Bullets are fired from the origin of the positive real line, one per second, with independent speeds sampled uniformly from a discrete set. Collisions result in mutual annihilation. We show that a bullet with the second largest speed survives with positive probability, while a bullet with the smallest speed does not. This also holds for exponential spacings between firing times. Our results imply that the middle-velocity particle survives with positive probability in a two-sided version of the bullet process with three speeds known to physicists as ballistic annihilation.
Brittany Dygert. Christoph Kinzel. Matthew Junge. Annie Raymond. Erik Slivken. Jennifer Zhu. "The bullet problem with discrete speeds." Electron. Commun. Probab. 24 1 - 11, 2019. https://doi.org/10.1214/19-ECP238