Fragment size distributions in random fragmentations with cutoff

We consider the following fragmentation model with cutoff: a fragment with initial size x0>1 splits into b>1 daughter fragments with random sizes, the partition law of which has exchangeable distribution. In subsequent steps, fragmentation proceeds independently for each sub-fragments whose sizes are bigger than some cutoff value xc=1 only. This process naturally terminates with probability 1. The size of a fragment is the random mass attached to a leaf of a "typical" path of the full (finite) fragmentation tree. The height's law of typical paths is first analyzed, using analytic and renewal processes techniques. We then compute fragments' size limiting distribution (x0[short up arrow][infinity]), for various senses of a typical path. Next, we exhibit some of its statistical features, essentially in the case of the exchangeable Dirichlet partition model.