A Gibbsian Random Tree with Nearest Neighbour Interaction

Abstract

We revisit the random tree model with nearest-neighbour interaction as described in Dunlop and Mardin (J Stat Phys 189:38, 2022), enhancing growth. When the underlying free Bienaymé–Galton–Watson (BGW) model is sub-critical, the (non-Markov) model with interaction exhibits a phase transition between sub- and super-critical regimes. In the critical regime, using tools from dynamical systems, we show that the partition function of the model approaches a limit at rate $$n^{-1}$$ in the generation number n. In the critical regime with almost sure extinction, we also prove that the mean number of external nodes in the tree at generation n decays like $$n^{-2}$$ . Finally, we give a spin representation of the random tree, opening the way to tools from the theory of Gibbs states, including FKG inequalities. We extend the construction in Dunlop and Mardin (J Stat Phys 189:38, 2022), when the law of the branching mechanism of the free BGW process has unbounded support.