On exponentially height-penalized random trees

Abstract

Given $n \in \mathbb{N}$ and $\mu \in \mathbb{R}$, a $\textit{$\mu$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-\mu h(t)}$, where $t$ ranges over fixed plane trees with $n$ vertices, and $h(t)$ is the height of $t$. Fix a sequence $(\mu_n)_{n \ge 1}$ of real numbers, and for $n \ge 1$ let $\mathbf{T}_n$ be a $\mu$-height-biased tree of size $n$. Durhuus and \"Unel (2023) described the asymptotic behaviour of $h(\mathbf{T}_n)$ when $\mu_n \equiv \mu \in \mathbb{R}$ is fixed. In this work, we extend their results to arbitrary sequences of positive parameters depending on $n$. Most notably, we show that such a tree behaves like a height-biased Continuum Random Tree (CRT) when $\mu_n$ is of order $1/\sqrt{n}$; that its height is asymptotically $(2\pi^2n/\mu_n)^{1/3}$ when $\mu_n$ is of larger order than $1/\sqrt{n}$ and of smaller order than $n$; and that its height converges to a fixed constant when $\mu_n$ is of order at least $n$, with some random jumps under specific conditions on $\mu_n$. We additionally prove various results on second order behaviours, and large deviation principles for the height, for different regimes of $\mu_n$. Finally, we describe new statistics of these trees, covering their widths, their root degrees, and the local structure around their roots.