Scaling of the cumulative weights of the invasion percolation cluster on a branching process tree

We analyse the scaling of the weights added by invasion percolation on a uniformly weighted branching process tree. In this paper, we are interested in the invasion percolation cluster (IPC), obtained by performing invasion percolation for n steps and letting n → ∞. The volume scaling of the IPC was discussed in detail in [12] and in this work, we extend this analysis to the scaling of the cumulative weights. We assume a power-law offspring distribution on the branching process tree with exponent α. For α > 2 and α ∈ (1, 2), we observe a natural law-of-large-numbers result, where the cumulative weights scale similar to the volume, but converge to a different limit. However, for α < 1, where the weights added by invasion percolation vanish, the scaling changes significantly. For α ∈ (1/2, 1), the cumulative weights scale exponentially but with a different exponent than the volume scaling, while for α ∈ (0, 1/2) the cumulative weights are summable without any scaling. Such a phase transition at α = 1/2 is novel and unexpected as there is no significant change in the neighbourhood scaling of the IPC around α = 1/2.