The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree

By Bénédicte Haas, Grégory Miermont. 2004

103 citations recorded by the source corpus.

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 Holder exponents of the height functions.

Subfields: Condensed Matter Physics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f9f58b8db7dc74266f1b1e9561278cd5afe935c8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

In Finance

In Mathematical Physics

Citations in the corpus

Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set (2011, background): “The following summarizes the parts of Theorem 1 and Lemma 5 in [4] that we will need.”
Probabilistic and fractal aspects of Lévy trees (2005, methodology): “In this form, the branching property has been recently used by Miermont [25] to investigate self-similar fragmentations of the stable tree.”
The local weak limit of the minimum spanning tree of the complete graph (2013, background): “Since the CRT has both Minkowski dimension 2 [28] and Hausdorff dimension 2 [28, 38], this shows that the law of M is mutually singular with that of the CRT, or any rescaled version thereof.”
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, background): “Haas and Miermont [17] and Bertoin [3] observed the following.”
Random trees and applications (2005, methodology): “Although we content ourselves with a simple application to solutions with boundary blow-up, the results of Section 3 provide most of the background that is necessary to understand the recent deep applications of the Brownian snake to the classification and probabilistic representation of solutions…”
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models (2006, methodology): “1 of [26] where it is more precisely proved that any fragmentation tree with an infinite dislocation measure – which is the case here – can be encoded into such continuous function).”
Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees (2010, methodology): “2 Continuum fragmentation trees This section is devoted to a more detailed description of the limiting self-similar fragmentation tree Tγ,ν [23].”
Combinatorial Stochastic Processes (2006, unknown): “4 and [185, 310, 311].”
A new family of Markov branching trees: the alpha-gamma model (2008, background): “In [16], Haas and Miermont’s self-similar continuum random trees (CRTs) [15] are shown to be scaling limits for a wide class of Markov branching models.”
Random real trees (2006, methodology): “Remark The formula for dim(T ) has also been derived by Haas and Miermont [33] independently of [22].”
Spinal partitions and invariance under re-rooting of continuum random trees (2007, methodology): “In a setting of continuous time and partitions of N we postulate a selfsimilarity condition, which leads to self-similar continuum random trees [25, 26].”