Convergence of uniform noncrossing partitions toward the Brownian triangulation

-- Jérémie Bettinelli (LIX, équipe Combi)

Time: 11:00 -- Location: Salle Philippe Flajolet du LIX

We give a short proof that a uniform noncrossing partition of the regular \(n\)-gon weakly converges toward Aldous's Brownian triangulation of the disk, in the sense of the Hausdorff topology. This result was first obtained by Curien and Kortchemski, using a more complicated encoding. Thanks to a result of Marchal on strong convergence of Dyck paths toward the Brownian excursion, we furthermore give an algorithm that allows to recursively construct a sequence of uniform noncrossing partitions for which the previous convergence holds almost surely. In addition, we also treat the case of uniform noncrossing pair partitions of even-sided polygons.

Category: seminars
Tags: Combi seminar combinatorics