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Jérémie Bettinelli

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

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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.

Cylinder 30k_200_500 [jpg1jpg2jpg33D pdfu3d&tex]   Cylinder 50k_200_300 [jpg1jpg2jpg3jpg43D pdfu3d&tex]

Quadrangulations with 2 boundary components
SpongeBob 30k_200_100_50 [jpg1jpg23D pdfu3d&tex]   SpongeBob 50k_400_200_150 [jpg1jpg2jpg3jpg43D pdfu3d&tex]

Quadrangulations with 3 boundary components

Cactus embedding

Brownian map Brownian map
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 half-edges
Brownian map Brownian map
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 half-edges

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