{
  "id": "1407.2979",
  "version": "v1",
  "published": "2014-07-10T22:11:11.000Z",
  "updated": "2014-07-10T22:11:11.000Z",
  "title": "Branching Brownian Motion Conditioned on Particle Numbers",
  "authors": [
    "Kabir Ramola",
    "Satya N. Majumdar",
    "Gregory Schehr"
  ],
  "comment": "19 pages, 5 figures",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study analytically the order and gap statistics of particles at time $t$ for the one dimensional branching Brownian motion, conditioned to have a fixed number of particles at $t$. The dynamics of the process proceeds in continuous time where at each time step, every particle in the system either diffuses (with diffusion constant $D$), dies (with rate $d$) or splits into two independent particles (with rate $b$). We derive exact results for the probability distribution function of $g_k(t) = x_k(t) - x_{k+1}(t)$, the distance between successive particles, conditioned on the event that there are exactly $n$ particles in the system at a given time $t$. We show that at large times these conditional distributions become stationary $P(g_k, t \\to \\infty|n) = p(g_k|n)$. We show that they are characterised by an exponential tail $p(g_k|n) \\sim \\exp[-\\sqrt{\\frac{|b - d|}{2 D}} ~g_k]$ for large gaps in the subcritical ($b < d$) and supercritical ($b > d$) phases, and a power law tail $p(g_k) \\sim 8\\left(\\frac{D}{b}\\right){g_k}^{-3}$ at the critical point ($b = d$), independently of $n$ and $k$. Some of these results for the critical case were announced in a recent letter [K. Ramola, S. N. Majumdar and G. Schehr, Phys. Rev. Lett. 112, 210602 (2014)].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-10T22:11:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "particle numbers",
      "dimensional branching brownian motion",
      "power law tail",
      "probability distribution function"
    ],
    "publication": {
      "doi": "10.1016/j.chaos.2014.12.013",
      "journal": "Chaos Solitons and Fractals",
      "year": 2015,
      "month": "May",
      "volume": 74,
      "pages": 79
    },
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015CSF....74...79R"
    }
  }
}