Random complex cross ratios

Abstract

We study the cross ratio of four independent points chosen uniformly on the Riemann sphere. A positive integral over a two-dimensional simplex gives its normalized planar density. We then evaluate this integral in terms of the Bloch–Wigner dilogarithm, providing a closed formula. We describe its symmetries, real-axis values, moment bounds and geometric interpretations. We use these to evaluate the mean ideal-tetrahedron volume directly from the density, recovering $π/6$, a recent theorem of Kabluchko and Schange. We then prove that the expected distance between two independently sampled hyperbolic geodesics in $\mathbb{H}^3$ is $4/3$, by evaluating the logarithmic potential of the cross-ratio distribution on the negative real axis. Further applications give the joint dihedral-angle density and an exact acute-tetrahedron probability, and a branched-cover probability law on the modular surface with exponentially small cusp tails.