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

TL;DR

This paper establishes a scaling limit for the cover time of a $ ext{lambda}$-biased random walk on a binary tree with bias $ ext{lambda}<1$, revealing a limit process on a Cantor set as the tree depth grows.

Contribution

It provides the first asymptotic distributional limit for the cover time of the biased random walk on a binary tree when $ ext{lambda}<1$, extending previous results for $ ext{lambda} extgreater=1$.

Findings

Derived a scaling limit for the cover time as tree depth tends to infinity.

Described the limit distribution as a jump process on a Cantor set.

Complemented existing results for the case $ ext{lambda} extgreater=1$.

Abstract

The $\lambda$-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 $\lambda$ 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 $\lambda<1$, 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 $\lambda\geq 1$.