Local limits of one-sided trees

TL;DR

This paper studies the limiting behavior of large one-sided trees with a probabilistic model, revealing a phase transition in their structure and growth rate depending on a parameter.

Contribution

It introduces a probabilistic model for one-sided trees and characterizes the phase transition in their structure and growth as the size tends to infinity.

Findings

Weak limit of the distribution on infinite trees established

Phase transition at a critical parameter value from single to multi-spine trees

Volume growth rate changes from linear to quadratic to cubic across the transition

Abstract

A finite \emph{one-sided tree} of height $h$ is defined as a rooted planar tree obtained by grafting branches on one side, say the right, of a spine, i.e. a linear path of length $h$ starting at the root, such that the resulting tree has no simple path starting at the root of length greater than $h$. We consider the distribution $\tau_N$ on the set of one-sided trees $T$ of fixed size $N$, such that the weight of $T$ is proportional to $e^{-\mu h(T)}$, where $\mu$ is a real constant and $h(T)$ denotes the height of $T$. We show that, for $N$ large, $\tau_N$ has a weak limit as a probability measure supported on infinite one-sided trees. The dependence of the limit measure $\tau$ on $\mu$ shows a transition at $\mu_0=-\ln 2$ from a single spine phase for $\mu\leq \mu_0$ to a multi-spine phase for $\mu> \mu_0$. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $\mu<\mu_0$, to quadratic growth at $\mu=\mu_0$, and to qubic growth for $\mu> \mu_0$.