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 $P_n\left(x\right)$ that at the $n$th stage, each of $2^n$ fragments is shorter than 1. We show that $P_n\left(x\right)$ approaches a traveling wave form, and the front position $x_n$ increases as $x_n\sim n^{\beta }{\rho }^n$ for large $n$ with $\rho =1.261\mathrm{}076\dots$ and $\beta =0.453\mathrm{}025\dots$. We also solve the $m$-section problem where each interval is broken into $m$ fragments and show that ${\rho }_m\approx m/\left(\ln m\right)$ and ${\beta }_m\approx 3/\left(2\ln m\right)$ for large $m$. Our approach establishes an intriguing connection between extreme value statistics and traveling wave propagation in the context of the fragmentation problem.