Tail asymptotics for extinction times of self-similar fragmentations

Self-similar fragmentation processes are random models for particles that are subject to successive fragmentations. When the index of self-similarity is negative the fragmentations intensify as the masses of particles decrease. This leads to a shattering phenomenon, where the initial particle is entirely reduced to dust - a set of zero-mass particles - in finite time which is what we call the extinction time. Equivalently, these extinction times may be seen as heights of continuous compact rooted trees or scaling limits of heights of sequences of discrete trees. Our objective is to set up precise bounds for the large time asymptotics of the tail distributions of these extinction times.