Independence Threshold for Collision Times of Many Planar Random Walks

Ziyang Liu

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arXiv · 2609.21937

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 $\mathbb Z^2$ 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 $\asymp (\log N)^{\frac{1}{3}}$, 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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Ziyang Liu. 2026-09-18. Independence Threshold for Collision Times of Many Planar Random Walks. https://arxiv.org/abs/2609.21937

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