Résumé : In this talk, we introduce a sequence of Markov chains on trees with a fixed number leaves and study its scaling limit as the number of leaves tends to infinity. The chains are derived from Marchal's algorithm to approximate stable trees, and are a parameterized extension of a Markov chain on binary trees introduced by Aldous in 1998. When considered in a suitable space of trees, we will see that the chains converge to a diffusion which is symmetric with respect to the law of a stable tree. Our result relies on the theory of algebraic measure trees, developed by Winter and Löhr in 2018 and used by them in 2020 to prove the convergence of Aldous's chain, resolving an open problem posed by Aldous in 2000. After a presentation of the framework and statement, we will focus on the spectral decomposition of the limiting diffusion's generator and semigroup, providing a simple combinatorial description of their eigenspaces, and motivating their study by connecting it to several open questions regarding the process.

Dernière modification : Thursday 27 March 2025

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