Trees with exponential height dependent weight

TL;DR

This paper analyzes the asymptotic properties of planar rooted random trees with exponential height-dependent weights, revealing phase transitions in their structure and growth rates as a function of a parameter.

Contribution

It determines the asymptotic behavior of the total weight and local limits of these weighted trees, identifying a phase transition at =0.

Findings

Transition from a single spine to multi-spine phase at =0

Change in volume growth rate from linear to quadratic to cubic across the transition

Asymptotic behavior of total weight for large tree size

Abstract

We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is of exponential form $e^{-\mu h}$. Defining the total weight for such trees of fixed size to be $Z^{(\mu)}_N$, we determine its asymptotic behaviour for large $N$, for arbitrary real values of $\mu$. Based on this we evaluate the local limit of the corresponding probability measures and find a transition at $\mu=0$ from a single spine phase to a multi-spine phase. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $\mu<0$ to the familiar quadratic growth at $\mu=0$ and to cubic growth for $\mu> 0$.