Amoeba Monte Carlo algorithms for random trees with controlled branching activity: efficient trial move generation and universal dynamics

The reptation Monte Carlo algorithm is a simple, physically motivated and efficient method for equilibrating semi-dilute solutions of linear polymers. Here we propose two simple generalizations for the analogue {\it Amoeba} algorithm for randomly branching chains, which allow to efficiently deal with random trees with controlled branching activity. We analyse the rich relaxation dynamics of Amoeba algorithms and demonstrate the existence of an unexpected scaling regime for the tree relaxation. In particular, our results suggests that the equilibration time for Amoeba algorithms scales in general like $N^2⟨n_{\mathrm{l}\mathrm{i}\mathrm{n}}{⟩}^{\mathrm{\Delta }}$, where $N$ denotes the number of tree nodes, $⟨n_{\mathrm{l}\mathrm{i}\mathrm{n}}⟩$ the mean number of linear segments the trees are composed of and $\mathrm{\Delta }\simeq 0.4$.