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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In this page, we look at uniform plane maps (maps drawn on the sphere). There is a classical correspondence (sometimes called the trivial bijection) between maps with n edges and quadrangulations (maps whose faces have degree 4) with n faces. To go from quadrangulations to maps, for each face, simply add a vertex in its middle and add four edges linking it to the vertices of the face, then remove the original edges.

It is believed that, in the limit, the map and the quadrangulation should look the same. We show here some examples of uniformly sampled maps with their corresponding quadrangulations (which are hence also uniformly sampled).