In this talk I will present the subject of my Ph.D. thesis which I worked on under the supervision of Thordur Jonsson at the University of Iceland.
I will introduce two models of random trees.
The first one is an equilibrium statistical mechanical model with a local action which depends only on the degrees of vertices in the tree. The model has been studied extensively, in different forms, by mathematicians with a history dating back to Galton and Watson in the 19th century who were interested in calculating the probability of the non-extinction of family names. Physicist became interested in the model in connection with simplicial gravity where they observed that a certain phase of the gravity model had tree-like features. The connection is in fact explicit in so-called 2D causal dynamical triangulations.
My contribution to this story was to prove that in a so-called non-generic phase, exactly one vertex of infinite degree emerges in the thermodynamic limit.
The second model is a new model of randomly growing trees referred to as the vertex splitting model. It originates from a related model of growing random trees which is equivalent to a model of random RNA folding. The model is appealing since it has very general growth rules and includes other previously studied random tree models as special cases.
