Abstract
We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. For the walk started from a leaf vertex and stopped upon hitting the root, we prove that, in the limit as the depth of the tree tends to infinity, the suitably scaled and centered maximal time spent at any leaf converges to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with square-root local-time process on the tree.
Funding Statement
This project has been supported in part by the NSF Grant DMS-1954343, ISF Grants No. 1382/17 and 2870/21 and BSF award 2018330.
Citation
Marek Biskup. Oren Louidor. "A limit law for the most favorite point of simplerandom walk on a regular tree." Ann. Probab. 52 (2) 502 - 544, March 2024. https://doi.org/10.1214/23-AOP1644
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