Résumé : We study random minimal factorizations of the n-cycle into transpositions, which are factorizations of the full cycle (1,...,n) as a product of n-1 transpositions. It is known since Denes (1959) that they are counted by the sequence nn-2 and bijections with Cayley trees and parking functions have been found. Moreover, these factorizations are naturally encoded (in two different ways) by sequences of non-crossing set of chords of the unit disk. We establish limit theorems for these sets of chords. The limiting objects are constructed from Levy’s excursion processes and interpolate between the circle and Aldous’ Brownian triangulation. One key step of the proof is to connect our model with conditioned bitype Galton-Watson trees with offspring distribution varying with $n$, and to find the limit of the contour function of those trees. If time allows, we will also describe the limit of the trajectory of a fixed element in this factorization (in some sense, this is a local limit result in space). This is a joint work with Igor Kortchemski.
|Dernière modification : Friday 18 October 2019||Contact : Cyril.Banderier at lipn.univ-paris13.fr|