We study the order statistics of one-dimensional branching Brownian motion in which particles either diffuse (with diffusion constant $D$), die (with rate $d$), or split into two particles (with rate $b$). At the critical point $b=d$, which we focus on, we show that at large time $t$ the particles are collectively bunched together. We find indeed that there are two length scales in the system: (i) the diffusive length scale $\sim \sqrt{Dt}$, which controls the collective fluctuations of the whole bunch, and (ii) the length scale of the gap between the bunched particles $\sim \sqrt{D/b}$. We compute the probability distribution function $\tilde{P}(g_k,t|n)$ of the $k$th gap $g_k=x_k-x_{k+1}$ between the $k$th and ($k+1$)th particles given that the system contains exactly $n>k$ particles at time $t$. We show that at large $t$, it converges to a stationary distribution $\tilde{P}(g_k,t\to \infty |n)=p(g_k|n)$ with an algebraic tail $p(g_k|n)\sim 8(D/b)g_k^{-3}$, for $g_k\gg 1$, independent of $k$ and $n$. We verify our predictions with Monte Carlo simulations.

• Received 18 March 2014

DOI:https://doi.org/10.1103/PhysRevLett.112.210602