Geodesic
Convergence of Uniform Noncrossing Partitions Toward the Brownian Triangulation
Séminaire lotharingien de combinatoire, 80B (2018)
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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. 

@article{SLC_2018_80B_a37,
     author = {J\'er\'emie Bettinelli},
     title = {Convergence of {Uniform} {Noncrossing} {Partitions} {Toward} the {Brownian} {Triangulation}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2018},
     volume = {80B},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a37/}
}
TY  - JOUR
AU  - Jérémie Bettinelli
TI  - Convergence of Uniform Noncrossing Partitions Toward the Brownian Triangulation
JO  - Séminaire lotharingien de combinatoire
PY  - 2018
VL  - 80B
UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a37/
ID  - SLC_2018_80B_a37
ER  - 
%0 Journal Article
%A Jérémie Bettinelli
%T Convergence of Uniform Noncrossing Partitions Toward the Brownian Triangulation
%J Séminaire lotharingien de combinatoire
%D 2018
%V 80B
%U http://geodesic.mathdoc.fr/item/SLC_2018_80B_a37/
%F SLC_2018_80B_a37
Jérémie Bettinelli. Convergence of Uniform Noncrossing Partitions Toward the Brownian Triangulation. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a37/