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PreprintCC BY 4.0arXiv

Independence Threshold for Collision Times of Many Planar Random Walks

  • Ziyang Liu
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Abstract

We study collision times of many independent simple random walks on ℤ2 through the joint moment generating function of their pairwise collision local times. For a fixed number of walks, these collision times are known to be asymptotically independent after a suitable logarithmic normalisation. We investigate the extent to which this independence persists when the number of walks grows. For N being the walk length, our results identify a threshold asymp (log N)1/3, on which the transition from independence to dependence happens. The proofs combine chaos expansion techniques and a correlation inequality, which is the result of a local limit theorem.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026independence,
  title = {{Independence Threshold for Collision Times of Many Planar Random Walks}},
  author = {Ziyang Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21937},
  eprint = {2609.21937},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21937},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Independence Threshold for Collision Times of Many Planar Random Walks
AU  - Ziyang Liu
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21937
UR  - https://arxiv.org/abs/2609.21937
AB  - We study collision times of many independent simple random walks on ℤ^2 through the joint moment generating function of their pairwise collision local times. For a fixed number of walks, these collision times are known to be asymptotically independent after a suitable logarithmic normalisation. We investigate the extent to which this independence persists when the number of walks grows. For N being the walk length, our results identify a threshold asymp (log N)^1/3, on which the transition from independence to dependence happens. The proofs combine chaos expansion techniques and a correlation inequality, which is the result of a local limit theorem.
ER  -