Convergence of uniform noncrossing partitions toward the Brownian triangulation

TL;DR

This paper proves that uniform noncrossing partitions of polygons converge to the Brownian triangulation, providing a simpler proof and an algorithm for almost sure convergence, extending to pair partitions.

Contribution

It offers a simplified proof of convergence and introduces an algorithm for almost sure convergence of noncrossing partitions to the Brownian triangulation.

Findings

Uniform noncrossing partitions converge to the Brownian triangulation in Hausdorff topology.

An algorithm is provided for recursive construction with almost sure convergence.

The results extend to noncrossing pair partitions of even-sided polygons.

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.