Phase Transition in a Random Fragmentation Problem with Applications to Computer Science

We study a fragmentation problem where an initial object of size x is broken into m random pieces provided x > x0 where x0 is an atomic cut-off. Subsequently the fragmentation process continues for each of those daughter pieces whose sizes are bigger than x0. The process stops when all the fragments have sizes smaller than x0. We show that the fluctuation of the total number of splitting events, characterized by the variance, generically undergoes a nontrivial phase transition as one tunes the branching number m through a critical value m = mc. For m < mc, the fluctuations are Gaussian where as for m > mc they are anomalously large and non-Gaussian. We apply this general result to analyze two different search algorithms in computer science.