Can one condition a killed random walk to survive?
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916946777210880 |
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| author | Rey, Lucas Teixeira, Augusto |
| author_facet | Rey, Lucas Teixeira, Augusto |
| contents | We consider the simple random walk on $\mathbb{Z}^d$ killed with probability $p(|x|)$ at site $x$ for a function $p$ decaying at infinity. Due to recurrence in dimension $d=2$, the killed random walk (KRW) dies almost surely if $p$ is positive, while in dimension $d \geq 3$ it is known that the KRW dies almost surely if and only if $\int_0^{\infty}rp(r)dr = \infty$, under mild technical assumptions on $p$. In this paper we consider, for any $d \geq 2$, functions $p$ for which the KRW dies almost surely and we ask ourselves if the KRW conditioned to survive is well-defined. More precisely, given an exhaustion $(Λ_R)_{R \in \mathbb{N}}$ of $\mathbb{Z}^d$, does the KRW conditioned to leave $Λ_R$ before dying converges in distribution towards a limit which does not depend on the exhaustion? We first prove that this conditioning is well-defined for $p(r) = o(r^{-2})$, and that it is not for $p(r) = \min(1, r^{-α})$ for $α\in (14/9,2)$. This question is connected to branching random walks and the infinite snake. More precisely, in dimension $d=4$, the infinite snake is related to the KRW with $p(r) \asymp (r^2\log(r))^{-1}$, therefore our results imply that the infinite snake conditioned to avoid the origin in four dimensions is well-defined. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_12328 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Can one condition a killed random walk to survive? Rey, Lucas Teixeira, Augusto Probability 05C81, 82B41, 82C41, 60J80 We consider the simple random walk on $\mathbb{Z}^d$ killed with probability $p(|x|)$ at site $x$ for a function $p$ decaying at infinity. Due to recurrence in dimension $d=2$, the killed random walk (KRW) dies almost surely if $p$ is positive, while in dimension $d \geq 3$ it is known that the KRW dies almost surely if and only if $\int_0^{\infty}rp(r)dr = \infty$, under mild technical assumptions on $p$. In this paper we consider, for any $d \geq 2$, functions $p$ for which the KRW dies almost surely and we ask ourselves if the KRW conditioned to survive is well-defined. More precisely, given an exhaustion $(Λ_R)_{R \in \mathbb{N}}$ of $\mathbb{Z}^d$, does the KRW conditioned to leave $Λ_R$ before dying converges in distribution towards a limit which does not depend on the exhaustion? We first prove that this conditioning is well-defined for $p(r) = o(r^{-2})$, and that it is not for $p(r) = \min(1, r^{-α})$ for $α\in (14/9,2)$. This question is connected to branching random walks and the infinite snake. More precisely, in dimension $d=4$, the infinite snake is related to the KRW with $p(r) \asymp (r^2\log(r))^{-1}$, therefore our results imply that the infinite snake conditioned to avoid the origin in four dimensions is well-defined. |
| title | Can one condition a killed random walk to survive? |
| topic | Probability 05C81, 82B41, 82C41, 60J80 |
| url | https://arxiv.org/abs/2406.12328 |