A random planar map is a finite connected graph embedded in the sphere, considered up to orientation-preserving
homeomorphism, and sampled from a specified probability law. A root is a distinguished oriented
edge. Unlike a planar graph,
a planar map retains the cyclic order of incident half-edges at each vertex.
Loops and multiple edges
may be allowed, depending on the model. For a map with 
edges, its corner probability
measure assigns mass 
to a vertex 
, counting a graph
loop twice in the degree. Its graph
distance counts every primal edge, irrespective
of a decoration.
Critical Fortuin-Kasteleyn planar maps and tree-weighted planar maps have different sampling laws. Liouville quantum gravity provides candidate continuum surfaces for these decorated maps (Sheffield 2016).
