Scale-free random branching tree in supercritical phase

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