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Fix a positive integer $n$. Every second, a particle is sent along a straight line from a fixed position in a fixed direction, at a random integer speed chosen uniformly in $\{1,\ldots, n\}$ meters per second. The particles move in a continuous manner (we assume everything is happening in $\mathbb{R}^3$). If two (or more) particles ever collide, they both get annihilated.

In terms of $n$, what is the probability $p_n$ that some particle survives forever? Even if there is no exact formula for $p_n$, can the values of $\lim\inf_{n\in \mathbb{N}}p_n$ and $\lim\sup_{n\in \mathbb{N}}p_n$ be determined (hopefully they are the same value)?

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    $\begingroup$ What if three or more particles collide at the same time? $\endgroup$
    – LeechLattice
    Sep 14, 2022 at 8:01
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    $\begingroup$ What if a particle is released at time 0 traveling at speed 1; and a new particle is released at time 1, traveling at speed 10. Do they “collide” between time 1 and time 2 or do they not collide because they never occupy the same site at the same (integer) time. If the former, I think it should be pretty easy to show that almost surely nothing survives. If the latter, I think the answer should be the same, but the proof may be a bit more involved. $\endgroup$
    – Anthony Quas
    Sep 14, 2022 at 9:42
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    $\begingroup$ And to simplify particles with speed $0$ can be omitted. They all get annihilted at position 0. As a consequence speed should be chosen uniformly in $\{1,\ldots,n\}$. $\endgroup$
    – Dieter Kadelka
    Sep 14, 2022 at 11:30
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    $\begingroup$ Brittany Dygert, Christoph Kinzel, Matthew Junge, Annie Raymond, Erik Slivken. Jennifer Zhu. "The bullet problem with discrete speeds." Electronic Communications in Probability 24 1--11, 2019. doi.org/10.1214/19-ECP238 scholarworks.umass.edu/cgi/… $\endgroup$
    – aorq
    Sep 14, 2022 at 12:49
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    $\begingroup$ Also related (bullets with continuous speeds): "The combinatorics of the colliding bullets" by Broutin and Marckert: doi.org/10.1002/rsa.20869 (and its references and citations). $\endgroup$
    – James Martin
    Sep 14, 2022 at 12:53

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