A random planar map can be parabolic (like the Euclidean plane — random walk returns) or hyperbolic (like the Poincaré disk — random walk escapes to infinity). The distinction is fundamental: it determines the behavior of harmonic functions, the recurrence of random walk, and the large-scale geometry of the surface. But certifying which type a given random map belongs to has required deep structural analysis of each case.
Timár provides a single test: look at a half-plane.
Consider an invariant ergodic percolation on the random map — a random subset of edges or faces. If both the percolation and its dual have infinite clusters in the full map, the question is whether they can coexist when restricted to a half-plane. The answer determines the geometry.
If they cannot coexist in any half-plane: the map is parabolic. If they can: the map is hyperbolic.
The criterion is a generalization of a recent result by Klausen and Kravitz, who proved half-plane non-coexistence for Z². On the integer lattice, this is equivalent to planarity arguments and the classical result that Z² is recurrent. Timár's extension shows the principle is universal across unimodular random planar maps.
The elegance is in the reduction. Parabolicity and hyperbolicity are global properties of the map — they describe the asymptotic behavior of the Laplacian, the growth of balls, the decay of the Green function. Half-plane coexistence of a percolation is a local-to-mesoscale property — it asks about the behavior in a restricted region. The theorem says the local test fully determines the global geometry.
The half-plane is doing the geometric work. In a parabolic map, the half-plane is “too thin” to support simultaneous infinite clusters in both the percolation and its dual. In a hyperbolic map, there is enough room.