Extremal process of the local time of simple random walk on a regular tree
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arXiv
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| Format: | Preprint |
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2025
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| author | Abe, Yoshihiro Biskup, Marek |
| author_facet | Abe, Yoshihiro Biskup, Marek |
| contents | We study a continuous-time simple random walk on a regular rooted tree of depth $n$ in two settings: either the walk is started from a leaf vertex and run until the tree root is first hit or it is started from the root and run until it has spent a prescribed amount of time there. In both cases we show that the extremal process associated with centered square-root local time on the leaves tends, as $n\to\infty$, to a decorated Poisson point process with a random intensity measure. While the intensity measure is specific to the local-time problem at hand, the decorations are exactly those for the tree-indexed Markov chain (a.k.a. Branching Random Walk or Gaussian Free Field) with normal step distribution. The proof demonstrates the latter by way of a Lindeberg-type swap of the decorations of the two processes which itself relies on a well-known isomorphism theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09592 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal process of the local time of simple random walk on a regular tree Abe, Yoshihiro Biskup, Marek Probability 60G50, 60G70, 05C81 We study a continuous-time simple random walk on a regular rooted tree of depth $n$ in two settings: either the walk is started from a leaf vertex and run until the tree root is first hit or it is started from the root and run until it has spent a prescribed amount of time there. In both cases we show that the extremal process associated with centered square-root local time on the leaves tends, as $n\to\infty$, to a decorated Poisson point process with a random intensity measure. While the intensity measure is specific to the local-time problem at hand, the decorations are exactly those for the tree-indexed Markov chain (a.k.a. Branching Random Walk or Gaussian Free Field) with normal step distribution. The proof demonstrates the latter by way of a Lindeberg-type swap of the decorations of the two processes which itself relies on a well-known isomorphism theorem. |
| title | Extremal process of the local time of simple random walk on a regular tree |
| topic | Probability 60G50, 60G70, 05C81 |
| url | https://arxiv.org/abs/2506.09592 |