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 $\sim t^{D_{\mathrm{f}}/2}$ 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 $\ell = \sqrt{D/\beta}$ 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 $g\ll \ell$ and
$\sim g^{-D_{\mathrm{f}}}$ for $g \gg \ell$. 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 $\simeq 4 (g/\sqrt{n})^{-2}$, 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.
