The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree
Bénédicte Haas, Université Pierre et Marie Curie
Grégory Miermont, DMA, Ecole Normale Supérieure, et Université Paris VI
Abstract
We encode a certain class of stochastic fragmentation processes,
namely self-similar fragmentation processes with a negative index
of self-similarity, into a metric family tree which belongs to the
family of Continuum Random Trees of Aldous. When the splitting
times of the fragmentation are dense near 0, the tree can in turn
be encoded into a continuous height function, just as the Brownian
Continuum Random Tree is encoded in a normalized Brownian
excursion. Under mild hypotheses, we then compute the Hausdorff
dimensions of these trees, and the maximal Hölder exponents of
the height functions.
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