The combinatorics of the colliding bullets. Issue 2 (19th June 2019)

• Record Type: Journal Article
• Authors: Broutin, Nicolas; Marckert, Jean‐François
• Is Part Of: Random structures & algorithms. Volume 56:Issue 2(2020)
• Page Start: 401
• Page End: 431
• Publication Date: 2019-06-19
• Subjects: annihilation -- interacting particles; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20869
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let ( V i )1 ≤ i ≤ n be an i.i.d. family of random variables with uniform distribution on [0, 1]; shoot n bullets one after another at times 1, 2, …, n, where the i th bullet has speed V i . When two bullets collide, they both annihilate. We give the distribution of the number of surviving bullets, and in some generalization of this model. While the distribution is relatively simple (and we found a number of bold claims online), our proof is surprisingly intricate and mixes combinatorial and geometric arguments; we argue that any rigorous argument must very likely be rather elaborate.