Local geometry of random geodesics on negatively curved surfaces

Abstract

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.

This version extends the results of the previous version to surfaces with possibly variable negative curvature