Résumé : We consider simply generated random plane trees with $n$ vertices and $k_n$ leaves, sampled from a sequence of weights. Motivated by questions on random planar maps, we will focus on the asymptotic behaviour of the largest degree. Precisely we will give conditions on both the number of leaves and the weight sequence that ensure the convergence in distribution of the associated Łukasiewicz path (or depth-first walk) to the Brownian excursion. This should also provide a first step towards the convergence of the height or contour function of the trees. The proof scheme is to reduce step by step to simpler and simpler objects and we will discuss excursion and bridge paths, non decreasing paths conditioned by their tip, and finally estimates of the form of the local limit theorem which may be of independent interest. Based on a joint work with Igor Kortchemski.

 [Slides.pdf] [arXiv] [vidéo]

Dernière modification : Thursday 21 November 2024

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