Branching Brownian Motion Conditioned on Particle Numbers

Kabir Ramola; Satya N. Majumdar; Gregory Schehr

doi:10.1016/j.chaos.2014.12.013

paper

Branching Brownian Motion Conditioned on Particle Numbers

arXiv:1407.2979 · doi:10.1016/j.chaos.2014.12.013

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 $p (g_{k} ∣ n) \sim exp [- \frac{∣ b - d ∣}{2 D} g_{k}]$ for large gaps in the subcritical ( $b < d$ ) and supercritical ( $b > d$ ) phases, and a power law tail $p (g_{k}) \sim 8 (\frac{D}{b}) g_{k}^{- 3}$ 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)].

19 pages, 5 figures

Branching Brownian Motion Conditioned on Particle Numbers

Abstract

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