The bullet problem with discrete speeds

TL;DR

This paper analyzes a bullet collision process with bullets fired at discrete speeds, revealing survival probabilities and extending insights into ballistic annihilation phenomena.

Contribution

It introduces a novel analysis of the bullet process with discrete speeds, highlighting survival probabilities and extending to non-uniform firing measures.

Findings

Second fastest bullet survives with positive probability

Slowest bullet does not survive

Results extend to exponential spacings and certain non-uniform measures

Abstract

Bullets are fired, one per second, with independent speeds sampled uniformly from a discrete set. Collisions result in mutual annihilation. We show that the second fastest bullet survives with positive probability, while a slowest bullet does not. This also holds for exponential spacings between firing times, and for certain non-uniform measures that place less probability on the second fastest bullet. Our results provide new insights into a two-sided version of the bullet process known to physicists as ballistic annihilation.