Appearance of vertices of infinite order in a model of random trees

TL;DR

This paper analyzes a statistical model of random trees with a spine and leaves, revealing two phases: one with an infinite spine and finite vertex order, and another with a finite spine and a single infinite-order vertex.

Contribution

It introduces a phase transition in a tree model, characterizes the spectral dimension in both phases, and establishes the existence of a Gibbs measure for the system.

Findings

Two distinct phases with different spine behaviors

Spectral dimension calculated for both phases

Existence of a Gibbs measure proven

Abstract

We study an equilibrium statistical mechanical model of tree graphs which are made up of a linear subgraph (the spine) to which leaves are attached. We prove that the model has two phases, a generic phase where the spine becomes infinitely long in the thermodynamic limit and all vertices have finite order and a condensed phase where the spine is finite with probability one and a single vertex of infinite order appears in the thermodynamic limit. We calculate the spectral dimension of the graphs in both phases and prove the existence of a Gibbs measure. We discuss generalizations of this model and the relationship with models of nongeneric random trees.