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 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) similar to n(beta)rho (n) for large n with rho = 1.261076... and beta = 0.453025.... We also solve the nz-section problem where each interval is broken into m fragments and show that rho (m) approximate to m/(lnm) and beta (m) approximate to 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.