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 ℍ3 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.
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BibTeX
@misc{marshall2026random,
title = {{Random complex cross ratios}},
author = {Tim Marshall and Gaven Martin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07670},
eprint = {2611.07670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07670},
}
RIS
TY - UNPB
TI - Random complex cross ratios
AU - Tim Marshall
AU - Gaven Martin
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07670
UR - https://arxiv.org/abs/2611.07670
AB - 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 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.
ER -