Scaling limit for the cover time of the $λ$-biased random walk on a binary tree with $λ<1$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909523419070464 |
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| author | Croydon, David A. |
| author_facet | Croydon, David A. |
| contents | 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 $λ\geq 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20776 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scaling limit for the cover time of the $λ$-biased random walk on a binary tree with $λ<1$ Croydon, David A. Probability 05C81 (primary), 60J27, 60J50, 60J75 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 $λ\geq 1$. |
| title | Scaling limit for the cover time of the $λ$-biased random walk on a binary tree with $λ<1$ |
| topic | Probability 05C81 (primary), 60J27, 60J50, 60J75 |
| url | https://arxiv.org/abs/2410.20776 |