Clusters in the critical branching Brownian motion

Brownian particles that are replicated and annihilated at equal rate have strongly correlated positions, forming a few compact clusters separated by large gaps. We characterize the distribution of the particles at a given time, using a definition of clusters in terms a coarse-graining length recently introduced by some of us. We show that, in a non-extinct realization, the average number of clusters grows as $\sim t^{D_{\mathrm{f}}\mathrm{/}2}$ where $D_{\mathrm{f}}\approx 0.22$ is the Haussdoff dimension of the boundary of the super-Brownian motion, found by Mueller, Mytnik, and Perkins. We also compute the distribution of gaps between consecutive particles. We find two regimes separated by the characteristic length scale $\mathrm{\ell }=\sqrt{D\mathrm{/}\beta }$ where $D$ is the diffusion constant and $\beta$ the branching rate. The average number of gaps greater than $g$ decays as $\sim g^{D_{\mathrm{f}}-2}$ for $g\ll \mathrm{\ell }$ and $\sim g^{-D_{\mathrm{f}}}$ for $g\gg \mathrm{\ell }$. Finally, conditioned on the number of particles $n$, the above distributions are valid for $g\ll \sqrt{n}$; the average number of gaps greater than $g\gg \sqrt{n}$ is much less than one, and decays as $\simeq 4(g\mathrm{/}\sqrt{n}{)}^{-2}$, in agreement with the universal gap distribution predicted by Ramola, Majumdar, and Schehr. Our results interpolate between a dense super-Brownian motion regime and a large-gap regime, unifying two previously independent approaches.