Statistical aspects of random fragmentations

Three random fragmentation of an interval processes are investigated. For each of them, there is a splitting probability and a probability not to split at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. Some of their statistical features are studied in each case among which fragments’ size distribution, partition function, structure of the underlying random fragmentation tree, occurrence of a phase transition. In the first homogeneous model, splitting probability does not depend on fragments’ size at each step. In the next two fragmentation models, splitting probability is fragments’ length dependent. In the first such models, fragments further split with probability one if their sizes exceed some cutoff value only; in a second model considered, splitting probability of finite-size objects is assumed to increase algebraically with fragments’ size at each step. The impact of these dependencies on statistical properties of the resulting random partitions are studied. Several examples are supplied.