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

We study a random bisection problem where an 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 nth stage, each of 2ⁿ 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∼n^βρⁿ for large n with ρ = 1.261076… and β = 0.453025…. We also solve the m-section problem where each interval is broken into m fragments and show that ρ_m≈m/(lnm) and β_m≈3/(2lnm) for large m. Our approach establishes an intriguing connection between extreme value statistics and traveling wave propagation in the context of the fragmentation problem.