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.

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