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September 2021 A geometric representation of fragmentation processes on stable trees
Paul Thévenin
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Ann. Probab. 49(5): 2416-2476 (September 2021). DOI: 10.1214/21-AOP1512


We provide a new geometric representation of a family of fragmentation processes by nested laminations which are compact subsets of the unit disk made of noncrossing chords. We specifically consider a fragmentation, obtained by cutting a random stable tree at random points, which split the tree into smaller subtrees. When coding each of these cutpoints by a chord in the unit disk, we separate the disk into smaller connected components, corresponding to the smaller subtrees of the initial tree. This geometric point of view allows us in particular to highlight a new relation between the Aldous–Pitman fragmentation of the Brownian continuum random tree and minimal factorizations of the n-cycle, that is, factorizations of the permutation (12n) into a product of (n1) transpositions, proving this way a conjecture of Féray and Kortchemski. We discuss various properties of these new lamination-valued processes, and we notably show that they can be coded by explicit Lévy processes.

Funding Statement

This work was supported by ANR GRAAL (ANR-14-CE25-0014).


I would like to thank my advisor Igor Kortchemski for asking the questions at the origin of this paper and for his help, comments, suggestions and corrections. I would also like to thank Bénédicte Haas, Cyril Marzouk and Loïc Richier for fruitful discussions and comments on the paper and Gerónimo Uribe Bravo for his useful remarks on Theorem 5.1.


Download Citation

Paul Thévenin. "A geometric representation of fragmentation processes on stable trees." Ann. Probab. 49 (5) 2416 - 2476, September 2021.


Received: 1 March 2020; Revised: 1 January 2021; Published: September 2021
First available in Project Euclid: 24 September 2021

Digital Object Identifier: 10.1214/21-AOP1512

Primary: 60C05 , 60F17
Secondary: 05A05

Keywords: Fragmentation processes , lamination of the disk , Lévy process , minimal factorization , permutation , Random trees , Scaling limit , stable tree

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 5 • September 2021
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