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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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Positive genus

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We continue to consider more general surfaces. Instead of the sphere with or without holes, we now look at surfaces of positive genus, that is, connected sums of tori. In plain English, the surface of genus 1 is a bike tube. To obtain the surface of genus 2, patch together 2 bike tubes as follows: cut a small disk out of each tube and glue the boundaries of these two disks together. Then, you may add to this another tube to get the surface of genus 3 and so on.

These are the objects we studied in The Topology of Scaling Limits of Positive Genus Random Quadrangulations, the main result being that, after a proper scaling, a uniform quadrangulation of a surface of fixed genus converges, in some sense, to a random metric space homeomorphic to the same surface.

Torus 20k [jpg1jpg2pdf1pdf23D pdfu3d&tex]   Torus 50k [jpg1jpg2jpg3jpg4pdf1pdf2pdf3pdf43D pdfu3d&tex]

Genus 1
Double torus 10k [jpg1jpg2jpg3jpg4pdf1pdf2pdf3pdf43D pdfu3d&tex]   Double torus 30k [jpg1jpg2pdf1pdf23D pdfu3d&tex]

Genus 2
Triple torus 10k [jpg1jpg2pdf1pdf23D pdfu3d&tex]   Triple torus 30k [jpg1jpg2jpg3jpg4jpg5jpg6pdf1pdf2pdf3pdf4pdf5pdf63D pdfu3d&tex]

Genus 3

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