On exponentially height-penalized random trees

TL;DR

The paper analyzes random plane trees biased by height via P(T_n=t) ∝ e^{-μ h(t)}. It extends the Brownian-regime scaling and the non-Brownian regime analysis of height, width, and local structure for μ_n depending on n, establishing a tilted Brownian CRT limit in the Brownian window and detailed height scaling h(T_n) with Gaussian, discrete, and Bernoulli fluctuations as μ grows. It develops a comprehensive partition-function framework to derive first- and second-order height results, large deviations, and Bernoulli height jumps, and couples these with a multiscale width analysis to show w(T_n) ∼ (μ n^2)^{1/3} in the intermediate regime. Local limits reveal a transition from Kesten’s tree to a Poisson-tree-like structure with condensation at the root for larger μ, including root-degree asymptotics and local-limit descriptions. Overall, the work combines encoding by walks, partition-function asymptotics, and multiscale excursion analysis to map the global and local geometry of height-biased trees across regimes, linking discrete models to continuum limits and offering detailed probabilistic characterizations across scales.

Abstract

Given $n \in \mathbb{N}$ and $μ\in \mathbb{R}$, a $\textit{$μ$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-μh(t)}$, where $t$ ranges over fixed plane trees with $n$ vertices, and $h(t)$ is the height of $t$. Fix a sequence $(μ_n)_{n \ge 1}$ of real numbers, and for $n \ge 1$ let $\mathbf{T}_n$ be a $μ$-height-biased tree of size $n$. Durhuus and Ünel (2023) described the asymptotic behaviour of $h(\mathbf{T}_n)$ when $μ_n \equiv μ\in \mathbb{R}$ 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 $μ_n$ is of order $1/\sqrt{n}$; that its height is asymptotically $(2π^2n/μ_n)^{1/3}$ when $μ_n$ is of larger order than $1/\sqrt{n}$ and of smaller order than $n$; and that its height converges to a fixed constant when $μ_n$ is of order at least $n$, with some random jumps under specific conditions on $μ_n$. We additionally prove various results on second order behaviours, and large deviation principles for the height, for different regimes of $μ_n$. Finally, we describe new statistics of these trees, covering their widths, their root degrees, and the local structure around their roots.

Figures (3)

• Figure 1: A representation of the bijection between a rooted plane tree with $n$ nodes and its contour --- an excursion of length $2n-1$. Each step of the excursion is represented by an arrow, both on the walk and on the tree, and corresponds to a single side of an edge (left or right). To highlight this relation, the nodes of the tree are shown at the bottom of the walk in the order in which they would be encountered when contouring around the tree.
• Figure 2: In the above picture for an element $w$ of $\mathcal{B}^{(n)}_x$, displayed as the red path, we apply the map $\phi_{x}$ to reflect the segment of the path $w$ from the last passage $\tau$ at $\lfloor x/2 \rfloor$ to $n$ with respect to $y=\left\lfloor x/2\right\rfloor$ to obtain the green segment. The image $\phi_{x}(w)$ is thus the concatenation of the red path up to time $\tau$ together with the green segment.
• Figure 3: A representation of the multiscale decomposition used for the proof of Theorem \ref{['thm:width']}. We bound the height with $h_n^\delta=(1+\delta)(2\pi^2n/\mu)^{1/3}$ and split the contour of a uniform tree into $r\sim(\mu^2n)^{1/3}$ bridges of length $\Delta_t\sim2(n/\mu)^{2/3}$. This way the length of each bridge is of the same order as the square of its fluctuations, thus are objects from the Brownian world, admitting good enough width tails for our purposes, thanks to Proposition \ref{['prop:bridge_subg_tail']}.

Theorems & Definitions (84)

• Theorem 1.1: Scaling limit in the Brownian regime
• Remark 1.2: On the process $B_\alpha$ being well-defined
• Theorem 1.3: Width in the Brownian regime
• Theorem 1.4: Asymptotic height in the intermediate regime
• Theorem 1.5: Gaussian fluctuations for the height
• Theorem 1.6: Discrete central limit theorem for the height
• Remark 1.7
• Theorem 1.8: Large deviations of the height
• Theorem 1.9: Width in the non-Brownian regime
• Conjecture 1.10: LLN for the width in the non-Brownian regime