Uniformly random planar map - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:41:10Z http://mathoverflow.net/feeds/question/107493 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107493/uniformly-random-planar-map Uniformly random planar map Zach Hamaker 2012-09-18T17:43:30Z 2012-09-19T01:23:58Z <p>Is there a way to sample a planar map uniformly at random? I am aware of the Cori-Vauquelin-Schaeffer 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.</p> http://mathoverflow.net/questions/107493/uniformly-random-planar-map/107495#107495 Answer by Nathann Cohen for Uniformly random planar map Nathann Cohen 2012-09-18T17:54:26Z 2012-09-18T17:54:26Z <p>Something you may potentially be interested in : <a href="http://www.lix.polytechnique.fr/~fusy/Articles/FusyAofa.pdf" rel="nofollow">http://www.lix.polytechnique.fr/~fusy/Articles/FusyAofa.pdf</a></p> <p>Nathann</p> http://mathoverflow.net/questions/107493/uniformly-random-planar-map/107521#107521 Answer by Omer for Uniformly random planar map Omer 2012-09-19T01:23:58Z 2012-09-19T01:23:58Z <p>Planar quadrangulation with \$n\$ faces are in bijection with planar maps with \$n\$ edges. The quadrangulation is bi-partite, 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. </p> <p>In fact, this is one of the reasons quadrangulations are a particularly interesting class of planar maps.</p>