The scaling limit of random outerplanar maps



Third take on a talk about outerplanar maps, and perhaps the most complete account I ever gave of this paper; the slides are the same those of the Journée YSP, with an extra section that describes the core algorithm. Sorry about the annoying video format; an incredibly heavy pdf file is available here.

A planar map is outerplanar if all of its vertices belong to a single (outer)face. The scaling limit for various classes of large planar maps has been shown to be the so-called “Brownian Map”; under certain conditions, however, different asymptotic behaviours may emerge, and some classes of planar maps with a macroscopic face admit Aldous’ CRT, or a scalar multiple thereof, as the scaling limit. We shall see that this is the case for outerplanar maps as well: using a bijection by Bonichon, Gavoille and Hanusse one can show that uniform outerplanar maps with n vertices, suitably rescaled by a factor 1/\sqrt{n}, converge to 7\sqrt{2}/9 times the CRT.

Leave a Reply