Scaling limit for the cover time of the $λ$-biased random walk on a binary tree with $λ<1$

TL;DR

This work determines the first-order distributional scaling limit of the cover time for the $\lambda$-biased random walk on binary trees in the regime $\lambda\in(0,1)$. By embedding the trees in a common real-tree framework and performing time changes, the authors construct a Brownian motion on a tree and a trace jump process on the Cantor boundary, then couple finite-tree walks to this limit object. They prove that the properly rescaled cover time converges in distribution to a random variable $\bar{\tau}_{\mathrm{cov}}(X^\Sigma)$ associated with a jump process $X^\Sigma$ on the middle-thirds Cantor set, with convergence of moments established via Gaussian-field methods. This random scaling limit contrasts with the deterministic limits observed for $\lambda\ge1$ and highlights the fractal boundary's role in cover-time fluctuations, connecting stochastic processes on trees to resistance forms and time-changed Brownian motion on fractal-like spaces.

Abstract

The $λ$-biased random walk on a binary tree of depth $n$ is the continuous-time Markov chain that has unit mean holding times and, when at a vertex other than the root or a leaf of the tree in question, has a probability of jumping to the parent vertex that is $λ$ times the probability of jumping to a particular child. (From the root, it chooses one of the two children with equal probability.) For this process, when $λ<1$, we derive an $n\rightarrow \infty$ scaling limit for the cover time, that is, the time taken to visit every vertex. The distributional limit is described in terms of a jump process on a Cantor set that can be seen as the asymptotic boundary of the tree. This conclusion complements previous results obtained when $λ\geq 1$.

Figures (4)

• Figure 1: The binary tree $T_n$ with $n=4$ and the 'middle-thirds' Cantor set capturing the boundary of $T_n$ as $n\rightarrow\infty$.
• Figure 2: A plot of the value of $\exp(\lim_{n\rightarrow\infty}n^{-1}\log\|\tau_{\mathrm{cov}}(X^n)\|_1$, which determines the geometric growth rate of the cover time, for the $\lambda$-biased random walk on a binary tree.
• Figure 3: A stylised representation of the supports of the measures $\mu_T$, $\mu_{\Sigma}$, $\mu_n$, $\bar{\mu}_n$ and $\tilde{\mu}_n$. In particular, the first sketch shows the support of $\mu_T$ as the whole of $T$, with the solid black triangles representing the parts of $T$ below vertices in $\Sigma_n$ (with $n=4$). The second shows the support of $\mu_{\Sigma}$ as the boundary set $\Sigma$. Third, the measure $\mu_n$ is supported on the vertices of $T_n$. Fourth, as will be introduced in the subsequent section, $\bar{\mu}_n$ is the restriction of $\mu_n$ to $\bar{\Sigma}_n=\cup_{m=n-\log n}^n\Sigma_m$. And finally, $\tilde{\mu}_n$ is the uniform measure on the subset $\tilde{\Sigma}_n$ of $\Sigma$, which has size $2^n$.
• Figure 4: Connections between the processes introduced in Section \ref{['sec3']}. The solid arrows show links between processes, either by construction or convergence. (We omit the arrows corresponding to $\bar{A}^n$ and $\tilde{A}^n$, which link $X^T$ to $\bar{X}^n$ and $\tilde{X}^n$, respectively, and also the convergence of $\tilde{X}^n$ to $X^\Sigma$, as given by Lemma \ref{['lem41']}.) The dotted lines represent links between the cover times of the various processes, as proved in Section \ref{['sec4']}. Specifically, Lemma \ref{['lem42']} shows the rescaled cover times of $X^n$ and $\bar{X}^n$ are close, Lemmas \ref{['lem43']} and \ref{['lem44']} demonstrate the rescaled cover time of $\bar{X}^n$ is asymptotically between that of $\tilde{X}^n$ and ${X}^\Sigma$, and Lemma \ref{['lem46']} establishes that the cover time of $\tilde{X}^n$ converges to that of ${X}^\Sigma$.

Theorems & Definitions (32)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2