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 of 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 similar to t(Df/2) where D-f asymptotic to 0.22 is the Hausdorff dimension of the boundary of the super-Brownian motion (SBM), 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 2 = D/0 where D is the diffusion constant and 0 the branching rate. The average number of gaps greater than g decays as similar to g(Df)-2 for g << 2 and similar to g(-Df) for g >> 2. Finally, conditioned on the number of particles n, the above distributions are valid for g << root n; the average number of gaps greater than g >> root n is much less than one, and decays as ?4(g/root n)(-2), in agreement with the universal gap distribution predicted by Ramola, Majumdar, and Schehr. Our results interpolate between a dense SBM regime and a large-gap regime, unifying two previously independent approaches.

Automated summary

Brownian particles that are replicated and annihilated at equal rate exhibit strong positional correlations, forming compact clusters with large gaps in between. The distribution of particles at a given time is characterized using a coarse-graining length definition of clusters. The average number of clusters grows as similar to t(Df/2) in a non-extinct realization, and the distribution of gaps between consecutive particles shows two regimes separated by a characteristic length scale 2 = D/0. The importance of this study is rated 8 out of 10.