Is there a way to sample a planar map uniformly at random? I am aware of the CoriVauquelinSchaeffer bijection that can be used to sample and study uniformly random quadrangulations. There are other results in the literature that allow for sampling other classes of planar maps, but I haven't seen any results for the entire class of planar maps.
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Something you may potentially be interested in : http://www.lix.polytechnique.fr/~fusy/Articles/FusyAofa.pdf Nathann 


Planar quadrangulation with $n$ faces are in bijection with planar maps with $n$ edges. The quadrangulation is bipartite, so colour its vertices white/black. inside each face, add a diagonal edge between the two black vertices. Then delete all white vertices and incident edges. The result is a map with $n$ edges, and it is not hard to see that this is a bijection. In fact, this is one of the reasons quadrangulations are a particularly interesting class of planar maps. 

