Branching Brownian motion conditioned on particle numbers

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 gk(t)=xk(t)-xk+1(t)gk(t)=xk(t)-xk+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(gk,t→∞|n)=p(gk|n)P(gk,t→∞|n)=p(gk|n). We show that they are characterized by an exponential tail p(gk|n)∼exp-|b-d|2Dgk for large gaps in the subcritical (bdb>d) phases, and a power law tail p(gk)∼8Dbgk-3 at the critical point (b=db=d), independently of n and k. Some of these results for the critical case were announced in a recent letter (Ramola et al., 2014).