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Jason Miller (mathematician) studies random planar geometry, including Schramm–Loewner evolution, the Gaussian free field, Liouville quantum gravity and random planar maps. His Cambridge research profile verifies these interests.^[ref]

His Random Planar Geometry by Jason Miller (2020) develops discrete-to-continuum connections through trees, maps and random curves. The trivial bijection between planar maps and quadrangulations is a concrete example: vertices of one type encode original map vertices and the other type encodes faces. Recovering the map from this bipartition explains why the correspondence is reversible.^[ref]

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.^[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.

The trivial bijection between planar maps and quadrangulations sends a rooted planar map with $n\ge 1$ edges to a rooted planar map that is a planar quadrangulation with $n$ faces. Colour its original vertices black. Insert a white vertex in each face and connect it to every incident corner, keeping repeated incidences as separate edges. Erase the original edges. Each original edge now occupies one quadrangular face, even when its endpoints or incident faces coincide.

Fix the root convention by taking the first new outgoing edge clockwise after the original root edge at its origin. The resulting quadrangulation is a bipartite graph. Its vertex bipartition is recovered by colouring the root origin black and alternating colours along edges. Draw a black diagonal in every quadrangular face, then erase the white vertices and the quadrangulation edges. Root the recovered map at the first outgoing black diagonal counterclockwise after the quadrangulation root. These constructions are inverse.^[ref]

The bipartition is a partition of vertices, not of faces. It distinguishes the original vertices from the inserted face vertices and makes the inverse construction possible. The bijection transfers enumeration and random sampling questions between general planar maps and planar quadrangulations.