A Fortuin-Kasteleyn planar map is a random planar map decorated by a random-cluster model
configuration of open edges. In the critical rooted
spherical model with 
edges and parameter 
,
a pair 
has probability proportional
to
 |
where 
is a subset of the edges
and 
counts interfaces between primal and dual clusters (Sheffield 2016, §4). Here
consists of the dual edges corresponding to edges
outside 
,
and both component counts include isolated vertices.
The map itself is random, rather than a fixed lattice carrying random edge
states.
