Local geometry of random geodesics on negatively curved surfaces

Athreya, Jayadev; Lalley, Steve; Sapir, Jenya; Wroten, Matthew

Annales Henri Lebesgue, Volume 4 (2021), pp. 187-226.

We show that the tessellation of a compact, negatively curved surface induced by a long random 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.

Nous montrons que la tessellation d’une surface compacte de courbure strictement négative induite par un long segment géodésique aléatoire ressemble localement à un processus de Poisson en droites, après rééchelonnement. Ceci implique que les statistiques globales de la tessellation (par exemple, la proportion de triangles) convergent vers celles du processus de Poisson en droites limite.

Classification: 37D40, 37E35, 37B10
Mots-clés : self-intersection, random tessellation, geodesic, hyperbolic surface, Poisson line process

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