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PreprintCC BY 4.0arXiv

Random walk in a non-homogeneous random environment on some random trees and the non-negative integers

  • Tamer Oraby
  • András Telcs
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Abstract

In this paper, we consider a random walk in a non-homogeneous random environment on two types of random trees and non-negative integers. The random trees here have progeny distributions that depend on the parent's generation. We provide a sharp threshold for the type of the random walk in a random environment on the random trees and the non-negative integers, up to the critical case. Then, we find the speed of the random walk on the non-negative integers and on random spherically symmetric trees. We use the method based on the relationship between the mean exit time from a vertex and speed. Meanwhile, we discuss several special cases that could be derived from those results. We also include many limiting formulas for sequences and series in the appendix, which are helpful for this study and others.

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Cite

BibTeXRIS
BibTeX
@misc{oraby2026random,
  title = {{Random walk in a non-homogeneous random environment on some random trees and the non-negative integers}},
  author = {Tamer Oraby and András Telcs},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30648},
  eprint = {2609.30648},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30648},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Random walk in a non-homogeneous random environment on some random trees and the non-negative integers
AU  - Tamer Oraby
AU  - András Telcs
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30648
UR  - https://arxiv.org/abs/2609.30648
AB  - In this paper, we consider a random walk in a non-homogeneous random environment on two types of random trees and non-negative integers. The random trees here have progeny distributions that depend on the parent's generation. We provide a sharp threshold for the type of the random walk in a random environment on the random trees and the non-negative integers, up to the critical case. Then, we find the speed of the random walk on the non-negative integers and on random spherically symmetric trees. We use the method based on the relationship between the mean exit time from a vertex and speed. Meanwhile, we discuss several special cases that could be derived from those results. We also include many limiting formulas for sequences and series in the appendix, which are helpful for this study and others.
ER  -