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Random trees, laminations of the disk and factorizations

Abstract : This work is devoted to the study of asymptotic properties of large random combinatorial structures. Three particular structures are the main objects of our interest: trees, factorizations of permutations and configurations of noncrossing chords in the unit disk (or laminations).First, we are specifically interested in the number of vertices with fixed degree in Galton-Watson trees that are conditioned in different ways, for example by their number of vertices with even degree, or by their number of leaves. When the offspring distribution of the tree is critical and in the domain of attraction of a stable law, we notably prove the asymptotic normality of these quantities. We are also interested in the spread of these vertices with fixed degree in the tree, when one explores it from left to right.Then, we consider configurations of chords that do not cross in the unit disk. Such configurations notably code trees in a natural way. We define in particular a nondecreasing sequence of laminations coding a fragmentation of a given tree, that is, a way of cutting this tree at points chosen randomly. This geometric point of view then allows us to study some properties of a factorization of the cycle $(1 , 2 , cdots , n)$ as a product of $n-1$ transpositions, chosen uniformly at random, by coding it in the disk by a random lamination and by remarking a connection between this model and a Galton-Watson tree conditioned by its total number of vertices. Finally, we present a generalization of these results to random factorizations of the same cycle, that are not necessarily as a product of transpositions anymore, but may involve cycles of larger lengths. We highlight this way a connection between conditioned Galton-Watson trees, factorizations of large permutations and the theory of fragmentations.
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Submitted on : Monday, August 31, 2020 - 11:23:11 AM
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Paul Thevenin. Random trees, laminations of the disk and factorizations. Probability [math.PR]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAX021⟩. ⟨tel-02925984⟩

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