Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908332365709312 |
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| author | Devulder, Alexis |
| author_facet | Devulder, Alexis |
| contents | We consider $d$ random walks $\big(S_n^{(j)}\big)_{n\in\mathbb{N}}$, $1\leq j \leq d$, in the same random environment $ω$ in $\mathbb{Z}$, and a recurrent simple random walk $(Z_n)_{n\in\mathbb{N}}$ on $\mathbb{Z}$. We assume that, conditionally on the environment $ω$, all the random walks are independent and start from even initial locations. Our assumption on the law of the environment is such that a single random walk in the environment $ω$ is transient to the right but subballistic, with parameter $0<κ<1/2$. We show that - for every value of $d$ - there are almost surely infinitely many times for which all these random walks, $(Z_n)_{n\in\mathbb{N}}$ and $\big(S_n^{(j)}\big)_{n\in\mathbb{N}}$, $1\leq j \leq d$, are simultaneously at the same location, even though one of them is recurrent and the $d$ others ones are transient. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment Devulder, Alexis Probability We consider $d$ random walks $\big(S_n^{(j)}\big)_{n\in\mathbb{N}}$, $1\leq j \leq d$, in the same random environment $ω$ in $\mathbb{Z}$, and a recurrent simple random walk $(Z_n)_{n\in\mathbb{N}}$ on $\mathbb{Z}$. We assume that, conditionally on the environment $ω$, all the random walks are independent and start from even initial locations. Our assumption on the law of the environment is such that a single random walk in the environment $ω$ is transient to the right but subballistic, with parameter $0<κ<1/2$. We show that - for every value of $d$ - there are almost surely infinitely many times for which all these random walks, $(Z_n)_{n\in\mathbb{N}}$ and $\big(S_n^{(j)}\big)_{n\in\mathbb{N}}$, $1\leq j \leq d$, are simultaneously at the same location, even though one of them is recurrent and the $d$ others ones are transient. |
| title | Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment |
| topic | Probability |
| url | https://arxiv.org/abs/2504.15999 |