Condensation in nongeneric trees

TL;DR

This paper investigates nongeneric planar trees, establishing the existence of a Gibbs measure on infinite trees, characterizing the degree distribution, and analyzing spectral dimensions with respect to vertex weights.

Contribution

It proves the existence of a Gibbs measure on infinite nongeneric trees and characterizes their degree distribution and spectral dimension properties.

Findings

Exactly one vertex of infinite degree in the limit

Degree divergence rate of (1-m)N for finite trees

Spectral dimension depends on vertex weight decay as 2β - 2

Abstract

We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability $m<1$. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like $(1-m)N$ where $N$ is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by $2\beta -2$ if the weight $w_n$ of a vertex of degree $n$ is asymptotic to $n^{-\beta}$.