Séminaire Lotharingien de Combinatoire, 80B.38 (2018), 12 pp.

Jérémie Bettinelli

Convergence of Uniform Noncrossing Partitions Toward the Brownian Triangulation

Abstract. 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 \& 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.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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