= Random planar map
A <random planar map> is a <planar map> sampled from a specified <probability distribution>. A common finite model chooses uniformly among rooted maps with $n$ edges, where a root is a distinguished oriented edge. Sampling maps means sampling embeddings as well as abstract <graphs>; distinct cyclic arrangements of edges can describe different maps.<Random Planar Geometry by Jason Miller (2020)>[opening sections, rooted maps and finite random-map models]{ref}
For a <connected> map on the <sphere>, the <Euler formula for a connected planar graph> gives $V-E+F=2$. Thus conditioning on $E=n$ fixes $V+F=n+2$ but permits random numbers of vertices and faces. One may instead restrict every face to degree three or four, giving random triangulations or quadrangulations. These are different probability models and need not yield the same finite-size statistics. Distances in large random maps motivate probabilistic models of random geometry.
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