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

David A. Croydon

Bernoulli2026https://doi.org/10.3150/25-bej1897article
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0.50

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→∞ 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.

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https://doi.org/https://doi.org/10.3150/25-bej1897

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@article{david2026,
  title        = {{Scaling limit for the cover time of the λ-biased random walk on a binary tree with λ<1}},
  author       = {David A. Croydon},
  journal      = {Bernoulli},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.3150/25-bej1897},
}

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