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PreprintCC BY-NC-SA 4.0arXiv

Favorite sites of one-dimensional transient random walk

  • Zechun Hu
  • Renming Song
  • Guangshuo Zhou
  • Qianqian Zhou
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Abstract

In this paper, we study favorite sites of one-dimensional transient random walk on β„€ beyond the simple case. Denote by 𝒦(n) the set of favorite sites at time n. The goal of this paper is twofold. First, we study # 𝒦(n), the cardinality of 𝒦(n), and show that, under some mild conditions, for any positive integer k, P(#𝒦(n)=k, i.o.)=1, and limsupnβ†’βˆž#𝒦(n)/loglog n=βˆ’1/log Ξ³, where Ξ³ is the probability that the random walk never returns to the starting point. Second, we study the escape rate of favorite sites, and give the limsup and liminf behaviors, where the liminf behavior is characterized via an integral test. To the best of our knowledge, this integral test is the first result of this nature for random walks (in non-random environments).

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Cite

BibTeXRIS
BibTeX
@misc{hu2026favorite,
  title = {{Favorite sites of one-dimensional transient random walk}},
  author = {Zechun Hu and Renming Song and Guangshuo Zhou and Qianqian Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02727},
  eprint = {2610.02727},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02727},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Favorite sites of one-dimensional transient random walk
AU  - Zechun Hu
AU  - Renming Song
AU  - Guangshuo Zhou
AU  - Qianqian Zhou
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.02727
UR  - https://arxiv.org/abs/2610.02727
AB  - In this paper, we study favorite sites of one-dimensional transient random walk on β„€ beyond the simple case. Denote by 𝒦(n) the set of favorite sites at time n. The goal of this paper is twofold. First, we study # 𝒦(n), the cardinality of 𝒦(n), and show that, under some mild conditions, for any positive integer k, P(#𝒦(n)=k, i.o.)=1, and limsup_nβ†’βˆž#𝒦(n)/loglog n=βˆ’1/log Ξ³, where Ξ³ is the probability that the random walk never returns to the starting point. Second, we study the escape rate of favorite sites, and give the limsup and liminf behaviors, where the liminf behavior is characterized via an integral test. To the best of our knowledge, this integral test is the first result of this nature for random walks (in non-random environments).
ER  -