The critical velocity of the bullet process appears pathwise
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914172829171712 |
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| author | Meisel, Josh |
| author_facet | Meisel, Josh |
| contents | In the bullet process, a gun fires bullets in the same direction at independent random speeds, and with independent random time delays between firings. When two bullets collide, they vanish. The critical velocity $v_c$ is the slowest speed the first bullet can take and still have positive probability of surviving forever. We characterize the critical velocity via a random variable determined by the sequence of speeds and delays, which we show almost surely equals $v_c$. In turn we prove other facts about the process, including that infinitely many bullets survive when the velocity distribution has finite support. Along the way we answer a question from Broutin--Marckert (2020), showing that if a bullet survives, it does so in all but finitely many truncations of the process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14219 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The critical velocity of the bullet process appears pathwise Meisel, Josh Probability 60K35 In the bullet process, a gun fires bullets in the same direction at independent random speeds, and with independent random time delays between firings. When two bullets collide, they vanish. The critical velocity $v_c$ is the slowest speed the first bullet can take and still have positive probability of surviving forever. We characterize the critical velocity via a random variable determined by the sequence of speeds and delays, which we show almost surely equals $v_c$. In turn we prove other facts about the process, including that infinitely many bullets survive when the velocity distribution has finite support. Along the way we answer a question from Broutin--Marckert (2020), showing that if a bullet survives, it does so in all but finitely many truncations of the process. |
| title | The critical velocity of the bullet process appears pathwise |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2501.14219 |