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Jérémie Bettinelli
École polytechnique
Laboratoire d'informatique (LIX)
91128 Palaiseau Cedex
FRANCE
Email: firstname « . » lastname « at » normalesup « . » org
Office: 2023
Phone: (+33) (0)1 77 57 80 61



Topologically, a disk is just a sphere with one "hole." One might also consider spheres with multiple holes (cylinders, pairs of pants, etc.) and quadrangulations of these surfaces (with a boundary). We showed in Geodesics in Brownian surfaces (Brownian maps) that, for the right scaling, when the length of every boundary component is of order the square root of the number of faces, the topology is conserved in the limit when the number of faces grows to infinity. In other words, if we consider quadrangulations with p boundary components having more and more faces, we will obtain a topological sphere with p holes in the limit.

Quadrangulations with 2 boundary components 

Quadrangulations with 3 boundary components 
Cactus embedding





n = 30 000, p = 200 and q = 500 
n = 50 000, p = 200 and q = 300 

Cactus embedding of a uniform plane quadrangulation with n faces and 2 boundary components with 2p and 2q halfedges 





n = 30 000, p = 200, q = 100 and r = 50 
n = 50 000, p = 400, q = 200 and r = 150 

Cactus embedding of a uniform plane quadrangulation with n faces and 2 boundary components with 2p and 2q halfedges 