Poissonization-based collision threshold derivation for random walks on lattices
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915274106601472 |
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| author | Burton, Zachary |
| author_facet | Burton, Zachary |
| contents | In this expository note, we give a short derivation of the expected number of collisions between two independent simple random walkers on integer lattices. Adapting a Poissonization technique introduced by Lange, we express the collision probability as the return probability of the continuous-time difference walk, given by a modified Bessel function. Analyzing its asymptotic decay yields a clean, self-contained proof that the expected number of collisions in $\mathbb{Z}^d$ is finite if and only if $d\geq3$. We also provide a general formula for the asymptotic number of collisions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02973 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poissonization-based collision threshold derivation for random walks on lattices Burton, Zachary Probability 60J27 (primary) 60G50, 33C10 (secondary) In this expository note, we give a short derivation of the expected number of collisions between two independent simple random walkers on integer lattices. Adapting a Poissonization technique introduced by Lange, we express the collision probability as the return probability of the continuous-time difference walk, given by a modified Bessel function. Analyzing its asymptotic decay yields a clean, self-contained proof that the expected number of collisions in $\mathbb{Z}^d$ is finite if and only if $d\geq3$. We also provide a general formula for the asymptotic number of collisions. |
| title | Poissonization-based collision threshold derivation for random walks on lattices |
| topic | Probability 60J27 (primary) 60G50, 33C10 (secondary) |
| url | https://arxiv.org/abs/2505.02973 |