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 Pₙ(x) that at the nth stage, each of 2ⁿ fragments is shorter than 1. We show that Pₙ(x) approaches a traveling wave form, and the front position xₙ increases as xₙ~n^βρⁿ for large n with ρ0ex0ex=0ex0ex1.261076 and β0ex0ex=0ex0ex0.453025. We also solve the m-section problem where each interval is broken into m fragments and show that ρₘ≈m/(lnm) and βₘ≈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.