Branching Brownian Motion Conditioned on Particle Numbers

TL;DR

This paper analytically investigates the distribution of particle gaps in a conditioned one-dimensional branching Brownian motion, revealing exponential and power-law tail behaviors depending on the criticality of the process.

Contribution

It provides exact analytical results for the gap distribution conditioned on fixed particle number, including stationary distributions and tail behaviors at criticality.

Findings

Conditional gap distributions become stationary at large times.

Exponential tail in subcritical and supercritical phases.

Power-law tail at the critical point.

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)].