Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem

Abstract

We study a random bisection problem where an initial interval of length x is cut into two random fragments at the first stage, then each of these two fragments is cut further, etc. We compute the probability P_n(x) that at the n-th stage, each of the 2^n fragments is shorter than 1. We show that P_n(x) approaches a traveling wave form, and the front position x_n increases as x_n\sim n^βρ^n for large n. We compute exactly the exponents ρ=1.261076... and β=0.453025.... as roots of transcendental equations. We also solve the m-section problem where each interval is broken into m fragments. In particular, the generalized exponents grow as ρ_m\approx m/(\ln m) and β_m\approx 3/(2\ln m) in the large m limit. Our approach establishes an intriguing connection between extreme value statistics and traveling wave propagation in the context of the fragmentation problem.