$. When $X_{\\text{min}}$ is significantly smaller than the expectation value of the position of the rightmost lead particle, i.e. when $X$ is effectively unconstrained, we check that both the mean and the typical values of $n$ grow exponentially with $\\Delta x$, up to a linear prefactor and to finite-$T$ corrections. When $X_{\\text{min}}$ is picked far ahead of the latter but within a region extending over a size of order $\\sqrt{T}$ to its right, the mean value of the particle number still grows exponentially with $\\Delta x$, but its typical value is lower by a multiplicative factor consistent with $e^{-\\zeta\\Delta x^{2/3}}$, where $\\zeta$ is a number of order unity. These numerical results bring strong support to recent analytical calculations and conjectures in the infinite-time limit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-09T19:01:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "monte carlo study",
      "branching random walks",
      "large times",
      "tip region",
      "results bring strong support"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}