Random maps and continuum random 2-dimensional geometries

Abstract

In the recent years, much progress has been made in the mathematical understanding of the scaling limit of random maps, making precise the sense in which random embedded graphs approach a model of continuum surface. In particular, it is now known that many natural models of random plane maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map. Other models, favoring large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid $O(n)$ model on quadrangulations).