Time of appearance of a large gap in a dynamic Poisson point process
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917124062052352 |
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| author | Foxall, Eric Soubrier, Clément |
| author_facet | Foxall, Eric Soubrier, Clément |
| contents | We study the distribution of the 'gap time', the first time that a large gap appears, in the spatial birth and death point process on $[0,1]$ in which particles are added uniformly in space at rate $λ$ and are removed independently at rate $1$, as a function of the parameter $λ$ and the specified gap size function $w_λ$ as $λ\to\infty$. If $w_λ$ is a large enough multiple of the typical largest gap $(\log(λ)+O(1))/λ$ and the initial distribution has a high enough local density of particles and not too many particles in total, then the gap time, scaled by its expected value, converges in distribution to exponential with mean $1$. If in addition $\limsup_λw_λ< 1$ then the expected time scales like $e^{λw_λ}/(λ^2 w_λ(1-w_λ))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time of appearance of a large gap in a dynamic Poisson point process Foxall, Eric Soubrier, Clément Probability 60K35, 60G55, 60F10 We study the distribution of the 'gap time', the first time that a large gap appears, in the spatial birth and death point process on $[0,1]$ in which particles are added uniformly in space at rate $λ$ and are removed independently at rate $1$, as a function of the parameter $λ$ and the specified gap size function $w_λ$ as $λ\to\infty$. If $w_λ$ is a large enough multiple of the typical largest gap $(\log(λ)+O(1))/λ$ and the initial distribution has a high enough local density of particles and not too many particles in total, then the gap time, scaled by its expected value, converges in distribution to exponential with mean $1$. If in addition $\limsup_λw_λ< 1$ then the expected time scales like $e^{λw_λ}/(λ^2 w_λ(1-w_λ))$. |
| title | Time of appearance of a large gap in a dynamic Poisson point process |
| topic | Probability 60K35, 60G55, 60F10 |
| url | https://arxiv.org/abs/2512.04218 |