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) approximately n(beta)rho(n) for large n with rho = 1.261 076ellipsis and beta = 0.453 025ellipsis. We also solve the m-section problem where each interval is broken into m fragments and show that rho(m) approximately m/(lnm) and beta(m) approximately 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.