Clusters in the critical branching Brownian motion

Abstract

Abstract 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 ∼ t D f / 2 where D f ≈ 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 ℓ = D / β where D is the diffusion constant and β the branching rate. The average number of gaps greater than g decays as ∼ g D f − 2 for g ≪ ℓ and ∼ g − D f for g ≫ ℓ . Finally, conditioned on the number of particles n, the above distributions are valid for g ≪ n ; the average number of gaps greater than g ≫ n is much less than one, and decays as ≃ 4 ( g / 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.