2024/10/28 by David A. Croydon, Croydon, David A.
Decision Sciences · Mathematics · #05C81 (primary) #60J27 #60J50 #60J75 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2410.20776
openalex publication_date 2024/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→ ∞ 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≥ 1.