For a unimodular random trivalent hyperbolic polyhedron, the expected local angle defect equals 2 pi minus pi/3 times the expected face degree, and its sign (zero or negative) exactly determines parabolic versus hyperbolic conformal type.
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A Gauss-Bonnet-Type Dichotomy for Unimodular Random Infinite Trivalent Hyperbolic Polyhedra
For a unimodular random trivalent hyperbolic polyhedron, the expected local angle defect equals 2 pi minus pi/3 times the expected face degree, and its sign (zero or negative) exactly determines parabolic versus hyperbolic conformal type.