A Gibbsian random tree with nearest neighbour interaction

TL;DR

This paper analyzes a Gibbsian random tree model with nearest-neighbour interaction, revealing phase transitions, asymptotic behaviors, and a spin representation, extending previous models to unbounded branching laws.

Contribution

It introduces a phase transition analysis and a spin representation for the Gibbsian random tree with unbounded branching, extending prior work.

Findings

Identifies a phase transition between sub- and super-critical regimes.

Shows the partition function approaches a limit at rate n^{-1} in the critical regime.

Proves the mean number of external nodes decays like n^{-2} in the critical regime with extinction.

Abstract

We revisit the random tree model with nearest-neighbour interaction as described in previous work, enhancing growth. When the underlying free Bienaym\'e-Galton-Watson (BGW) model is sub-critical, we show that 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 previous work when the law of the branching mechanism of the free BGW process has unbounded support.