Scale-free random branching trees in supercritical phase

We study the size and the lifetime distributions of scale-free random branching trees in which k branches are generated from a node at each time step with probability q_k ∼ k^-γ. In particular, we focus on finite-size trees in a supercritical phase, where the mean branching number C ≤ ∑_kkq_k is larger than 1. The tree-size distribution p(s) exhibits a crossover behaviour when 2 < γ < 3. A characteristic tree size s_c exists such that for s ≪ s _c, p(s) ∼ s^-γ/(γ-1) and for s ≫ s _c, p(s) ∼ s^-3/2exp(-s/s_c), where s _c scales as ∼(C - 1)^{-(γ-1)/(γ-2)}. For γ > 3, it follows the conventional mean-field solution, p(s) ∼ s^-3/2exp(-s/s_c) with s_c ∼ (C - 1) ^-2. The lifetime distribution is also derived. It behaves as ℓ(t) ∼ t^{-(γ-1)/(γ-2)} for 2 < γ < 3, and ∼t^-2 for γ > 3 when branching step t ≪ t_c ∼ (C - 1)^-1, and ℓ(t) ∼ exp(-t/t_c) for all γ > 2 when t ≫ t_c. The analytic solutions are corroborated by numerical results.