Self-similar fragmentations derived from the stable tree II: splitting at nodes

By Grégory Miermont. 2004

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Abstract.We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Lévy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin.

Subfields: Condensed Matter Physics, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/68640052e734ed55fb1eb33ac418165897f6f380

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In Condensed Matter Physics

In Management Science and Operations Research

In Mathematical Physics

Citations in the corpus

Gibbs fragmentation trees (2007, background): “ons of N, associated with a homogeneous fragmentation process. This is closely related to (17) via Kingman’s paintbox representation κ= R S↓ κ sν(ds). The extended range was ﬁrst observed by Miermont [13] in the special case θ= −1 (related to the stable trees of Duquesne and Le Gall [5]). We refer t”
The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree (2004, unknown): “Fragmentations of CRT’s that are difierent from the Brownian one and that follow the same kind of construction have been studied in [ 23 ].”
A new family of Markov branching trees: the alpha-gamma model (2008, background): “Denote the discrete tree without edge lengths by ˜ T n – it has exchangeable leaf labels. e Then R n is the almost sure scaling limit of the reduced trees R ( T n , [ k ]), by Proposition 7 in [16]. e e On the other hand, if we denote by T n the (non-exchangeably) labelled trees obtained via the…”
Random real trees (2006, methodology): “A form of the branching property was used by Miermont [ 53 ] in his study of fragmentations of the stable tree.”
Self-similar fragmentations derived from the stable tree I (2003, methodology): “And as a matter of fact, the fragmentation F + considered in the companion paper [ 23 ] has also an explicit semigroup.”
The asymptotic behavior of fragmentation processes (2003, background): “We further refer to [5] for an alternative construction of the same fragmentation from the Brownian excursion, to [7] for another binary and non discrete self-similar fragmentation with negative index α = −1/2, which is also naturally connected to the Brownian excursion, and to [23,24] for other mor”